A193330 -id:A193330

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page 1

A193432

Number of divisors of n^2 + 1.

+10
26

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 4, 4, 8, 2, 4, 2, 8, 6, 4, 2, 8, 4, 8, 2, 4, 2, 8, 4, 4, 4, 8, 6, 8, 4, 4, 2, 8, 6, 4, 2, 6, 4, 12, 4, 4, 4, 16, 4, 4, 4, 4, 4, 8, 2, 8, 2, 16, 4, 4, 4, 4, 4, 8, 4, 4, 2, 8, 8, 4, 6, 4, 8, 16, 2, 8, 4, 8, 4, 4, 4, 8, 6, 16, 2

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

OFFSET

0,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = A000005(A002522(n)). - Michel Marcus, Mar 16 2018

EXAMPLE

a(7) = 6 because 7^2 + 1 = 50 and the 6 divisors are {1, 2, 5, 10, 25, 50}.

MAPLE

with(numtheory):for n from 0 to 110 do:n1:=nops(divisors(n^2+1)):s:=0:for m from 1 to n1 do: s:=s+1:od: printf(`%d, `, s):od:

MATHEMATICA

Array[DivisorSigma[0, #^2 + 1] &, 85, 0] (* Michael De Vlieger, Mar 17 2018 *)

PROG

(PARI) a(n) = numdiv(n^2+1); \\ Michel Marcus, Mar 16 2018

CROSSREFS

Cf. A000005, A002522, A128428, A193330, A193433.

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jul 28 2011

STATUS

approved

A128428

Number of distinct prime factors of n^2+1.

+10
9

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 1, 3, 2, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 2, 2, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 2, 2, 3, 2, 2, 1, 3, 2, 2, 2, 2, 3, 4, 1, 3, 2, 3, 2, 2, 2, 3, 2, 4, 1, 2, 2, 3, 2, 3, 1, 3, 2, 3, 1, 2, 2, 3, 3, 3, 2, 2, 2

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

OFFSET

1,3

COMMENTS

a(n) is also the number of distinct prime factors, up to multiplication by units, of n + i in the ring of Gaussian integers. - Jason Kimberley, Dec 17 2011

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A001221(n^2+1).

EXAMPLE

a(3) = 2 because 3^2+1 = 2*5.

MAPLE

a:= n-> nops(select(isprime, numtheory[divisors](n^2+1))):

seq(a(n), n=1..100); # Alois P. Heinz, Dec 06 2020

MATHEMATICA

a[n_]:=Length[FactorInteger[n^2 + 1]]

PROG

(PARI) a(n)=omega(n^2+1) \\ Charles R Greathouse IV, Jul 31 2011

CROSSREFS

Cf. A193330 (counted with multiplicity).

KEYWORD

nonn

AUTHOR

Kent Horvath (kenthorvath(AT)gmail.com), May 10 2007

STATUS

approved

A193929

Number of prime factors of n^4 + 1, counted with multiplicity.

+10
3

0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 2, 2, 3, 2, 4, 3, 3, 1, 3, 1, 3, 2, 3, 2, 3, 1, 3, 1, 2, 2, 4, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 2, 1, 3, 3, 4, 2, 4, 1, 2, 1, 4, 2, 4, 2, 3, 1, 3, 1, 3, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 2, 2, 3, 2, 1, 3, 2, 3, 3, 4, 2, 2, 2, 2, 2, 3, 1

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

OFFSET

0,4

COMMENTS

This is to A193330 as A002523(n) = n^4+1 is to A002522(n) = n^2 + 1. a(n) = 2 when n^4+1 is prime, iff n is in A037896.

LINKS

Amiram Eldar, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = A001222(A002523(n)) = bigomega(n^4+1) or Omega(n^4+1).

EXAMPLE

a(9) = 3 because 9^4+1 = 6562 = 2 * 17 * 193, which has 3 prime factors, counted with multiplicity

MATHEMATICA

Join[{0}, Table[Total[Transpose[FactorInteger[n^4 + 1]][[2]]], {n, 100}]] (* T. D. Noe, Aug 10 2011 *)

Join[{0}, Table[PrimeOmega[n^4+1], {n, 120}]] (* Harvey P. Dale, Sep 25 2012 *)

PROG

(PARI) a(n) = bigomega(n^4+1); \\ Michel Marcus, Feb 09 2020

(Magma) [0] cat [&+[p[2]: p in Factorization(n^4+1)]:n in [1..120]]; // Marius A. Burtea, Feb 09 2020

CROSSREFS

Cf. A001222, A002523, A193432, A193562.

KEYWORD

nonn,easy

AUTHOR

Jonathan Vos Post, Aug 09 2011

STATUS

approved

A257366

Smallest integer m such that m^2 + 1 has exactly n prime factors, counted with multiplicity.

+10
3

1, 3, 7, 43, 57, 307, 1068, 2943, 12943, 45807, 110443, 670807, 2733307, 25670807, 113561432, 123327057, 657922943, 17213170807, 7200891693, 148802454193

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

OFFSET

1,2

COMMENTS

Or, first occurrences of n in A193330.

Is the sequence finite?

a(n) exists for arbitrarily large n, and in particular a(n+k) < A185952(n) by the Chinese Remainder Theorem and the fact that -1 is a square mod the primes in A002313, for some k >= 0. Probably a(n) exists for each n. - Charles R Greathouse IV, Apr 21 2015

From Jon E. Schoenfield, Jun 14-15 2015: (Start)

Numbers of the form m^2+1 cannot be divisible by 3; they may be divisible by 2 (but not 2^2), and the only other numbers they can have as prime factors are 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, ..., i.e., the terms of A002144. This explains why 5 tends to appear with such high multiplicity in the factorizations of a(n)^2+1, as numbers with a higher multiplicity of the prime factor 5 are more likely to be small enough to be the smallest integer with n prime factors than numbers whose n prime factors (counted with multiplicity) are mostly larger than 5. Having 2 as one of the n prime factors is also an advantage, which accounts for the predominance of odd terms, yielding even values of m^2+1.

For k>1, if m^2+1 is divisible by 5^k, then there are only two possible residues of m mod 5^k; e.g., if m^2+1 is divisible by 5^4, then m mod 5^4 must be either 182 or 443. Thus it is not coincidental that the last few digits of some terms also appear as the last few digits of other terms, e.g., terms ending in 443 or 443+500 = 943, or in 182+125 = 307 or 182+625 = 807. (End)

LINKS

Table of n, a(n) for n=1..20.

EXAMPLE

a(1)=1 because 1^2+1=2(prime),

a(2)=3 because 3^2+1=10=2*5,

a(3)=7 because 7^2+1=50=2*2*5,

...............

a(14)=25670807 because 25670807^2+1=2*5^11*149*45289.

MATHEMATICA

Table[m = 1; While[PrimeOmega[m^2 + 1] != n, m++]; m, {n, 12}] (* Michael De Vlieger, Apr 21 2015 *)

PROG

(PARI) a(n)=my(m); while(bigomega(m++^2+1)!=n, ); m \\ Charles R Greathouse IV, Apr 21 2015

CROSSREFS

Cf. A002144, A180278, A193330.

KEYWORD

nonn,more

AUTHOR

Zak Seidov, Apr 21 2015

EXTENSIONS

a(15)-a(17) from Jon E. Schoenfield, Jun 14 2015

a(18)-a(19) from Jon E. Schoenfield, Jun 15 2015

a(20) from Jon E. Schoenfield, Jul 10 2015

STATUS

approved

A194003

Number of prime factors of n^8 + 1, counted with multiplicity.

+10
2

0, 1, 1, 3, 1, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 4, 3, 3, 2, 6, 2, 4, 3, 3, 2, 2, 2, 4, 3, 3, 2, 4, 6, 3, 2, 2, 4, 3, 3, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 5, 2, 3, 2, 4, 4, 4, 3, 6, 2, 5, 2, 2, 2, 5, 2, 5, 4, 4, 3, 4, 3, 5, 4, 2, 3, 4, 2, 4

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

OFFSET

0,4

COMMENTS

This is to A193330 as A002523(n) = n^4+1 is to A002522(n) = n^2 + 1, and as A060890(n) = n^8+1 is to A002522(n) = n^2 + 1. a(n) = 1 when n^8+1 is prime, iff n is in {1, 2, 4} unless there is a larger Fermat prime than 65537.

LINKS

Amiram Eldar, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = A001222(A060890(n)) = bigomega(n^8+1) or Omega(n^8+1)

EXAMPLE

a(10) = 2 because 10^8 + 1 = 100000001 = 17 * 5882353 has 2 prime factors.

a(40) = 6 because 40^8 + 1 = 6553600000001 = 17^2 * 113 * 337 * 641 * 929 has 6 prime factors (with multiplicity) and is the smallest example not squarefree.

MATHEMATICA

Join[{0}, Table[Total[Transpose[FactorInteger[n^8 + 1]][[2]]], {n, 50}]]

PrimeOmega[Range[0, 90]^8+1] (* Harvey P. Dale, May 27 2018 *)

PROG

(PARI) a(n) = bigomega(n^8+1); \\ Michel Marcus, Feb 09 2020

(Magma) [0] cat [&+[p[2]: p in Factorization(n^8+1)]:n in [1..90]]; // Marius A. Burtea, Feb 09 2020

CROSSREFS

Cf. A001222, A060890, A002523, A193432, A193562, A193929.

KEYWORD

nonn,easy

AUTHOR

Jonathan Vos Post, Aug 10 2011

STATUS

approved

A272044

Numbers n such that n and n^2+1 have the same number of prime factors (including multiplicities).

+10
2

2, 9, 15, 18, 22, 25, 27, 34, 35, 39, 46, 49, 51, 58, 62, 63, 65, 69, 70, 75, 85, 86, 95, 98, 105, 106, 121, 125, 132, 138, 141, 145, 147, 148, 153, 158, 159, 166, 169, 172, 174, 178, 194, 201, 202, 205, 209, 212, 214, 219, 221, 226, 254, 262, 274, 282, 285, 289, 298, 299

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

OFFSET

1,1

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

FORMULA

Numbers n such that bigomega(n) = bigomega(n^2+1).

MAPLE

with(numtheory): A272044:=n->`if`(bigomega(n)=bigomega(n^2+1), n, NULL): seq(A272044(n), n=1..500); # Wesley Ivan Hurt, Apr 19 2016

MATHEMATICA

Select[Range@ 300, PrimeOmega[#^2 + 1] == PrimeOmega@ # &] (* Michael De Vlieger, Apr 19 2016 *)

PROG

(PARI) is(n)=bigomega(n)==bigomega(n^2+1) \\ Charles R Greathouse IV, Apr 18 2016

CROSSREFS

Cf. A001222 (bigomega), A193330.

KEYWORD

nonn

AUTHOR

Benjamin Przybocki, Apr 18 2016

STATUS

approved

A261609

Number of composite divisors of n^2+1.

+10
1

0, 0, 1, 0, 1, 0, 3, 1, 1, 0, 1, 1, 4, 0, 1, 0, 4, 3, 1, 0, 4, 1, 4, 0, 1, 0, 4, 1, 1, 1, 4, 3, 4, 1, 1, 0, 4, 3, 1, 0, 3, 1, 8, 1, 1, 1, 11, 1, 1, 1, 1, 1, 4, 0, 4, 0, 12, 1, 1, 1, 1, 1, 4, 1, 1, 0, 4, 5, 1, 3, 1, 4, 11, 0, 4, 1, 4, 1, 1, 1, 4, 3, 11, 0, 1, 1

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

OFFSET

1,7

LINKS

Michel Lagneau, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A055212(A002522(n)).

EXAMPLE

a(7) = 3 because the composite divisors of 7^2+1 are 10, 25, 50.

MATHEMATICA

Table[ Count[ PrimeQ[ Divisors[n^2+1] ], False] - 1, {n, 1, 105} ]

CROSSREFS

Cf. A002522, A055212, A128428, A193330, A193432.

KEYWORD

nonn,easy

AUTHOR

Michel Lagneau, Aug 26 2015

STATUS

approved

A193759

Array, by antidiagonals, A(k,n) is the number of prime factors of n^(2^k) + 1, counted with multiplicity.

+10
0

0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 3, 1, 2, 0, 1, 2, 2, 1, 2, 1, 0, 1, 2, 2, 2, 3, 1, 3, 0, 1, 2, 2, 2, 3, 2, 2, 2, 0, 1, 2, 6, 2, 4, 3, 3, 2, 2, 0, 1, 3, 5, 2, 4, 3, 3, 3, 3, 1, 3, 0, 1, 4, 7, 3, 4, 3, 4, 3, 2, 2, 2, 1, 0, 1, 5

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

OFFSET

0,10

COMMENTS

The main diagonal A(n,n) = number of prime factors of n^(2^n) + 1, counted with multiplicity, begins 0, 1, 1, 3, 2, 4, 3, 6, 6.

LINKS

Table of n, a(n) for n=0..82.

EXAMPLE

A(4,5) = 3 because 1+5^16 = 152587890626 = 2 * 2593 * 29423041, which has 3 prime factors. The array begins:

================================================================

....|n=0|n=1|n=2|n=3|n=4|n=5|n=6|n=7|n=8|n=9|.10|.11|comment

====|===|===|===|===|===|===|===|===|===|===|===|===|===========

k=0.|.0.|.1.|.1.|.2.|.1.|.2.|.1.|.3.|.2.|.2.|.1.|.3.|A001222

k=1.|.0.|.1.|.1.|.2.|.1.|.2.|.1.|.3.|.2.|.2.|.1.|.2.|A193330

k=2.|.0.|.1.|.1.|.2.|.1.|.2.|.1.|.2.|.2.|.3.|.2.|.2.|A193929

k=3.|.0.|.1.|.1.|.3.|.1.|.3.|.2.|.3.|.3.|.2.|.2.|.3.|A194003

k=4.|.0.|.1.|.1.|.2.|.2.|.3.|.3.|.3.|.3.|.2.|.5.|.3.|not in OEIS

k=5.|.0.|.1.|.2.|.2.|.2.|.4.|.3.|.4.|.3.|.2.|.4.|.4.|not in OEIS

================================================================

CROSSREFS

Cf. A001222, A002523, A060890, A193330, A193929, A194003.

KEYWORD

nonn,hard,tabl

AUTHOR

Jonathan Vos Post, Aug 11 2011

EXTENSIONS

Edited by Alois P. Heinz, Aug 11 2011

More terms from Max Alekseyev, Sep 09 2011

STATUS

approved

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