A303554 -id:A303554

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A340596

Number of co-balanced factorizations of n.

+10
30

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

COMMENTS

We define a factorization of n into factors > 1 to be co-balanced if it has exactly A001221(n) factors.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

EXAMPLE

The a(n) co-balanced factorizations for n = 12, 24, 36, 72, 120, 144, 180:

2*6 3*8 4*9 8*9 3*5*8 2*72 4*5*9

3*4 4*6 6*6 2*36 4*5*6 3*48 5*6*6

2*12 2*18 3*24 2*2*30 4*36 2*2*45

3*12 4*18 2*3*20 6*24 2*3*30

6*12 2*4*15 8*18 2*5*18

2*5*12 9*16 2*6*15

2*6*10 12*12 2*9*10

3*4*10 3*3*20

3*4*15

3*5*12

3*6*10

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Table[Length[Select[facs[n], Length[#]==PrimeNu[n]&]], {n, 100}]

PROG

(PARI) A340596(n, m=n, om=omega(n)) = if(1==n, (0==om), sumdiv(n, d, if((d>1)&&(d<=m), A340596(n/d, d, om-1)))); \\ Antti Karttunen, Jun 10 2024

CROSSREFS

Positions of terms > 1 are A126706.

Positions of 1's are A303554.

The version for unlabeled multiset partitions is A319616.

The alt-balanced version is A340599.

The balanced version is A340653.

The cross-balanced version is A340654.

The twice-balanced version is A340655.

A001055 counts factorizations.

A045778 counts strict factorizations.

A316439 counts factorizations by product and length.

Other balance-related sequences:

- A010054 counts balanced strict partitions.

- A047993 counts balanced partitions.

- A098124 counts balanced compositions.

- A106529 lists Heinz numbers of balanced partitions.

- A340597 lists numbers with an alt-balanced factorization.

- A340598 counts balanced set partitions.

- A340600 counts unlabeled balanced multiset partitions.

Cf. A003963, A006141, A050320, A112798, A117409, A324518, A339846, A339890, A340607, A340656, A340657.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 15 2021

EXTENSIONS

Data section extended up to a(120) by Antti Karttunen, Jun 10 2024

STATUS

approved

A340654

Number of cross-balanced factorizations of n.

+10
23

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 2, 5, 1, 1, 1, 3, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

COMMENTS

We define a factorization of n into factors > 1 to be cross-balanced if either (1) it is empty or (2) the maximum image of A001222 over the factors is A001221(n).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

EXAMPLE

The cross-balanced factorizations for n = 12, 24, 36, 72, 144, 240:

2*6 4*6 4*9 2*4*9 4*4*9 8*30

3*4 2*2*6 6*6 2*6*6 4*6*6 12*20

2*3*4 2*2*9 3*4*6 2*2*4*9 5*6*8

2*3*6 2*2*2*9 2*2*6*6 2*4*30

3*3*4 2*2*3*6 2*3*4*6 2*6*20

2*3*3*4 3*3*4*4 2*8*15

2*2*2*2*9 3*4*20

2*2*2*3*6 3*8*10

2*2*3*3*4 4*5*12

2*10*12

2*3*5*8

2*2*2*30

2*2*3*20

2*2*5*12

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Table[Length[Select[facs[n], #=={}||PrimeNu[n]==Max[PrimeOmega/@#]&]], {n, 100}]

PROG

(PARI) A340654(n, m=n, om=omega(n), mbo=0) = if(1==n, (mbo==om), sumdiv(n, d, if((d>1)&&(d<=m), A340654(n/d, d, om, max(mbo, bigomega(d)))))); \\ Antti Karttunen, Jun 19 2024

CROSSREFS

Positions of terms > 1 are A126706.

Positions of 1's are A303554.

The co-balanced version is A340596.

The version for unlabeled multiset partitions is A340651.

The balanced version is A340653.

The twice-balanced version is A340655.

A001055 counts factorizations.

A045778 counts strict factorizations.

A316439 counts factorizations by product and length.

A320655 counts factorizations into semiprimes.

Other balance-related sequences:

- A010054 counts balanced strict partitions.

- A047993 counts balanced partitions.

- A098124 counts balanced compositions.

- A106529 lists Heinz numbers of balanced partitions.

- A340597 have an alt-balanced factorization.

- A340598 counts balanced set partitions.

- A340599 counts alt-balanced factorizations.

- A340652 counts unlabeled twice-balanced multiset partitions.

- A340656 have no twice-balanced factorizations.

- A340657 have a twice-balanced factorization.

Cf. A003963, A117409, A303975, A320656, A324518, A339846, A339890, A340608.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 15 2021

EXTENSIONS

Data section extended up to a(105) by Antti Karttunen, Jun 19 2024

STATUS

approved

A335460

Number of (1,2,1) or (2,1,2)-matching permutations of the prime indices of n.

+10
21

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 8, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,24

COMMENTS

Depends only on sorted prime signature (A118914).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

LINKS

Table of n, a(n) for n=1..87.

Wikipedia, Permutation pattern

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

EXAMPLE

The a(n) compositions for n = 12, 24, 48, 36, 60, 72:

(121) (1121) (11121) (1212) (1213) (11212)

(1211) (11211) (1221) (1231) (11221)

(12111) (2112) (1312) (12112)

(2121) (1321) (12121)

(2131) (12211)

(3121) (21112)

(21121)

(21211)

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Length[Select[Permutations[primeMS[n]], MatchQ[#, {___, x_, ___, y_, ___, x_, ___}/; x!=y]&]], {n, 100}]

CROSSREFS

Positions of zeros are A303554.

The (1,2,1)-matching part is A335446.

The (2,1,2)-matching part is A335453.

Replacing "or" with "and" gives A335462.

Permutations of prime indices are counted by A008480.

Unsorted prime signature is A124010. Sorted prime signature is A118914.

STC-numbers of permutations of prime indices are A333221.

(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.

Patterns matched by standard compositions are counted by A335454.

(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.

Dimensions of downsets of standard compositions are A335465.

Cf. A056239, A056986, A112798, A158005, A181796, A335451, A335452, A335463.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 20 2020

STATUS

approved

A304779

The "rootless" zeta function. Dirichlet inverse of the function defined by r(n) = (-1)^Omega(n) if n is 1 or not a perfect power and r(n) = 0 otherwise.

+10
9

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 2, 2, 5, 1, 1, 1, 2, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

COMMENTS

Omega(n) = A001222(n) is the number of prime factors of n counted with multiplicity.

First occurrence of k: 1, 12, 48, 60, 36, 3072, 72, 420, 240, 786432, 3145728, 144, 216, ..., . - Robert G. Wilson v, Jul 22 2018

Records: 1, 2, 5, 7, 12, 13, 15, 18, 26, 37, 38, 57, 60, 67, 81, 96, 142, 165, 199, 221, 234, ..., . - Robert G. Wilson v, Jul 22 2018

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

a(1) = 1 and a(n > 1) = -Sum_{d|n, d not a perfect power} (-1)^Omega(d) * a(n/d).

MATHEMATICA

a[n_]:=a[n]=If[n==1, 1, -Sum[(-1)^PrimeOmega[d]*a[n/d], {d, Select[Rest[Divisors[n]], GCD@@FactorInteger[#][[All, 2]]==1&]}]];

Array[a, 100]

PROG

(PARI) A304779(n) = if(1==n, 1, -sumdiv(n, d, if((d>1)&&!ispower(d), ((-1)^bigomega(d))*A304779(n/d), 0))); \\ Antti Karttunen, Jul 22 2018

CROSSREFS

Positions of entries greater than 1 appear to be A126706.

Cf. A000005, A000012, A000961, A001221, A001222, A001597, A005117, A007916, A008683, A008966, A091050, A303554, A304326, A304362, A304819.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 18 2018

EXTENSIONS

More terms from Antti Karttunen, Jul 22 2018

STATUS

approved

A374247

The greatest number of runs possible in a permutation of the prime factors of n (A373957) minus the number of distinct such factors (A001221).

+10
7

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,36

COMMENTS

If n has separable prime factors (A335433), then a(n) = A001222(n) - A001221(n) = A046660(n). A multiset is separable iff it has an anti-run permutation (meaning there are no adjacent equal parts).

LINKS

Table of n, a(n) for n=1..87.

FORMULA

a(n) = A373957(n) - A001221(n).

EXAMPLE

The runs of the 6 permutations of the prime factors of 36 are:

((2,2),(3,3))

((2),(3),(2),(3))

((2),(3,3),(2))

((3),(2,2),(3))

((3),(2),(3),(2))

((3,3),(2,2))

The longest length is 4, so a(36) = 4 - 2 = 2.

MATHEMATICA

prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];

Table[Max@@Table[Length[Split[y]], {y, Permutations[prifacs[n]]}]-PrimeNu[n], {n, 100}]

CROSSREFS

Positions of first appearances appear to be A026549.

Positions of nonzero terms are A126706, complement A303554.

This is an opposite version of A373957.

The sister-sequence A374246 uses A001222 instead of A001221.

This is the number of nonzero terms in row n of A374252.

A003242 counts run-compressed compositions, i.e., anti-runs.

A008480 counts permutations of prime factors, by number of runs A374252.

A027746 lists prime factors, row-sums A001414.

A027748 is run-compression of prime factors, row-sums A008472.

A304038 is run-compression of prime indices, row-sums A066328.

A333755 counts compositions by number of runs.

A335433 lists separable numbers, complement A335448.

A374250 maximizes sum of run-compression, for indices A373956.

Cf. A007947, A046660, A124767, A151821, A238130, A373951, A374251.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 07 2024

STATUS

approved

A372720

a(n) = A000005(n) - A008479(n).

+10
6

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 4, 1, 2, 3, 4, 1, 1, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 3, 4, 3, 7, 1, 5, 1, 3, 1, 10, 3, 3, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

COMMENTS

A095960(50) = 3, a(50) = 2.

a(162) = -2 is the first negative term.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A095960(n) for n in A303554, i.e., for squarefree n or prime powers n.

a(n) = A095960(n) for n in A360767, i.e., for nonsquarefree composite n such that omega(n) > 1 and A003557(n) < A119288(n), since A008479(n) is the number of terms k in row n of A010846 such that k <= A003557(n).

a(n) = A183093(n) - A355432(n).

EXAMPLE

Table of a(n), b(n) = A000005(n), and c(n) = A008479(n) for n <= 12:

n b(n) c(n) a(n)

------------------

1 1 1 0

2 2 1 1

3 2 1 1

4 3 2 1

5 2 1 1

6 4 1 3

7 2 1 1

8 4 3 1

9 3 2 1

10 4 1 3

11 2 1 1

12 6 2 4

a(12) = 4 since 12 has 6 divisors {1, 2, 3, 4, 6, 12}, and row 12 of A369609 has 2 terms {6, 12}.

a(18) = 3 since 18 has 6 divisors {1, 2, 3, 6, 9, 18}, and row 18 of A369609 has 3 terms {6, 12, 18}.

a(50) = 2 since 50 has 6 divisors {1, 2, 5, 10, 25, 50}, and row 50 of A369609 has 4 terms {10, 20, 40, 50}

a(162) = -2 since 162 has 10 divisors {1,2,3,6,9,18,27,54,81,162} but row 162 of A369609 has 12 terms {6,12,18,24,36,48,54,72,96,108,144,162}.

a(500) = 0 since 500 has as many divisors {1,2,4,5,10,20,25,50,100,125,250,500} as terms in row 500 of A369609 {10,20,40,50,80,100,160,200,250,320,400,500}.

MATHEMATICA

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], _?(Divisible[r, rad[#]] &)], {n, 120}]

PROG

(PARI) a(n) = my(f=factor(n)[, 1], s); forvec(v=vector(#f, i, [1, logint(n, f[i])]), if(prod(i=1, #f, f[i]^v[i])<=n, s++)); numdiv(n) - s; \\ after A008479 \\ Michel Marcus, Jun 03 2024

CROSSREFS

Cf. A000005, A003557, A008479, A095960, A119288, A162306, A183093, A303554, A355432, A360767, A369609.

KEYWORD

sign

AUTHOR

Michael De Vlieger, May 13 2024

STATUS

approved

A304653

a(n) = (-1)^Omega(n) if n is not a perfect power > 1, and 0 otherwise.

+10
3

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, -1, -1, 1, 1, 0, -1, -1, -1, -1, 1, 1, -1, 1, 0, 1, 0, -1, -1, -1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1, -1, 0, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

Omega(n) = A001222(n) is the number of prime factors of n counted with multiplicity.

LINKS

Table of n, a(n) for n=1..74.

EXAMPLE

Agrees with Moebius function A008683 except for numbers like 12, 18, .... Moebius(12) = 0, a(12) = (-1)^3 = -1. - N. J. A. Sloane, May 26 2018

MATHEMATICA

Table[If[GCD @@ FactorInteger[n][[All, 2]] == 1, (-1)^PrimeOmega[n], 0], {n, 100}]

CROSSREFS

An unsigned version is A132350.

Cf. A000005, A000961, A001221, A001222, A001597, A005117, A007916, A008683, A091050, A126706, A303554, A304326, A304362, A304779, A304817, A304819.

KEYWORD

sign

AUTHOR

Gus Wiseman, May 18 2018

STATUS

approved

A304768

Augmented integer conjugate of n. a(n) = (1/n) * A007947(n)^(1 + A051903(n)) where A007947 is squarefree kernel and A051903 is maximum prime exponent.

+10
2

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 18, 13, 14, 15, 2, 17, 12, 19, 50, 21, 22, 23, 54, 5, 26, 3, 98, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 250, 41, 42, 43, 242, 75, 46, 47, 162, 7, 20, 51, 338, 53, 24, 55, 686, 57, 58, 59, 450, 61, 62, 147, 2, 65, 66, 67

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Image is the weak numbers A052485, on which n -> a(n) is an involution whose fixed points are the squarefree numbers A005117.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

If n = Product_{i = 1..k} prime(x_i)^y_i, then a(n) = Product_{i = 1..k} prime(x_i)^(max{y_1,...,y_k} - y_i + 1).

MATHEMATICA

acj[n_]:=Module[{f, m}, f=FactorInteger[n]; m=Max[Last/@f]; Times@@Table[p[[1]]^(m-p[[2]]+1), {p, f}]];

Array[acj, 100]

PROG

(PARI) a(n) = {if(n==1, 1, my(f = factor(n), e = vecmax(f[, 2]) + 1); prod(i = 1, #f~, f[i, 1]^e) / n); } \\ Amiram Eldar, Feb 12 2023

CROSSREFS

Cf. A001597, A001694, A005117, A007916, A007947, A013929, A051903, A052410, A062759, A066638, A072774, A087320, A303554.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 18 2018

STATUS

approved

A304449

Numbers that are either squarefree or a perfect power.

+10
1

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

First differs from A072774 at a(105) = 144, A072774(105) = 145.

Apparently the 1 and the members of A062770. - R. J. Mathar, May 22 2018

LINKS

Table of n, a(n) for n=1..67.

FORMULA

Union of A005117 and A001597. Complement of A303946.

MATHEMATICA

Select[Range[150], SquareFreeQ[#]||GCD@@FactorInteger[#][[All, 2]]>1&]

PROG

(PARI) isok(n) = issquarefree(n) || ispower(n); \\ Michel Marcus, May 13 2018

(Python)

from math import isqrt

from sympy import mobius, integer_nthroot

def A304449(n):

def f(x): return int(n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))+sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))

m, k = n, f(n)

while m != k:

m, k = k, f(k)

return m # Chai Wah Wu, Aug 19 2024

CROSSREFS

Cf. A000009, A000720, A000837, A000837, A001222, A001597, A005117, A007916, A013929, A047966, A072774, A073576, A126706, A303554, A303946.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 12 2018

STATUS

approved

A363597

Union of prime powers and numbers that are not squarefree.

+10
1

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 100, 101, 103

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers that are prime powers p^m, m >= 0, or products of multiple powers of distinct primes p^m where at least 1 prime power p^m is such that m > 1.

Let N = A000027. Analogous to the following sequences:

A002808 = N \ {{1} U A000040} = {1} U A024619 U A013929,

A303554 = N \ A126706 = A000961 U A005117, and

A085961 = N \ {{1} U A246547} = {A005117 U A024619} \ {1}.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

FORMULA

Complement of A120944, i.e., A000027 \ A120944.

Union of A000961 and A013929.

Union of {1}, A000040, A126706, and A246547.

EXAMPLE

1 is in the sequence because it is the empty product.

Prime p is in the sequence because it is not a composite squarefree number.

Numbers k that have prime power factors p^m | k where at least one prime power factor is such that m > 1 are in the sequence because they are not squarefree composites. Examples include 8, 9, 12, 20, and 36.

MATHEMATICA

Select[Range[103], Nand[SquareFreeQ[#], CompositeQ[#]] &]

PROG

(PARI) isok(k) = (k==1) || isprimepower(k) || !issquarefree(k); \\ Michel Marcus, Aug 24 2023

(Python)

from math import isqrt

from sympy import mobius, primepi

def A363597(n):

if n==1: return 1

def f(x): return n-1+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x)

m, k = n-1, f(n-1)

while m != k:

m, k = k, f(k)

return m # Chai Wah Wu, Aug 02 2024

CROSSREFS

Cf. A000040, A000961, A013929, A120944, A126706, A246547.

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, Aug 15 2023

STATUS

approved

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