A331751 - OEIS

%I #35 Jul 20 2021 07:01:38

%S 2,6,27,28,84,270,496,1053,1120,1488,1625,1638,3360,3780,4875,8128,

%T 10530,24384,66960,147420,167400,406224,611226,775000,872960,943250,

%U 1097280,1245699,1255338,1303533,1464320,1686400,1740024,1922375,1952500,2011625,2193408,2325000,2611440,2618880,2829750,2941029,4392960

%N Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

%C Numbers k such that A097248(sigma(k)) is equal to A097248(2*k).

%C Numbers k such that A331750(k) is equal to 1+A048675(k), which in turn is equal to A048675(A225546(2*k)) = A048675(2*A225546(k)).

%C Among the first 60 terms, 15 are odd: 27, 1053, 1625, 4875, 1245699, 1303533, 1922375, 2011625, 2941029, 5767125, 6034875, 12733875, 17137575, 26316675, 29362905, and only 1053 = 3^4 * 13 is in A228058.

%C Note that the condition A090880(sigma(k)) == A090880(2*k) appears to be much more constrained.

%H Antti Karttunen, <a href="/A331751/b331751.txt">Table of n, a(n) for n = 1..60</a> (up to the term 33550336).

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>

%e For n = 1053 = 3^4 * 13^1, A331750(1053) = A331750(81) + A331750(13) = 32+9 = 41, while A048675(2*1053) = A048675(2)+A048675(81)+A048675(13) = 1+8+32 = 41 also, thus 1053 is included in this sequence.

%e For n = 3360 = 2^5 * 3^1 * 5^1 * 7^1, A331750(3360) = A331750(32)+A331750(3)+A331750(5)+A331750(7) = 12+2+3+3 = 20, while A048675(2*3360) = A048675(2)+A048675(32)+A048675(3)+A048675(5)+A048675(7) = 1+5+2+4+8 = 20 also, thus 3360 is included in this sequence.

%o (PARI)

%o A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i, 1]+1)^(f[i, 2]\2))*((f[i, 1])^(f[i, 2]%2))); };

%o A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };

%o isA331751(n) = (A097248(2*n)==A097248(sigma(n)));

%Y Cf. A000203, A048675, A090880, A097246, A097248, A331750, A331752, A332208.

%Y Cf. A000396 (a subsequence).

%K nonn

%O 1,1

%A _Antti Karttunen_, Feb 05 2020