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.

28125, 153125, 253125, 378125, 528125, 903125, 1128125, 1378125, 1653125, 2278125, 2628125, 3003125, 3341637, 3403125, 4278125, 4753125, 5253125, 5778125, 6903125, 7503125, 8128125, 8778125, 9282325, 10153125, 10878125, 11628125, 12403125, 12778713, 14028125, 14878125, 15753125, 16653125, 17578125, 18193357, 18528125

Offset: 1

Author

Antti Karttunen, Aug 17 2025

Keywords

• nonn
• new

Comments

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.

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.

a(1) = A228058(520); a(1..12) = 5^5 * A028375(2..13).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..811
• Jose A. B. Dris, Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017), pp. 12-20, arXiv preprint, arXiv:1610.01868 [math.NT], 2016.

Crossrefs

Intersection of A001694 (or A320966) and A228058.

Cf. A028375.

Programs

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));

PARI

isA386428(n) = (ispowerful(n) && isA228058(n)); \\ See A228058.