A048257 -id:A048257 - cp's OEIS frontend

A019423 Numbers whose sum of divisors is a fifth power.

1, 21, 31, 651, 889, 3210, 3498, 3710, 3882, 3910, 4310, 4922, 4982, 5182, 5457, 5885, 6035, 6095, 6307, 6797, 7117, 7327, 7597, 24573, 27559, 71193, 82110, 90510, 94981, 97410, 98671, 99301, 99510, 100110, 103362, 104622, 107778, 108438, 108822

Offset: 1

David W. Wilson

nonn

Marius A. Burtea, Table of n, a(n) for n = 1..3648 (terms 1..1000 from Donovan Johnson; a(1211) re-indexed and duplicate a(2311) removed by _Georg Fischer_, Mar 21 2022).
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A000203, A006532, A019422, A019424, A020477, A048257, A048258.

[n:n in [1..10000]| IsPower(SumOfDivisors(n),5)]; // Marius A. Burtea, Apr 17 2019

lista(nn) = {for (i=1, nn, s = sigma(i); if (s == 1 || ispower(s, 5), print1(i, ", ")););} \\ Michel Marcus, Jun 12 2013

A019424 Numbers whose sum of divisors is a sixth power.

Original entry on oeis.org

1, 2667, 3937, 17490, 19410, 22578, 24610, 24910, 25466, 25910, 26554, 26818, 27285, 29342, 29733, 29762, 31102, 31535, 32043, 32997, 33985, 35585, 36635, 37985, 39697, 41393, 41837, 42347, 44047, 45637, 45739, 45937, 46117, 172011, 253921, 640737

Offset: 1

David W. Wilson

nonn

			sigma(2667) = 1+3+7+21+127+381+889+2667 = 4096 = 4^6.
sigma(3937) = 1+31+127+3937 = 4096 = 4^6.

Marius A. Burtea, Table of n, a(n) for n = 1..2010 (terms 1..1000 from Donovan Johnson)
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A000203, A006532, A019422, A019423, A020477, A048257, A048258.

[n:n in [1..100000]| IsPower(SumOfDivisors(n),6)]; // Marius A. Burtea, Apr 17 2019

Select[Range[700000],IntegerQ[Surd[DivisorSigma[1,#],6]]&] (* Harvey P. Dale, Apr 19 2019 *)

c=0; for(n=1, 306455560, if(ispower(sigma(n), 6), c++; write("b019424.txt", c " " n))) /* Donovan Johnson, Jun 13 2013 */

A048258 Integers whose sum of divisors is an 8th power.

Original entry on oeis.org

1, 217, 57337, 600270, 621690, 669990, 685290, 693294, 693770, 699810, 725934, 747670, 769930, 774894, 782598, 805970, 813378, 823938, 835670, 839802, 854930, 865490, 873334, 895594, 918435, 920414, 923410, 931634, 935715, 959565, 965174, 969034, 969206

Offset: 1

Labos Elemer

nonn

			Divisors(217) = {1,7,31,217}, sum = 256 = 2^8.
Divisors(57337) = {1,7,8191,57337}, sum = 65536 = 4^8.
Divisors(1676377) = {1,647,2591,1676377}, sum = 1679616 = 6^8.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A006532, A020477, A019422, A019423, A019424, A048251-A048257.

Select[Range[10^6], IntegerQ@ Surd[DivisorSigma[1, #], 8] &] (* Michael De Vlieger, Aug 01 2017, after Harvey P. Dale at A048257 *)

isok(n) = ispower(sigma(n), 8); \\ Michel Marcus, Dec 30 2013

Sigma(1, a(n)) = x^8, where the initial values of x are 1, 2, 4, 6 (occurs 85 times), ...

A063869 Least k such that sigma(k)=m^n for some m>1.

Original entry on oeis.org

2, 3, 7, 217, 21, 2667, 93, 217, 381, 651, 2752491, 2667, 8191, 11811, 24573, 57337, 82677, 172011, 393213, 761763, 1572861, 2752491, 5332341, 11010027, 21845397, 48758691, 85327221, 199753347, 341310837, 677207307, 1398273429, 3220807683

Offset: 1

Labos Elemer, Aug 27 2001

nonn

For n=2 to 20 sigma(a(n)) = m^n with m=2 or m=4. Computed terms are products of Mersenne primes (A000668). Is this true for larger n? Validity of a(11) was tested individually.
The Nagell-Ljunggren conjecture implies that sigma(x) is never 3^n for n>1. If this is true, then m=2 and m=4 are the smallest possible solutions. When A063883(n)>0, we can take m=2 and, as explained by Brown, find k to be a product of Mersenne primes (i.e. one of the numbers in A046528). When A063883(n)=0, which is true for the n in A078426, then m=4 and we have a(n)=a(2n) because 4=2^2. - T. D. Noe, Oct 18 2006
Sierpiński says that he proved sigma(x) is never 3^r for r>1. Hence m=2 and m=4 are the smallest possible solutions. When A063883(n)>0, we can take m=2 and, as explained by Brown, find k to be a product of Mersenne primes (i.e. one of the numbers in A046528). When A063883(n)=0, which is true for the n in A078426, then m=4 and we have a(n)=a(2n) because 4=2^2. - T. D. Noe, Oct 18 2006

			For n = 11, sigma(a(n)) = sigma(2752491) = sigma(3 * 7 * 131071) = 4^11.

T. D. Noe, Table of n, a(n) for n=1..500
K. S. Brown, Sum of Divisors Equals a Power of 2
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964, page 165.

Cf. A006532, A020477, A019422, A019423, A019424, A048257, A048258, A000668, A046528.

d={2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253}; nn=3700; t=Table[Infinity, {nn}]; t[[1]]=2; u={0}; k=1; While[2+d[[k]]<=nn, mer=2^d[[k]]-1; Do[a=u[[i]]+d[[k]]; If[a<=nn, If[u[[i]]==0, t[[a]]=Min[t[[a]], mer], t[[a]]=Min[t[[a]], t[[u[[i]]]]*mer]]], {i, Length[u]}]; u=Union[u, u+d[[k]]]; k++ ]; Do[If[t[[i]]==Infinity, t[[i]]=t[[2i]]], {i, nn}]; t (* T. D. Noe, Oct 13 2006 *)
c[] = 0; c[1] = 2; r = 1; Do[S = If[# > 1, Rest@ Divisors@ #, 0] &[GCD @@ FactorInteger[DivisorSigma[1, i]][[All, -1]]]; If[Length[S] > 0, Map[If[c[#] == 0, Set[c[#], i]] &, S]; If[# > r, r = #] &@ Max@ S], {i, 2^22}]; TakeWhile[Array[c, r], # > 0 &] (* _Michael De Vlieger, May 23 2022 *)

a(n) = my(k=2); while (!ispower(sigma(k), n), k++); k; \\ Michel Marcus, May 23 2022

a(n) = Min{x : A000203(x)=m^n} for some m.

a(24) corrected by T. D. Noe, Oct 15 2006

A303999 Numbers whose sum of divisors is the seventh power of one of their divisors.

Original entry on oeis.org

1, 112890, 120054, 124338, 133998, 137058, 139962, 36705396, 39118548, 52166212, 4661585292, 4677211812, 4851457716, 4968055596, 6168611160, 6232929480, 6236525932, 6261521812, 6311227560, 6362855640, 6430524120, 6468862876, 6488003880, 6500134440, 6506266732

Offset: 1

Paolo P. Lava, May 04 2018

Subset of A048257.

			Divisors of 112890 are 1, 2, 3, 5, 6, 10, 15, 30, 53, 71, 106, 142, 159, 213, 265, 318, 355, 426, 530, 710, 795, 1065, 1590, 2130, 3763, 7526, 11289, 18815, 22578, 37630, 56445, 112890 and their sum is 279936 = 6^7.

Cf. A000203, A048257, A303123, A303993, A303994, A303995, A303996.

with(numtheory): P:=proc(q) local a,k,n;
for n from 1 to q do a:=sort([op(divisors(n))]);
for k from 1 to nops(a) do if sigma(n)=a[k]^7 then print(n); break; fi; od; od; end: P(10^9);

Select[Range[150000], IntegerQ[t = DivisorSigma[1, #]^(1/7)] && Mod[#, t] == 0 &] (* Giovanni Resta, May 04 2018 *)

isok(n) = (n==1) || (ispower(s=sigma(n), 7) && !(n % sqrtnint(s, 7))); \\ Michel Marcus, May 05 2018

a(11)-a(25) from Giovanni Resta, May 04 2018

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A019423 Numbers whose sum of divisors is a fifth power.

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A019424 Numbers whose sum of divisors is a sixth power.

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A048258 Integers whose sum of divisors is an 8th power.

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A063869 Least k such that sigma(k)=m^n for some m>1.

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A303999 Numbers whose sum of divisors is the seventh power of one of their divisors.

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