cp's OEIS Frontend

A386428 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, k > 0, r > 1, and gcd(p,r) = 1.

%I A386428 #21 Aug 19 2025 09:33:09
%S A386428 28125,153125,253125,378125,528125,903125,1128125,1378125,1653125,
%T A386428 2278125,2628125,3003125,3341637,3403125,4278125,4753125,5253125,
%U A386428 5778125,6903125,7503125,8128125,8778125,9282325,10153125,10878125,11628125,12403125,12778713,14028125,14878125,15753125,16653125,17578125,18193357,18528125
%N A386428 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, k > 0, r > 1, and gcd(p,r) = 1.
%C A386428 Powerful numbers (A001694) that satisfy Euler's criterion for odd perfect numbers (A228058). That is, terms of A228058 whose special factor is not a prime, but a prime power.
%C A386428 If N = q^k*n^2 (i.e., a number of the form A228058) is an odd perfect number with special prime q, then the assertion that k must be 1 is known as the Descartes-Frenicle-Sorli conjecture on odd perfect numbers. In other words, the conjecture stipulates that certainly this subsequence of A228058 does not contain any odd perfect numbers.
%C A386428 a(1) = A228058(520); a(1..12) = 5^5 * A028375(2..13).
%H A386428 Antti Karttunen, <a href="/A386428/b386428.txt">Table of n, a(n) for n = 1..811</a>
%H A386428 Jose A. B. Dris, <a href="https://doi.org/10.7546/nntdm.2018.24.3.62-67">Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017), pp. 12-20, <a href="https://arxiv.org/abs/1610.01868">arXiv preprint</a>, arXiv:1610.01868 [math.NT], 2016.
%o A386428 (PARI) isA386428(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||1==f[i,2]||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
%o A386428 (PARI) isA386428(n) = (ispowerful(n) && isA228058(n)); \\ See A228058.
%Y A386428 Intersection of A001694 (or A320966) and A228058.
%Y A386428 Cf. A028375.
%K A386428 nonn,new
%O A386428 1,1
%A A386428 _Antti Karttunen_, Aug 17 2025