A193433 -id:A193433 - 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

A193462 Sum of the distinct prime divisors of n^2+1.

Original entry on oeis.org

0, 2, 5, 7, 17, 15, 37, 7, 18, 43, 101, 63, 34, 24, 197, 115, 257, 36, 18, 183, 401, 32, 102, 60, 577, 315, 677, 80, 162, 423, 70, 52, 46, 116, 102, 615, 1297, 144, 22, 763, 1601, 31, 358, 44, 162, 1015, 102, 37, 466, 1203, 102, 1303, 546, 288, 2917, 108, 3137

Offset: 0

Michel Lagneau, Jul 28 2011

nonn

			a(7) = 7 because 7^2+1 = 2*5^2 and the sum of the 2 distinct prime divisors {2, 5} is 7.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A193432, A193433.

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

Join[{0},Table[Total[Transpose[FactorInteger[n^2+1]][[1]]],{n,60}]] (* Harvey P. Dale, Oct 18 2013 *)

A333169 a(n) = phi(n^2 + 1), where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 1, 4, 4, 16, 12, 36, 20, 48, 40, 100, 60, 112, 64, 196, 112, 256, 112, 240, 180, 400, 192, 384, 208, 576, 312, 676, 288, 624, 420, 832, 432, 800, 432, 1056, 612, 1296, 544, 1088, 760, 1600, 812, 1408, 720, 1776, 1012, 2016, 768, 1840, 1200, 2400, 1300, 2160

Offset: 0

Amiram Eldar, Mar 09 2020

nonn

			a(0) = phi(0^2 + 1) = phi(1) = 1.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 166.

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

Cf. A000010, A002522, A067720, A193432, A193433, A333167.

Mathematica
```
Table[EulerPhi[k^2 + 1], {k, 0, 100}]
```

a(n) = eulerphi(n^2+1); \\ Michel Marcus, Mar 10 2020

a(n) = A000010(A002522(n)).

A333167 a(n) = r_2(n^2 + 1), where r_2(k) is the number of ways of writing k as a sum of 2 squares (A004018).

Original entry on oeis.org

4, 4, 8, 8, 8, 8, 8, 12, 16, 8, 8, 8, 16, 16, 8, 8, 8, 16, 24, 8, 8, 16, 16, 16, 8, 8, 8, 16, 16, 8, 16, 16, 24, 16, 16, 8, 8, 16, 24, 8, 8, 12, 16, 24, 16, 8, 16, 32, 16, 8, 16, 8, 16, 16, 8, 16, 8, 32, 16, 8, 16, 8, 16, 16, 16, 8, 8, 16, 32, 8, 24, 8, 32, 32

Offset: 0

Amiram Eldar, Mar 09 2020

nonn

			a(0) = r_2(0^2 + 1) = r_2(1) = A004018(1) = 4.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 166.

Amiram Eldar, Table of n, a(n) for n = 0..10000
E. J. Scourfield, The divisors of a quadratic polynomial, Glasgow Mathematical Journal, Vol. 5, No. 1 (1961), pp. 8-20.

Cf. A004018, A002522, A193432, A193433, A333169.

Table[SquaresR[2, k^2 + 1], {k, 0, 100}]

a(n) = A004018(A002522(n)).

A067719 Numbers k such that sigma(k^2 + 1) == 0 (mod k).

Original entry on oeis.org

1, 2, 3, 9, 12, 21, 33, 72, 93, 196, 228, 252, 360, 475, 850, 1458, 1725, 1752, 2100, 2241, 2584, 3007, 3404, 4347, 4743, 5544, 5720, 6555, 6600, 9909, 10512, 14175, 15507, 16680, 19404, 26460, 29008, 29484, 36003, 36400, 37107, 46728, 88209, 88641, 89424, 94770

Offset: 1

Benoit Cloitre, Feb 05 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..250 (terms 1..100 from Harvey P. Dale)

Cf. A000203, A002522, A193433.

q:= n-> is(irem(numtheory[sigma](n^2+1), n)=0):
select(q, [$1..100000])[];  # Alois P. Heinz, Jan 26 2023

Select[Range[50000],Divisible[DivisorSigma[1,#^2+1],#]&] (* Harvey P. Dale, Nov 04 2011 *)

A333173 a(n) = r_4(n^2 + 1), where r_4(k) is the number of ways of writing k as a sum of 4 squares (A000118).

Original entry on oeis.org

8, 24, 48, 144, 144, 336, 304, 744, 672, 1008, 816, 1488, 1440, 2592, 1584, 2736, 2064, 4320, 3472, 4368, 3216, 6048, 4704, 7776, 4624, 7536, 5424, 10656, 7584, 10128, 7776, 12768, 10416, 15840, 10080, 14736, 10384, 19872, 14736, 18288, 12816, 20904, 16992, 28272

Offset: 0

Amiram Eldar, Mar 09 2020

nonn

			a(0) = r_4(0^2 + 1) = r_4(1) = A000118(1) = 8.

Amiram Eldar, Table of n, a(n) for n = 0..10000
R. Sitaramachandrarao and P. V. Krishnaiah, On the sums Sigma_{n<=x} A(f(n)) and Sigma_{p<=x} A(f(p)), Journal of Number Theory, Vol. 23, No. 2 (1986), pp. 149-168.

Cf. A000118, A002522 A193432, A193433, A333167, A333169.

Table[SquaresR[4, k^2 + 1], {k, 0, 100}]

a(n) = A000118(A002522(n)).

A067465 Numbers k such that sigma(k^2+1) is a perfect square.

Original entry on oeis.org

0, 13, 64, 67, 189, 344, 900, 2499, 3028, 4253, 5257, 6277, 9075, 11516, 12544, 18739, 18925, 47018, 49011, 62355, 72981, 112061, 125011, 125653, 152765, 167820, 168317, 174724, 179040, 225492, 240597, 251568, 261163, 302128, 302812, 345283

Offset: 1

Benoit Cloitre, Feb 23 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A000203, A000290, A002522, A006532, A193433.

Select[Range[0,400000],IntegerQ[Sqrt[DivisorSigma[1,#^2+1]]]&] (* Harvey P. Dale, May 14 2018 *)

for(n=0,500000,if(issquare(sigma(n^2+1)),print1(n,",")))

More terms from Rick L. Shepherd, Apr 16 2002

A234645 Sum of the divisors of n^3+1.

Original entry on oeis.org

1, 3, 13, 56, 84, 312, 256, 660, 800, 1332, 1344, 3458, 2240, 3792, 4836, 6572, 4356, 13440, 6160, 16800, 13312, 15192, 11136, 35685, 19840, 25284, 30976, 42560, 22740, 63648, 30464, 71820, 51792, 65664, 53952, 111440, 52136, 84480, 99008, 133560, 75264

Offset: 0

Vincenzo Librandi, Jan 01 2014

			a(4) = 84 because 4^3+1 = 65 and the sum of the 4 divisors {1, 5, 13, 65} is 84.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000203, A001093, A062835, A065764, A175926, A193433, A333240.

Magma
```
[SumOfDivisors(n^3+1): n in [0..50]];
```

Table[Total[Divisors[n^3 + 1]], {n, 0, 50}]
DivisorSigma[1,Range[0,40]^3+1] (* Harvey P. Dale, Jul 27 2021 *)

a(n) = sigma(n^3+1); \\ Michel Marcus, Jun 19 2015

a(n) = A000203(A001093(n)). - Michel Marcus, Jun 19 2015

Sum_{k=1..n} a(k) = c * n^4 + O((n*log(n))^3), where c = (83/288) * Product_{primes p == 1 (mod 3)} ((p^2+2)/(p^2-1)) * Product_{primes p == 2 (mod 3)} (p^2/(p^2-1)) = 0.449926279... . - Amiram Eldar, Dec 09 2024

A234860 Sum of the divisors of n^3 - 1.

Original entry on oeis.org

8, 42, 104, 224, 264, 780, 592, 1680, 1520, 2880, 1896, 5642, 2968, 5808, 8736, 9548, 7200, 15360, 8440, 19488, 19032, 23040, 14448, 49920, 23560, 31836, 32912, 53312, 34200, 77688, 38912, 70812, 62088, 74088, 67584, 152320, 56392, 107520, 99736

Offset: 2

Vincenzo Librandi, Jan 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 2..1001

Cf. A000203, A062835, A065764, A068601, A175926, A193433, A333240.

Magma
```
[SumOfDivisors(n^3-1): n in [2..50]];
```

Table[Total[Divisors[n^3 - 1]], {n, 2, 50}]

a(n) = sigma(n^3-1); \\ Amiram Eldar, Dec 09 2024

From Amiram Eldar, Dec 09 2024: (Start)
a(n) = A000203(A068601(n)).
Sum_{k=2..n} a(k) = c * n^4 + O((n*log(n))^3), where c = (83/288) * Product_{primes p == 1 (mod 3)} ((p^2+2)/(p^2-1)) * Product_{primes p == 2 (mod 3)} (p^2/(p^2-1)) = 0.4499262799... . (End)

Offset corrected by Amiram Eldar, Dec 09 2024

A333172 a(n) = Sum_{k=0..n} sigma(k^2 + 1), where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

1, 4, 10, 28, 46, 88, 126, 219, 303, 429, 531, 717, 897, 1221, 1419, 1761, 2019, 2559, 2993, 3539, 3941, 4697, 5285, 6257, 6835, 7777, 8455, 9787, 10735, 12001, 12973, 14569, 15871, 17851, 19111, 20953, 22251, 24735, 26577, 28863, 30465, 33078, 35202, 38736

Offset: 0

Amiram Eldar, Mar 09 2020

nonn

			a(0) = sigma(0^2 + 1) = sigma(1) = 1.
a(1) = sigma(0^2 + 1) + sigma(1^2 + 1) = sigma(1) + sigma(2) = 1 + 3 = 4.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 166.

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

Partial sums of A193433.

Cf. A000203, A002522, A006752.

Accumulate @ Table[DivisorSigma[1, k^2 + 1], {k, 0, 100}]

a(n) = sum(k=0, n, sigma(k^2+1)); \\ Michel Marcus, Mar 10 2020

a(n) ~ (5*G/Pi^2) * n^3, where G is Catalan's constant (A006752).

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A193432 Number of divisors of n^2 + 1.

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A193462 Sum of the distinct prime divisors of n^2+1.

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A333169 a(n) = phi(n^2 + 1), where phi is the Euler totient function (A000010).

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A333167 a(n) = r_2(n^2 + 1), where r_2(k) is the number of ways of writing k as a sum of 2 squares (A004018).

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A067719 Numbers k such that sigma(k^2 + 1) == 0 (mod k).

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A333173 a(n) = r_4(n^2 + 1), where r_4(k) is the number of ways of writing k as a sum of 4 squares (A000118).

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A067465 Numbers k such that sigma(k^2+1) is a perfect square.

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A234645 Sum of the divisors of n^3+1.

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