A354824 -id:A354824

Search: a354824 -id:a354824

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A346242

Dirichlet inverse of A324198, where A324198(n) = gcd(n, A276086(n)).

+10
20

1, -1, -3, 0, -1, 5, -1, 0, 6, -3, -1, -2, -1, 1, -9, 0, -1, -16, -1, 4, 3, 1, -1, 0, -24, 1, -12, 0, -1, 43, -1, 0, 3, 1, -5, 14, -1, 1, 3, 0, -1, -11, -1, 0, 54, 1, -1, 0, -6, 32, 3, 0, -1, 44, -3, -6, 3, 1, -1, -50, -1, 1, -24, 0, 1, -5, -1, 0, 3, -15, -1, -4, -1, 1, 96, 0, -5, -5, -1, 0, 24, 1, -1, 8, -3, 1, 3, 0, -1

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

OFFSET

1,3

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..30030

Index entries for sequences related to primorial base

FORMULA

a(n) = A346243(n) - A324198(n).

From Antti Karttunen, Jun 09 2022: (Start)

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A324198(n/d) * a(d).

For all n >= 1, A000035(a(n)) = A008966(n).

For all n >= 1, a(A045344(n)) = -1.

(End)

PROG

(PARI)

up_to = 65537;

DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.

A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };

v346242 = DirInverseCorrect(vector(up_to, n, A324198(n)));

A346242(n) = v346242[n];

CROSSREFS

Cf. A276086, A324198, A346243.

Cf. A008966 (parity of terms), A005117 (positions of odd terms), A013929 (of even terms), A045344 (of -1's, at least a subset of them), A354810 (of 0's), A354811 (of 1's), A354812 (of 2's), A354813 (of 3's), A354814 (of 4's), A354822 (of -2's).

Cf. also A354347, A354348, A354823, A354824.

KEYWORD

sign,base

AUTHOR

Antti Karttunen, Jul 13 2021

STATUS

approved

A354347

Dirichlet inverse of A345000, where A345000(n) = gcd(A003415(n), A003415(A276086(n))), with A003415 the arithmetic derivative, and A276086 the primorial base exp-function.

+10
14

1, -1, -1, -1, -1, 1, -1, -1, 0, 1, -1, 1, -1, 1, 1, -9, -1, -2, -1, 1, -3, 1, -1, 1, -4, -3, 0, 1, -1, -1, -1, 21, 1, 1, -1, -6, -1, 1, 1, 3, -1, 7, -1, -1, 0, -3, -1, 23, 0, 4, -3, 7, -1, 2, 1, 3, 1, 1, -1, -1, -1, 1, 8, 15, -1, -1, -1, 1, 1, 3, -1, 14, -1, 1, -46, -7, -1, 7, -1, 5, 0, 1, -1, 3, 1, -3, 1, -131

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

OFFSET

1,16

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..30030

Index entries for sequences related to primorial base

FORMULA

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A345000(n/d) * a(d).

For all n >= 1, A000035(a(n)) = A353627(n).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

A345000(n) = gcd(A003415(n), A003415(A276086(n)));

memoA354347 = Map();

A354347(n) = if(1==n, 1, my(v); if(mapisdefined(memoA354347, n, &v), v, v = -sumdiv(n, d, if(d<n, A345000(n/d)*A354347(d), 0)); mapput(memoA354347, n, v); (v)));

CROSSREFS

Cf. A003415, A276086, A345000.

Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms), A354815 (positions of 0's), A354816 (of -1's).

Cf. also A346242, A354348, A354823, A354824.

KEYWORD

sign

AUTHOR

Antti Karttunen, Jun 07 2022

STATUS

approved

A354823

Dirichlet inverse of A351083, where A351083(n) = gcd(n, A327860(n)), and A327860 is the arithmetic derivative of the primorial base exp-function.

+10
8

1, -1, -1, -1, -1, 1, -7, -5, 0, 1, -1, 1, -1, 13, -3, -1, -1, -2, -1, -7, 13, 1, -1, 9, -24, 1, 0, 7, -1, 7, -1, 33, 1, -15, 9, -6, -1, 1, -11, 27, -1, -25, -1, -1, 4, 1, -1, 7, 48, 24, 1, -1, -1, 2, -3, 59, 1, 1, -1, 19, -1, 1, -12, 23, 1, -1, -1, 33, 1, -23, -1, -2, -1, 1, 52, 1, 7, 23, -1, -67, 0, 1, -1, -25

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

OFFSET

1,7

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..30030

Index entries for sequences related to primorial base

FORMULA

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A351083(n/d) * a(d).

PROG

(PARI)

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };

A351083(n) = gcd(n, A327860(n));

memoA354823 = Map();

A354823(n) = if(1==n, 1, my(v); if(mapisdefined(memoA354823, n, &v), v, v = -sumdiv(n, d, if(d<n, A351083(n/d)*A354823(d), 0)); mapput(memoA354823, n, v); (v)));

CROSSREFS

Cf. A327860, A351083.

Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).

Cf. also A346242, A354347, A354348, A354824.

KEYWORD

sign

AUTHOR

Antti Karttunen, Jun 09 2022

STATUS

approved

A355692

Dirichlet inverse of A355442, gcd(A003961(n), A276086(n)), where A003961 is fully multiplicative with a(p) = nextprime(p), and A276086 is primorial base exp-function.

+10
2

1, -3, -1, 0, -1, 1, -1, 24, -4, 3, -1, 16, -1, 3, -3, -72, -1, 6, -1, 6, -3, 3, -1, -68, 0, 3, -116, 0, -1, 21, -1, 24, 1, 3, -5, 72, -1, 3, -3, -120, -1, 23, -1, 6, -158, 3, -1, 28, 0, -18, -3, 0, -1, 632, -5, -24, -3, 3, -1, -54, -1, 3, 16, 504, -5, -1, -1, 6, -3, 15, -1, -400, -1, 3, -236, 0, 1, 23, -1, 474, 136

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

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..11550

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to primorial base

FORMULA

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A355442(n/d) * a(d).

PROG

(PARI)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

A355442(n) = gcd(A003961(n), A276086(n));

memoA355692 = Map();

A355692(n) = if(1==n, 1, my(v); if(mapisdefined(memoA355692, n, &v), v, v = -sumdiv(n, d, if(d<n, A355442(n/d)*A355692(d), 0)); mapput(memoA355692, n, v); (v)));

CROSSREFS

Cf. A003961, A276086.

Cf. also A346242, A354347, A354348, A354823, A354824.

KEYWORD

sign

AUTHOR

Antti Karttunen, Jul 18 2022

STATUS

approved

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