A001157 -id:A001157 - cp's OEIS frontend

A179930 a(n) = gcd(n, A001157(n)).

1, 1, 1, 1, 1, 2, 1, 1, 1, 10, 1, 6, 1, 2, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 14, 1, 10, 1, 1, 1, 2, 5, 3, 1, 2, 1, 10, 1, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 60, 1, 2, 7, 1, 65, 2, 1, 2, 1, 10, 1, 1, 1, 2, 15, 2, 1, 2, 1, 2, 1, 2, 1, 84, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 1, 6

Offset: 1

N. J. A. Sloane, Jul 09 2011, following a suggestion from R. J. Mathar

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001157, A082063, A082069, A179931, A009194.

A179930 := proc(n) igcd( n, numtheory[sigma][2](n)) ; end proc:
seq(A179930(n),n=1..100) ;

Array[GCD[#, DivisorSigma[2, #]] &, 96] (* Michael De Vlieger, Nov 03 2017 *)

A179930(n) = gcd(n,sigma(n,2)); \\ Antti Karttunen, Nov 03 2017

A006530(a(n)) = A082063(n). - Reinhard Zumkeller, Jul 10 2011

A020639(a(n)) = A082069(n). - Antti Karttunen, Nov 03 2017

A318250 a(n) = (n - 1)! * sigma_2(n), where sigma_2(n) = sum of squares of divisors of n (A001157).

Original entry on oeis.org

1, 5, 20, 126, 624, 6000, 36000, 428400, 3669120, 47174400, 442713600, 8382528000, 81430272000, 1556755200000, 22666355712000, 445916959488000, 6067609067520000, 161837779783680000, 2317659281473536000, 66418224823222272000, 1216451004088320000000, 31165474724742758400000

Offset: 1

Ilya Gutkovskiy, Aug 22 2018

nonn

Cf. A000219, A001157, A038048, A318249.

Table[(n - 1)! DivisorSigma[2, n], {n, 1, 22}]
nmax = 22; Rest[CoefficientList[Series[Sum[x^k/(k (1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 22; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]

a(n) = (n-1)!*sigma(n,2); \\ Michel Marcus, Aug 22 2018

E.g.f.: Sum_{k>=1} x^k/(k*(1 - x^k)^2).
E.g.f.: -log(Product_{k>=1} (1 - x^k)^k).
E.g.f.: A(x) = log(B(x)), where B(x) = o.g.f. of A000219.
a(p^k) = (p^(2*k+2) - 1)*(p^k - 1)!/(p^2 - 1), where p is a prime.

A330495 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * sigma_2(k), where sigma_2 = A001157.

Original entry on oeis.org

1, 6, 37, 307, 2858, 32060, 405830, 5777354, 91400200, 1593023040, 30251766840, 622016655816, 13777150847952, 327040289212320, 8280040187137200, 222696435041359824, 6341359225470493440, 190609840724078576256, 6031297367477133540480, 200389374367707186619776

Offset: 1

Vaclav Kotesovec, Dec 16 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Cf. A330449, A330450, A330493, A330494.

Table[Sum[(-1)^(n-k) * StirlingS1[n, k] * (k-1)! * DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1/(1 - x)]^k/(k*(1 - Log[1/(1 - x)]^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]

a(n) = sum(k=1, n, (-1)^(n-k)*stirling(n, k, 1)*(k-1)!*sigma(k, 2)); \\ Michel Marcus, Dec 16 2019

E.g.f.: Sum_{k>=1} log(1/(1 - x))^k / (k * (1 - log(1/(1 - x))^k)^2).

a(n) ~ n! * zeta(3) * n * exp(n) / (exp(1) - 1)^(n+2).

A335763 Decimal expansion of Sum_{k>=1} sigma_2(k)/2^k where sigma_2(k) is the sum of squares of divisors of k (A001157).

Original entry on oeis.org

7, 0, 9, 9, 2, 8, 5, 1, 7, 8, 8, 9, 0, 9, 0, 7, 1, 1, 4, 0, 3, 3, 1, 2, 5, 0, 2, 2, 1, 6, 4, 7, 5, 3, 6, 6, 3, 1, 5, 7, 6, 0, 8, 8, 3, 3, 2, 1, 1, 8, 9, 5, 9, 7, 8, 8, 3, 9, 2, 3, 7, 7, 4, 2, 8, 8, 9, 1, 2, 8, 8, 9, 1, 1, 2, 2, 6, 4, 5, 8, 7, 1, 7, 3, 5, 5, 4

Offset: 1

Amiram Eldar, Jun 21 2020

			7.099285178890907114033125022164753663157608833211895...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Lambert Series.
Wikipedia, Lambert series.

Cf. A001157, A065442, A066766, A227989, A256936, A335764.

evalf(add( (1/2)^(n^2)*( n^2*(8^n) - ((n-1)^2 - 2)*4^n - ((n+1)^2 - 2)*(2^n) + n^2 )/(2^n - 1)^3, n = 1..20 ), 100); # Peter Bala, Jan 22 2021

RealDigits[Sum[n^2/(2^n - 1), {n, 1, 500}], 10, 100][[1]]

Equals Sum_{k>=1} k^2/(2^k - 1).

Faster converging series: Sum_{n >= 1} (1/2)^(n^2)*( n^2*(8^n) - ((n-1)^2 - 2)*4^n - ((n+1)^2 - 2)*(2^n) + n^2 )/(2^n - 1)^3. - Peter Bala, Jan 19 2021

A080401 Numbers k such that the sum of the squares of the divisors of k (A001157(k)) is squarefree.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 22, 23, 25, 29, 31, 32, 37, 38, 40, 44, 47, 48, 49, 50, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 88, 89, 92, 97, 98, 99, 101, 103, 109, 113, 116, 117, 118, 121, 122, 124, 127, 128, 131, 137

Offset: 1

Labos Elemer, Mar 19 2003

nonn

If m*k is in the sequence with m and k coprime, then m and k must be in the sequence. - Robert Israel, Mar 29 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A001157, A005117, A065300, A080402 (complement).

select(n -> numtheory:-issqrfree(numtheory:-sigma[2](n)), [$1..1000]); # Robert Israel, Mar 29 2019

Do[s=MoebiusMu[DivisorSigma[2, n]]; If[ !Equal[s, 0], Print[n]], {n, 1, 1000}]
Select[Range[200],SquareFreeQ[DivisorSigma[2,#]]&] (* Harvey P. Dale, Jun 17 2014 *)

isok(n) = issquarefree(sigma(n, 2)); \\ Michel Marcus, Mar 29 2019

abs(mu(sigma_2(a(n)))) = 1.

A080402 Numbers k such that the sum of the squares of the divisors of k (A001157(k)) is not squarefree.

Original entry on oeis.org

6, 7, 14, 15, 21, 24, 26, 27, 28, 30, 33, 34, 35, 36, 39, 41, 42, 43, 45, 46, 51, 54, 55, 56, 57, 60, 63, 65, 66, 69, 70, 74, 77, 78, 81, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 100, 102, 104, 105, 106, 107, 108, 110, 111, 112, 114, 115, 119, 120, 123, 125, 126, 129

Offset: 1

Labos Elemer, Mar 19 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A001157, A005117, A013929, A065300, A080401 (complement).

Do[s=MoebiusMu[DivisorSigma[2, n]]; If[Equal[s, 0], Print[n]], {n, 1, 1000}]
Select[Range[130],!SquareFreeQ[DivisorSigma[2,#]]&] (* Harvey P. Dale, Oct 10 2011 *)

is(k) = !issquarefree(sigma(k, 2)); \\ Amiram Eldar, Aug 12 2024

mu(sigma_2(a(n))) = 0.

A099979 Bisection of A001157: sigma_2(2n).

Original entry on oeis.org

5, 21, 50, 85, 130, 210, 250, 341, 455, 546, 610, 850, 850, 1050, 1300, 1365, 1450, 1911, 1810, 2210, 2500, 2562, 2650, 3410, 3255, 3570, 4100, 4250, 4210, 5460, 4810, 5461, 6100, 6090, 6500, 7735, 6850, 7602, 8500, 8866, 8410, 10500, 9250, 10370

Offset: 1

N. J. A. Sloane, Nov 19 2004

			a(1) = sigma_2(2*1) = 1 + 2^2 = 5.
a(2) = sigma_2(2*2) = 1 + 2^2 + 4^2 = 21.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma_2(k*n): A001157 (k=1), this sequence (k=2), A283237 (k=3).

with(numtheory): seq(sigma[2](2*n),n=1..50); # C. Ronaldo

Table[DivisorSigma[2,2n],{n,1,47}] (* Indranil Ghosh, Mar 03 2017 *)

a(n) = sigma(2*n,2); \\ Indranil Ghosh, Mar 03 2017

a(n) = A001157(2*n) = sigma_2(2*n).

Sum_{k=1..n} a(k) ~ 3 * zeta(3) * n^3 / 2. - Vaclav Kotesovec, Aug 07 2022

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 18 2005
Offset changed to 1 by Seiichi Manyama, Mar 03 2017
Examples added and name edited by M. F. Hasler, Mar 06 2017

A131905 Integers x such that sigma_2(k)=sigma_2(x) for some 0A001157=sigma_2 is the sum of squares of divisors.

Original entry on oeis.org

7, 26, 35, 47, 77, 91, 119, 130, 133, 141, 157, 161, 175, 182, 203, 215, 217, 249, 259, 282, 286, 287, 301, 329, 371, 385, 413, 423, 427, 434, 442, 455, 469, 471, 494, 497, 511, 517, 553, 581, 595, 598, 611, 623, 650, 651, 665, 679, 707, 721, 749, 754, 763, 785

Offset: 1

Peter Pein (petsie(AT)dordos.net), Jul 26 2007

			This sequence contains 35, because sigma_2(35) = 1^2+5^2+7^2+35^2 = 1+25+49+1225 = 1300, and the sum of the squares of the divisors of 30<35 is sigma_2(30) = 1+4+9+25+36+100+225+900 = 1300.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A069822, A131902-A131908.

N:= 100: # to get a(1)..a(N)
count:= 0: Res:= NULL:
for n from 1 while count < N do
  v:= numtheory:-sigma[2](n);
  if assigned(V[v]) then count:= count+1; Res:= Res, n;
  else V[v]:= n
  fi
od:
Res; # Robert Israel, Mar 30 2018

Clear[tmp]; First@Transpose[ Function[n, (If[Head[ #1] === tmp, #1 = n; Unevaluated[Sequence[]], {n, #1}] & )[tmp[DivisorSigma[2, n]]]] /@ Range[500]]
Module[{nn=800,ds2,c},ds2=DivisorSigma[2,Range[nn]];Table[c=TakeDrop[Take[ds2,n],-1];If[ MemberQ[c[[2]],c[[1,1]]],n,Nothing],{n,nn}]] (* Harvey P. Dale, May 22 2024 *)

isok(n) = {sn = sigma(n,2); for (k=1, n-1, if (sigma(k,2) == sn, return (1)););} \\ Michel Marcus, Apr 03 2015

a(n) = n-th element of {x: there exists some k with 0A001157=sigma_2 is the sum of squares of divisors.

a(37)-a(54) from Michel Marcus, Apr 03 2015

Edited by Danny Rorabaugh, Apr 03 2015

A169635 Integers m such that sigma_2(m) = sigma_2(m + 2) where sigma_2(m) is the sum of squares of divisors of m (A001157).

Original entry on oeis.org

24, 215, 280, 1079, 947519, 1362239, 2230271, 14939999, 19720007, 32509439, 45581759, 45841247, 49436927, 78436511, 82842911, 101014631, 166828031, 225622151, 225757799, 250999559, 377129087, 554998751, 619606439, 846765431, 1204092287, 1302170687, 1710035711

Offset: 1

Michel Lagneau, Apr 04 2010

nonn

The equation sigma_2(m) = sigma_2(m + k) has infinitely many solutions where k >= 2 and k is even (J.-M. De Koninck).
From Amiram Eldar, Apr 19 2024: (Start)
De Koninck's proof is based on the assumption of Schinzel's hypothesis H. If q, r = q + 2, s = q^2 + q + 1, and p = q^2 + 3*q + 3 are all primes, then p*q is a term (the values of q+1 are the terms of A268043).
The equation sigma_2(m) = sigma_2(m + 1) has only one solution: m = 6. (End)

			For m=24, sigma_2(24) = sigma_2(26) = 850.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 118, entry 1079.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B13, pp. 103-104.

Amiram Eldar, Table of n, a(n) for n = 1..156
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].
Jean-Marie De Koninck, On the solutions of sigma_2(n) = sigma_2(n + l), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.
Wikipedia, Schinzel's hypothesis H.

Cf. A001157, A002961, A007373, A175199, A268043.

with(numtheory):for n from 1 to 500000000 do:liste:= divisors(n) : s2 :=sum(liste[i]^2, i=1..nops(liste)):liste:=divisors(n+2):s3:=sum(liste[i]^2, i=1..nops(liste)):if s2 = s3 then print(n):else fi:od:

Select[Range[10^9], DivisorSigma[2,#] == DivisorSigma[2,#+2]&]

is(n) = sigma(n, 2) == sigma(n + 2, 2); \\ Amiram Eldar, Apr 19 2024

lista(mmax) = {my(s1 = sigma(1, 2), s2 = sigma(2, 2), s3, s4); forstep(m = 3, mmax, 2, s3 = sigma(m, 2); s4 = sigma(m+1, 2); if(s1 == s3, print1(m - 2, ", ")); if(s2 == s4, print1(m - 1, ", ")); s1 = s3; s2 = s4);} \\ Amiram Eldar, Apr 19 2024

a(25)-a(27) from Donovan Johnson, Apr 14 2013

A364269 a(n) = Sum_{k=1..n} k^3*sigma_2(k), where sigma_2 is A001157.

Original entry on oeis.org

1, 41, 311, 1655, 4905, 15705, 32855, 76375, 142714, 272714, 435096, 797976, 1171466, 1857466, 2734966, 4131702, 5556472, 8210032, 10692990, 15060990, 19691490, 26186770, 32635280, 44385680, 54557555, 69497155, 85637215, 108686815, 129222353, 164322353

Offset: 1

Seiichi Manyama, Oct 20 2023

Wikipedia, Faulhaber's formula.

Cf. A356125, A364268.

Cf. A000537, A001157.

Accumulate[Table[n^3*DivisorSigma[2, n], {n, 1, 30}]] (* 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=3, t=2) = sum(k=1, n, k^(s+t)*f(n\k, s));

def A364269(n): return sum(k*(k**2*(m:=n//k)*(m+1)>>1)**2 for k in range(1,n+1)) # Chai Wah Wu, Oct 20 2023

from math import isqrt
def A364269(n): return ((((s:=isqrt(n))*(s+1))**4*(1-s*(s+1<<1))>>2) + sum(((q:=n//k)*(q+1))**2*k**3*(3*k**2+(q*(q+1<<1)-1)) for k in range(1,s+1)))//12 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} k^5 * A000537(floor(n/k)).

a(n) ~ (zeta(3)/6) * n^6. - Amiram Eldar, Oct 20 2023

cp's OEIS Frontend

A179930 a(n) = gcd(n, A001157(n)).

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A318250 a(n) = (n - 1)! * sigma_2(n), where sigma_2(n) = sum of squares of divisors of n (A001157).

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A330495 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * sigma_2(k), where sigma_2 = A001157.

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A335763 Decimal expansion of Sum_{k>=1} sigma_2(k)/2^k where sigma_2(k) is the sum of squares of divisors of k (A001157).

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A080401 Numbers k such that the sum of the squares of the divisors of k (A001157(k)) is squarefree.

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A080402 Numbers k such that the sum of the squares of the divisors of k (A001157(k)) is not squarefree.

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A099979 Bisection of A001157: sigma_2(2n).

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A131905 Integers x such that sigma_2(k)=sigma_2(x) for some 0A001157=sigma_2 is the sum of squares of divisors.

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