A001157 -id:A001157 - cp's OEIS frontend

A066088 Number of distinct prime factors of sigma_2(n), where sigma_2(n) = A001157(n).

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

Offset: 1

Labos Elemer, Dec 04 2001

nonn

			sigma_2(12) = 144 + 36 + 16 + 9 + 4 + 1 = 210, so a(12)=4.

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A001157, A001221.

PrimeNu[DivisorSigma[2,Range[110]]] (* Harvey P. Dale, Feb 23 2015 *)

a(n) = { omega(sigma(n, 2)) } \\ Harry J. Smith, Nov 11 2009

a(n) = A001221(A001157(n)).

Offset changed from 0 to 1 by Harry J. Smith, Nov 11 2009

A081380 Numbers k such that the sets of prime factors (ignoring multiplicity) of A000203(k) = sigma(k) and of A001157(k) = sigma_2(k) are identical.

Original entry on oeis.org

1, 180, 1444, 12996, 23805, 36100, 52020, 60228, 64980, 68832, 95220, 301140, 324900, 344160, 481824, 1505700, 1718721, 1720800, 2275758, 2409120, 3755844, 6874884, 6879645, 7965153, 8593605, 11378790, 12045600, 15930306, 17405892

Offset: 1

Labos Elemer, Mar 26 2003

nonn

			n = 1444 = 2^2*19^2, sigma(1444) = 2667 = 3*7*127, sigma_2(1444) = 2744343 = 3^2*7^4*127, common factor set = {3,7,127}.

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 180, p. 56, Ellipses, Paris, 2008.

Amiram Eldar, Table of n, a(n) for n = 1..100 (terms 1..67 from Donovan Johnson)

Cf. A000203, A001157, A081377, A081378.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; Do[s=ba[DivisorSigma[1, n]]; s5=ba[DivisorSigma[2, n]]; If[Equal[s, s5], Print[n]], {n, 1, 1000000}]

is(n)=factor(sigma(n))[,1]==factor(sigma(n,2))[,1] \\ Charles R Greathouse IV, Feb 19 2013

More terms from Lekraj Beedassy, Jul 18 2008

a(16)-a(29) from Donovan Johnson, May 24 2009

A082902 a(n) = gcd(2^n, sigma(2,n)) = gcd(A000079(n), A001157(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 4, 1, 2, 1, 2, 2, 4, 2, 2, 2, 1, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 1, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 2, 1, 1, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 2, 1, 4, 4, 2, 2, 4, 4, 2, 1, 2, 2, 2, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 2, 1, 2, 1, 2, 4, 2, 2, 8

Offset: 1

Labos Elemer, Apr 22 2003

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

Cf. A000079, A001157, A006519.

a[n_] := 2^IntegerExponent[DivisorSigma[2, n], 2]; Array[a, 100] (* Amiram Eldar, Oct 01 2023 *)

A082902(n) = gcd(2^n, sigma(n, 2)); \\ Antti Karttunen, Sep 27 2018

a(n) = A006519(A001157(n)). - Antti Karttunen, Sep 27 2018

Multiplicative with a(2^e) = 1, and a(p^e) = A006519(e+1) for an odd prime p. - Amiram Eldar, Oct 01 2023

A099978 Bisection of A001157: a(n) = sigma_2(2n-1).

Original entry on oeis.org

1, 10, 26, 50, 91, 122, 170, 260, 290, 362, 500, 530, 651, 820, 842, 962, 1220, 1300, 1370, 1700, 1682, 1850, 2366, 2210, 2451, 2900, 2810, 3172, 3620, 3482, 3722, 4550, 4420, 4490, 5300, 5042, 5330, 6510, 6100, 6242, 7381, 6890, 7540, 8420, 7922, 8500

Offset: 1

N. J. A. Sloane, Nov 19 2004

			From _M. F. Hasler_, Mar 06 2017: (Start)
a(1) = sigma_2(2*1-1) = 1.
a(2) = sigma_2(2*2-1) = 1 + 3^2 = 10.
a(5) = sigma_2(2*5-1) = 1 + 3^2 + 9^2 = 91. (End)
G.f.: A(x) = x + 10*x^2 + 26*x^3 + 50*x^4 + 91*x^5 + 122*x^6 + 170*x^7 + 260*x^8 + 290*x^9 + 362*x^10 + 500*x^11 + 530*x^12 + ... where A(x) = x/(1 - x) + 3^2*x^2/(1 - x^3) + 5^2*x^3/(1 - x^5) + 7^2*x^4/(1 - x^7) + 9^2*x^5/(1 - x^9) + .... - _Paul D. Hanna_, Jun 23 2025

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

Cf. A099979(n) = sigma_2(2n), the other bisection of A001157.

Cf. A283224.

with(numtheory): A099978 := n->sigma[2](2*n-1): seq(A099978(n), n=1..60);

DivisorSigma[2, Range[1, 91, 2]] (* Amiram Eldar, Aug 17 2019 *)

A099978(n)=sigma(2*n-1,2) \\ M. F. Hasler, Mar 06 2017

a(n) = A001157(2n-1) = sigma_2(2n-1). - M. F. Hasler, Mar 06 2017
Sum_{k=1..n} a(k) ~ 7*zeta(3)*n^3/6. - Vaclav Kotesovec, Aug 07 2022
G.f.: Sum_{n>=1} (2*n-1)^2 * x^n / (1 - x^(2*n-1)). - Paul D. Hanna, Jun 23 2025

More terms from Emeric Deutsch, Dec 07 2004

Edited by M. F. Hasler, Mar 06 2017

A175199 a(n) is the smallest integer k such that sigma_2(k) = sigma_2(k + 2n), where sigma_2(k) is the sum of squares of divisors of k (A001157).

Original entry on oeis.org

24, 430, 645, 860, 120, 864, 168, 1720, 1935, 10790, 264, 2580, 2795, 1570, 16185, 3440, 408, 3870, 456, 21580, 2355, 4730, 552, 5160, 600, 5590, 5805, 3140, 696, 4320, 744, 6880, 7095, 1248, 840, 7740, 888, 8170, 8385, 43160, 984, 4710, 1032, 9460

Offset: 1

Michel Lagneau, Mar 03 2010

nonn

The equation sigma_2(n) = sigma_2(n + p) has infinitely many solutions where p >= 2 and p is even (J. M. De Koninck).

			For n=1, sigma_2(24) = sigma_2(26) = 850.
For n=2, sigma_2(430) = sigma_2(434) = 240500.
For n=3, sigma_2(645) = sigma_2(651) = 481000.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 827.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Amiram Eldar, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. M. De Koninck, On the solutions of sigma_2(n) = sigma_2(n + p), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000005, A000203, A001158, A001159.

Cf. A053807, A064602.

with(numtheory):for k from 2 by 2 to 200 do :indic:=0:for n from 1 to 100000 do:liste:= divisors(n) : s2 :=sum(liste[i]^2, i=1..nops(liste)):liste:=divisors(n+k):s3:=sum(liste[i]^2, i=1..nops(liste)):if s2 = s3 and indic=0 then print(k):print(n):indic:=1:else fi:od:od:

Edited by Robert Israel, Aug 02 2024

A175705 Convolution square of A001157 (the sum of squared divisors).

Original entry on oeis.org

1, 10, 45, 142, 362, 780, 1561, 2762, 4808, 7570, 12034, 17482, 26072, 35884, 50909, 67012, 92111, 116950, 155720, 193564, 250914, 304244, 389286, 461654, 578952, 680944, 839304, 970094, 1188924, 1354164, 1637145, 1858344, 2215866, 2485068

Offset: 1

Michel Lagneau, Aug 10 2010

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

Cf. A001157, A328490.

with(numtheory): T:=array(1..200):for p from 1 to 200 do: liste:=divisors(p) :s2:=sum(liste[i]^2,i=1..nops(liste)):T[p] :=s2 :od : for n from 1 to 100 do: printf(`%d, `, sum (T[k]*T[n+1-k],k=1..n)):od:

a[n_] := Sum[DivisorSigma[2, k] * DivisorSigma[2, n + 1 - k], {k, 1, n}]; Array[a, 34] (* Amiram Eldar, Jul 31 2019 *)

a(n) = Sum_{k=1..n} A001157(k)* A001157(n+1-k).
G.f.: (1/x)*(Sum_{k>=1} k^2*x^k/(1 - x^k))^2. - Ilya Gutkovskiy, Jan 01 2017
Conjecture: Sum_{k=1..n} a(k) ~ zeta(3)^2 * n^6 / 180. - Vaclav Kotesovec, Aug 20 2025

Definition slightly rephrased by R. J. Mathar, Aug 19 2010

A361063 Multiplicative with a(p^e) = sigma_2(e), where sigma_2 = A001157.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 10, 5, 1, 1, 5, 1, 1, 1, 21, 1, 5, 1, 5, 1, 1, 1, 10, 5, 1, 10, 5, 1, 1, 1, 26, 1, 1, 1, 25, 1, 1, 1, 10, 1, 1, 1, 5, 5, 1, 1, 21, 5, 5, 1, 5, 1, 10, 1, 10, 1, 1, 1, 5, 1, 1, 5, 50, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 5, 5, 1, 1, 1, 21, 21, 1, 1, 5

Offset: 1

Vaclav Kotesovec, Mar 01 2023

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

Cf. A001157, A049419, A361012, A361064.

Cf. A327837, A361013.

g[p_, e_] := DivisorSigma[2, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> sigma(x, 2), factor(n)[, 2])); \\ Amiram Eldar, Jan 07 2025

from math import prod
from sympy import factorint, divisor_sigma
def A361063(n): return prod(divisor_sigma(e,2) for e in factorint(n).values()) # Chai Wah Wu, Mar 01 2023

Dirichlet g.f.: Product_{primes p} (1 + Sum_{e>=1} sigma_2(e) / p^(e*s)).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma_2(e) - sigma_2(e-1)) / p^e) = 11.343154585178523783556367128387762286267199879648613456124659589127638983...

A364268 a(n) = Sum_{k=1..n} k^2*sigma_2(k), where sigma_2 is A001157.

Original entry on oeis.org

1, 21, 111, 447, 1097, 2897, 5347, 10787, 18158, 31158, 45920, 76160, 104890, 153890, 212390, 299686, 383496, 530916, 661598, 879998, 1100498, 1395738, 1676108, 2165708, 2572583, 3147183, 3744963, 4568163, 5276285, 6446285, 7370767, 8768527, 10097107

Offset: 1

Seiichi Manyama, Oct 20 2023

Wikipedia, Faulhaber's formula.

Cf. A356125, A364269.

Cf. A000330, A001157.

Accumulate[Table[n^2*DivisorSigma[2, n], {n, 1, 33}]] (* Amiram Eldar, Oct 20 2023 *)

f(n, m) = (subst(bernpol(m+1, x), x, n+1)-subst(bernpol(m+1, x), x, 0))/(m+1);
a(n, s=2, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));

def A364268(n): return sum(k**4*(m:=n//k)*(m+1)*((m<<1)+1)//6 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023

from math import isqrt
def A364268(n): return (((s:=isqrt(n))*(s+1)*(2*s+1))**2*(1-3*s*(s+1))//6 + sum((q:=n//k)*(q+1)*(2*q+1)*k**2*(5*k**2+3*q*(q+1)-1) for k in range(1,s+1)))//30 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^4 * A000330(floor(n/k)).

a(n) ~ (zeta(3)/5) * n^5. - Amiram Eldar, Oct 20 2023

A066293 a(n) = A000203(n)^2 - A001157(n) = sigma(n)^2 - sigma_2(n).

Original entry on oeis.org

0, 4, 6, 28, 10, 94, 14, 140, 78, 194, 22, 574, 26, 326, 316, 620, 34, 1066, 38, 1218, 524, 686, 46, 2750, 310, 914, 780, 2086, 58, 3884, 62, 2604, 1084, 1466, 1004, 6370, 74, 1790, 1436, 5890, 82, 6716, 86, 4494, 3718, 2534, 94, 11966, 798, 5394, 2284

Offset: 1

Labos Elemer, Dec 12 2001

nonn

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

Cf. A000203, A001157, A002117, A072861.

a[n_] := DivisorSigma[1, n]^2 - DivisorSigma[2, n]; Array[a, 50] (* Amiram Eldar, Jul 31 2019 *)

a(n) = sigma(n)^2 - sigma(n, 2); \\ Michel Marcus, Mar 22 2020

For p prime, a(p) = 2p.
From Amiram Eldar, Mar 17 2024: (Start)
a(n) = A072861(n) - A001157(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/2 = 0.601028451579... . (End)

A066294 a(n) = A000203(n)^2 - A001157(n) - 2n = sigma(n)^2 - sigma_2(n) - 2n.

Original entry on oeis.org

-2, 0, 0, 20, 0, 82, 0, 124, 60, 174, 0, 550, 0, 298, 286, 588, 0, 1030, 0, 1178, 482, 642, 0, 2702, 260, 862, 726, 2030, 0, 3824, 0, 2540, 1018, 1398, 934, 6298, 0, 1714, 1358, 5810, 0, 6632, 0, 4406, 3628, 2442, 0, 11870, 700, 5294, 2182, 5930, 0, 10192, 1902, 10038, 2666, 3774, 0, 22644, 0, 4282, 6140, 10540, 2506

Offset: 1

Labos Elemer, Dec 12 2001

sign

For primes p, a(p) = 0, otherwise positive, except for n = 1 where it is negative.

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

Cf. A000203, A001157.

a[n_] := DivisorSigma[1,n]^2 - DivisorSigma[2,n] - 2n; Array[a, 65] (* Amiram Eldar, Jul 31 2019 *)

a(n) = sigma(n)^2 - sigma(n, 2) - 2*n; \\ Michel Marcus, Mar 22 2020

cp's OEIS Frontend

A066088 Number of distinct prime factors of sigma_2(n), where sigma_2(n) = A001157(n).

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A081380 Numbers k such that the sets of prime factors (ignoring multiplicity) of A000203(k) = sigma(k) and of A001157(k) = sigma_2(k) are identical.

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A082902 a(n) = gcd(2^n, sigma(2,n)) = gcd(A000079(n), A001157(n)).

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A099978 Bisection of A001157: a(n) = sigma_2(2n-1).

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A175199 a(n) is the smallest integer k such that sigma_2(k) = sigma_2(k + 2n), where sigma_2(k) is the sum of squares of divisors of k (A001157).

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A175705 Convolution square of A001157 (the sum of squared divisors).

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A361063 Multiplicative with a(p^e) = sigma_2(e), where sigma_2 = A001157.

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