A193330 -id:A193330 - cp's OEIS frontend

A193432 Number of divisors of n^2 + 1.

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

Offset: 0

Michel Lagneau, Jul 28 2011

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Anton Shakov, Polynomials in Z[x] whose divisors are enumerated by SL_2(N_0), arXiv:2405.03552 [math.NT], 2024.

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

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:

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

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

from sympy import divisor_count
def A193432(n): return divisor_count(n**2+1) # Chai Wah Wu, Apr 17 2025

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

A128428 Number of distinct prime factors of n^2+1.

Original entry on oeis.org

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

Offset: 1

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

nonn

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

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

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

Cf. A193330 (counted with multiplicity).

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]]
```

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

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

A257366 Smallest integer m such that m^2 + 1 has exactly n prime factors, counted with multiplicity.

Original entry on oeis.org

1, 3, 7, 43, 57, 307, 1068, 2943, 12943, 45807, 110443, 670807, 2733307, 25670807, 113561432, 123327057, 657922943, 17213170807, 7200891693, 148802454193, 1120482141693

Offset: 1

Zak Seidov, Apr 21 2015

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)

			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.

Cf. A002144, A180278, A193330.

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

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

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
a(21) from Max Alekseyev, Jan 08 2025

A193929 Number of prime factors of n^4 + 1, counted with multiplicity.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Aug 09 2011

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.

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

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

Cf. A001222, A002523, A193432, A193562.

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

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 *)

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

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

A194003 Number of prime factors of n^8 + 1, counted with multiplicity.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Aug 10 2011

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.

			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.

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

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

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

Join[{0}, Table[Total[Transpose[FactorInteger[n^8 + 1]][[2]]], {n, 50}]]
PrimeOmega[Range[0,90]^8+1] (* Harvey P. Dale, May 27 2018 *)

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

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

A272044 Numbers n such that n and n^2+1 have the same number of prime factors (including multiplicities).

Original entry on oeis.org

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

Offset: 1

Benjamin Przybocki, Apr 18 2016

nonn

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

Cf. A001222 (bigomega), A193330.

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

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

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

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

A261609 Number of composite divisors of n^2+1.

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Aug 26 2015

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

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

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

Table[ Count[ PrimeQ[ Divisors[n^2+1] ], False] - 1, {n, 1, 105} ]
Table[Count[Divisors[n^2+1],?CompositeQ],{n,100}] (* _Harvey P. Dale, Dec 16 2024 *)

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

A193759 Array, by antidiagonals, A(k,n) is the number of prime factors of n^(2^k) + 1, counted with multiplicity.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Aug 11 2011

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.

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

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

Edited by Alois P. Heinz, Aug 11 2011

More terms from Max Alekseyev, Sep 09 2011

cp's OEIS Frontend

A193432 Number of divisors of n^2 + 1.

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A128428 Number of distinct prime factors of n^2+1.

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A257366 Smallest integer m such that m^2 + 1 has exactly n prime factors, counted with multiplicity.

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A193929 Number of prime factors of n^4 + 1, counted with multiplicity.

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A194003 Number of prime factors of n^8 + 1, counted with multiplicity.

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A272044 Numbers n such that n and n^2+1 have the same number of prime factors (including multiplicities).

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A261609 Number of composite divisors of n^2+1.

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