A001159 -id:A001159 - cp's OEIS frontend

A347175 Sum of 4th powers of odd divisors of n that are <= sqrt(n).

1, 1, 1, 1, 1, 1, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 1, 1, 82, 626, 1, 82, 1, 1, 707, 1, 1, 82, 1, 626, 82, 1, 1, 82, 626, 1, 82, 1, 1, 707, 1, 1, 82, 2402, 626, 82, 1, 1, 82, 626, 2402, 82, 1, 1, 707, 1, 1, 2483, 1, 626, 82, 1, 1, 82, 3027, 1, 82, 1, 1, 707

Offset: 1

Ilya Gutkovskiy, Aug 21 2021

			a(18) = 82 as the odd divisors of 18 are the divisors of 9 which are 1, 3 and 9. Of those, 1 and 3 are <= sqrt(18) so we find the sum of fourth powers of 1 and 3 then add them i.e., a(18) = 1^4 + 3^4 = 82. - _David A. Corneth_, Feb 24 2024

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001159, A051001, A069288, A069289, A347143, A347172, A347173, A347174.

Table[DivisorSum[n, #^4 &, # <= Sqrt[n] && OddQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[(2 k - 1)^4 x^((2 k - 1)^2)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = {
	my(s = sqrtint(n), res);
	n>>=valuation(n, 2);
	d = divisors(n);
	for(i = 1, #d,
		if(d[i] <= s,
			res += d[i]^4
		,
			return(res)
		)
	); res
} \\ David A. Corneth, Feb 24 2024

G.f.: Sum_{k>=1} (2*k - 1)^4 * x^((2*k - 1)^2) / (1 - x^(2*k - 1)).

A046870 Numbers k such that sigma_1(k) divides sigma_4(k).

Original entry on oeis.org

1, 4, 9, 16, 20, 25, 36, 49, 50, 64, 81, 100, 117, 121, 144, 169, 180, 196, 225, 242, 256, 289, 324, 325, 361, 400, 441, 450, 468, 484, 500, 529, 576, 578, 605, 625, 650, 676, 729, 784, 800, 841, 900, 961, 980, 1024, 1025, 1058, 1089, 1156, 1225, 1280, 1296

Offset: 1

Labos Elemer

nonn

sigma_1(n) is the sum of the divisors of n [same as sigma(n)] (A000203).
sigma_2(n) is the sum of the squares of the divisors of n (A001157).
sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

			k = a(18) = 196 of which divisor power sums for k=0,1,2,3,4 are 9,399,51471, 8613489, 1574446419. sigma_1(k) = 399 and sigma_4(k) = 51471*30589=399*129*30589. Thus both sigma_2(k) and sigma_1(k) divide sigma_4(k).

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

Cf. A000203, A001157, A001159.

Has large overlap with A020487.

Select[Range[1300], Divisible @@ DivisorSigma[{4, 1}, #] &] (* Amiram Eldar, Jun 15 2024 *)

is(k) = {my(f = factor(k)); !(sigma(f, 4) % sigma(f)); } \\ Amiram Eldar, Jun 15 2024

A050448 a(n) = Sum_{d|n, d==1 (mod 4)} d^4.

Original entry on oeis.org

1, 1, 1, 1, 626, 1, 1, 1, 6562, 626, 1, 1, 28562, 1, 626, 1, 83522, 6562, 1, 626, 194482, 1, 1, 1, 391251, 28562, 6562, 1, 707282, 626, 1, 1, 1185922, 83522, 626, 6562, 1874162, 1, 28562, 626, 2825762, 194482, 1, 1, 4107812, 1, 1, 1

Offset: 1

N. J. A. Sloane, Dec 23 1999

nonn

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

Cf. A001159, A050449, A050450, A050451.

a[n_] := DivisorSum[n, #^4 &, Mod[#, 4] == 1 &]; Array[a, 50] (* Amiram Eldar, Jul 08 2023 *)

a(n) = sumdiv(n, d, if ((d%4)==1, d^4)); \\ Michel Marcus, Aug 16 2021

Offset corrected by Sean A. Irvine, Aug 15 2021

A066102 Number of distinct prime factors of sigma_4(n), the sum of the 4th powers of divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Dec 04 2001

nonn

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

Cf. A001159, A001221.

Table[PrimeNu[DivisorSigma[4, n]], {n, 1, 50}] (* G. C. Greubel, May 08 2017 *)

a(n) = { omega(sigma(n, 4)) } \\ Harry J. Smith, Nov 14 2009

a(n) = omega(sigma_4(n)) = A001221(A001159(n)).

A068021 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=4.

Original entry on oeis.org

1, 31, 121, 651, 781, 4333, 2801, 11811, 11011, 26481, 16105, 96957, 30941, 92613, 100771, 200787, 88741, 412087, 137561, 579201, 354923, 520221, 292561, 1812477, 508431, 993153, 925771, 2003477, 732541, 3996003, 954305, 3309747, 2006851

Offset: 1

Vladeta Jovovic, Feb 08 2002

nonn

Cf. A067692, A068020, A068022-A068027, A000203, A001157, A001158, A001159.

CIP4 = CycleIndexPolynomial[SymmetricGroup[4], Array[x, 4]]; a[n_] := CIP4 /. x[k_] -> DivisorSigma[k, n]; Array[a, 33] (* Jean-François Alcover, Nov 04 2016 *)

1/4!*(sigma[1](n)^4 + 6*sigma[1](n)^2*sigma[2](n) + 8*sigma[1](n)*sigma[3](n) + 3*sigma[2](n)^2 + 6*sigma[4](n)).

A076230 Numbers k such that sigma_2(k)/k and sigma_4(k)/k are integers.

Original entry on oeis.org

1, 84, 435708, 986076, 1441188, 6066396, 7407036, 16763292, 18735444, 78863148, 260855028, 318502548, 1340673516, 3391115364, 10829086632, 45076171140, 45582899544, 57648961188, 77314641768, 128293717860, 149098104540, 174975241896, 440359518060, 461191689504

Offset: 1

Labos Elemer, Oct 03 2002

nonn

			For k = 6066396, sigma_2(k)/k = 9156979, sigma_4(k)/k = 241153415598179286943.

Giovanni Resta, Table of n, a(n) for n = 1..41 (terms < 5*10^12)

Intersection of A046762 and A046764.

Cf. A001157, A001159.

[n: n in [1..10000000]|IsIntegral(DivisorSigma(2,n)/n) and IsIntegral(DivisorSigma(4,n)/n)]; // Marius A. Burtea, Aug 28 2019

Select[Range[10^7], And @@ Divisible[DivisorSigma[{2, 4}, #], #] &] (* Amiram Eldar, Jan 20 2020 *)

isok(n) = !(sigma(n, 2) % n) && !(sigma(n, 4) % n); \\ Michel Marcus, Aug 28 2019

More terms from T. D. Noe, Apr 11 2006
a(11)-a(12) from Michel Marcus, Aug 28 2019
a(13)-a(18) from Amiram Eldar, Jan 20 2020
a(19)-a(24) from Giovanni Resta, Jan 21 2020

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

A229326 Total sum of 4th powers of parts in all partitions of n.

Original entry on oeis.org

0, 1, 18, 101, 392, 1119, 2904, 6407, 13578, 26218, 49218, 86782, 150860, 249723, 408810, 647170, 1013278, 1545029, 2337738, 3460218, 5086658, 7350874, 10549872, 14929931, 21009874, 29205500, 40385036, 55289000, 75309056, 101692923, 136710130, 182377824

Offset: 0

Alois P. Heinz, Sep 20 2013

nonn

The bivariate g.f. for the partition statistic "sum of 4th powers of the parts" is G(t,x) = 1/Product_{k>=1}(1 - t^{k^4}*x^k). The g.f. g at the Formula section has been obtained by evaluating dG/dt at t=1. - Emeric Deutsch, Dec 06 2015

Convolution of A001159 and A000041. - Vaclav Kotesovec, May 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO], 2008.

Column k=4 of A213191.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], `if`(i>n, b(n, i-1),
      ((g, h)-> g+h+[0, h[1]*i^4])(b(n, i-1), b(n-i, i)))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..40);
# second Maple program:
g := (sum(k^4*x^k/(1-x^k), k = 1..100))/(product(1-x^k, k = 1..100)): gser := series(g, x = 0, 45): seq(coeff(gser, x, m), m = 1 .. 40); # Emeric Deutsch, Dec 06 2015

(* T = A066633 *) T[n_, n_] = 1; T[n_, k_] /; k < n := T[n, k] = T[n-k, k] + PartitionsP[n-k]; T[, ] = 0; a[n_] := Sum[T[n, k]*k^4, {k, 1, n}]; Array[a, 32, 0] (* Jean-François Alcover, Dec 15 2016 *)
Table[Sum[DivisorSigma[4, k]*PartitionsP[n-k], {k, 1, n}], {n, 0, 40}] (* Vaclav Kotesovec, May 27 2018 *)

a(n) = Sum_{k=1..n} A066633(n,k) * k^4.
G.f.: g(x) = (Sum_{k>=1} k^4*x^k/(1-x^k))/Product_{q>=1} (1-x^q). - Emeric Deutsch, Dec 06 2015
a(n) ~ 216*sqrt(2)*Zeta(5)/Pi^5 * exp(Pi*sqrt(2*n/3)) * n^(3/2). - Vaclav Kotesovec, May 28 2018

A236328 a(n) = sigma(n,1) + sigma(n,2) + ... + sigma(n,n).

Original entry on oeis.org

1, 8, 42, 374, 3910, 57210, 960806, 19261858, 435877581, 11123320196, 313842837682, 9729290348244, 328114698808286, 11967567841654606, 469172063576559644, 19676848703371278522, 878942778254232811954, 41661071646298278566886, 2088331858752553232964218

Offset: 1

Colin Barker, Jan 22 2014

Sigma(n,k) is the sum of the k-th powers of the divisors of n.
The sequence seems to be strictly increasing. - Chayim Lowen, Aug 05 2015.
This is true. Moreover, subsequent ratios a(n+1)/a(n) steadily grow for n>3. The difference of subsequent ratios tends to the limit e = 2.718... The reason is that a(n) roughly behaves like n^n; already the second largest term in the sum is smaller by a factor 2^n (for even n) or by a factor 3^n (for n=6k+3) etc. - M. F. Hasler, Aug 16 2015

			a(4) = sigma(4,1) + sigma(4,2) + sigma(4,3) + sigma(4,4) = 7 + 21 + 73 + 273 = 374.

Robert Israel, Table of n, a(n) for n = 1..353

Cf. A000203, A001157, A001158, A001159, A001160, A109974.
Cf. A000005, A065805.
Cf. A236329.

seq(add(numtheory:-sigma[k](n), k=1..n), n=1..50); # Robert Israel, Aug 04 2015

Table[Sum[DivisorSigma[i, n], {i, n}], {n, 19}] (* Michael De Vlieger, Aug 06 2015 *)
f[n_] := Sum[DivisorSigma[i, n], {i, n}]; (* OR *) f[n_] := Block[{d = Rest@Divisors@n}, n + Total[(d^(n + 1) - d)/(d - 1)]]; (* then *) Array[f, 19] (* Robert G. Wilson v, Aug 06 2015 *)

vector(30, n, sum(k=1, n, sigma(n, k)))

vector(30, n, n + sumdiv(n, d, if (d>1,(d^(n+1)-d)/(d-1)))) \\ Michel Marcus, Aug 04 2015

a(n) = n + Sum_{d|n, d>1} d*(d^n-1)/(d-1). - Chayim Lowen, Aug 02 2015
a(n) >= n*(n^n+n-2)/(n-1) for n>1. - Chayim Lowen, Aug 05 2015
a(n) = A065805(n)-A000005(n). - Chayim Lowen, Aug 11 2015
a(n) ~ n^n. - Vaclav Kotesovec, Aug 04 2025

A252040 Least number k such that the sum of the n-th powers of divisors of k is prime, or 0 if there is no such k.

Original entry on oeis.org

2, 2, 4, 2, 3418801, 0, 64, 2, 4, 0, 21611482313284249, 0, 38580382095460899155325996786287338133521, 0, 0, 2, 2908327322588596409721563830760333292497745890881, 0, 366742604046618684582110328334808145525757868881, 0, 0, 0, 264638735999480827156185738343112840094453729311824181089

Offset: 1

Derek Orr, Dec 12 2014

nonn

a(n) = 2 if and only if 2^n + 1 is in A019434.
From Jinyuan Wang, Jan 30 2021: (Start)
a(n) = 0 if n > 1 is not a prime power. Proof: note that sigma_n(k) = Product_{i=1..m} (1 + p_i^n + ... + p_i^(n*e_i)), where k = Product_{i=1..m} p_i^e_i. We only need to prove when n > 1 is not a prime power and e > 1, s = Sum_{i=0..e-1} p^(n*i) = (p^(n*e) - 1)/(p^n - 1) is composite. If e is prime, then s is divisible by (p^(e^(t+1)) - 1)/(p^(e^t) - 1), where t is the e-adic valuation of n. If e is composite, then s is divisible by (p^(n*q) - 1)/(p^n - 1), where q is a prime factor of e.
Corollary: k must be of the form p^(e - 1) when n = e^t, where p and e are primes. Therefore, a(2^t) = 0 if 2^2^t + 1 is composite. (End)

			2 has two divisors, 2 and 1. 2^3 + 1^3 = 9 is not prime.
3 has two divisors, 3 and 1. 3^3 + 1^3 = 28 is not prime.
4 has three divisors, 4, 2, and 1. 4^3 + 2^3 + 1^3 = 73 is prime. So, a(3) = 4.

Jinyuan Wang, Table of n, a(n) for n = 1..50

Cf. A000203, A001157, A001158, A001159, A001160, A019434.

Cf. A023194 (sigma(n) is prime), A063783 (sigma_3(n) is prime).

a252040[n_Integer] := If[PrimePowerQ[n] && (p=First@ First@ FactorInteger[n])>2, q=2; While[!PrimeQ[DivisorSigma[n, q^(p-1)]], q=NextPrime[q]]; q^(p-1), 2*Boole[PrimeQ[2^n+1]]]; a252040 /@ Range[10] (* Michael De Vlieger, Dec 13 2014 *) (* modified by Jinyuan Wang, Jan 30 2021 *)

a(n) = if(isprimepower(n, &p) && p>2, my(q=2); while(!ispseudoprime(sigma(q^(p-1), n)), q=nextprime(q+1)); q^(p-1), 2*isprime(2^n+1)); \\ Modified by Jinyuan Wang, Jan 25 2021

Name edited by and more terms from Jinyuan Wang, Jan 30 2021

cp's OEIS Frontend

A347175 Sum of 4th powers of odd divisors of n that are <= sqrt(n).

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A046870 Numbers k such that sigma_1(k) divides sigma_4(k).

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A050448 a(n) = Sum_{d|n, d==1 (mod 4)} d^4.

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A066102 Number of distinct prime factors of sigma_4(n), the sum of the 4th powers of divisors of n.

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A068021 Z(S_m; sigma[1](n), sigma[2](n),..., sigma[m](n)) where Z(S_m; x_1,x_2,...,x_m) is the cycle index of the symmetric group S_m and sigma[k](n) is the sum of k-th powers of divisors of n; m=4.

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A076230 Numbers k such that sigma_2(k)/k and sigma_4(k)/k are integers.

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Magma

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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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A229326 Total sum of 4th powers of parts in all partitions of n.

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