This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A063869 #43 Jun 22 2025 05:35:11
%S A063869 2,3,7,217,21,2667,93,217,381,651,2752491,2667,8191,11811,24573,57337,
%T A063869 82677,172011,393213,761763,1572861,2752491,5332341,11010027,21845397,
%U A063869 48758691,85327221,199753347,341310837,677207307,1398273429,3220807683
%N A063869 Least k such that sigma(k)=m^n for some m>1.
%C A063869 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.
%C A063869 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
%C A063869 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
%H A063869 T. D. Noe, <a href="/A063869/b063869.txt">Table of n, a(n) for n=1..500</a>
%H A063869 K. S. Brown, <a href="http://www.mathpages.com/home/kmath048.htm">Sum of Divisors Equals a Power of 2</a>
%H A063869 W. Sierpiński, <a href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.dl-catalog-556369c7-b6cc-4a5b-be36-bfc8e0ca7cfa">Elementary Theory of Numbers</a>, Warszawa 1964, page 165.
%F A063869 a(n) = Min{x : A000203(x)=m^n} for some m.
%e A063869 For n = 11, sigma(a(n)) = sigma(2752491) = sigma(3 * 7 * 131071) = 4^11.
%t A063869 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 *)
%t A063869 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 *)
%o A063869 (PARI) a(n) = my(k=2); while (!ispower(sigma(k), n), k++); k; \\ _Michel Marcus_, May 23 2022
%Y A063869 Cf. A006532, A020477, A019422, A019423, A019424, A048257, A048258, A000668, A046528.
%K A063869 nonn
%O A063869 1,1
%A A063869 _Labos Elemer_, Aug 27 2001
%E A063869 a(24) corrected by _T. D. Noe_, Oct 15 2006