A331752 -id:A331752 - cp's OEIS frontend

A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

2, 6, 27, 28, 84, 270, 496, 1053, 1120, 1488, 1625, 1638, 3360, 3780, 4875, 8128, 10530, 24384, 66960, 147420, 167400, 406224, 611226, 775000, 872960, 943250, 1097280, 1245699, 1255338, 1303533, 1464320, 1686400, 1740024, 1922375, 1952500, 2011625, 2193408, 2325000, 2611440, 2618880, 2829750, 2941029, 4392960

Offset: 1

Antti Karttunen, Feb 05 2020

nonn

Numbers k such that A097248(sigma(k)) is equal to A097248(2*k).
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)).
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.
Note that the condition A090880(sigma(k)) == A090880(2*k) appears to be much more constrained.

			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.
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.

Antti Karttunen, Table of n, a(n) for n = 1..60 (up to the term 33550336).
Index entries for sequences where any odd perfect numbers must occur

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

Cf. A000396 (a subsequence).

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))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
isA331751(n) = (A097248(2*n)==A097248(sigma(n)));

A332208 Numbers k such that the squarefree kernel of sigma(k) is equal to the squarefree kernel of 2*k.

Original entry on oeis.org

6, 28, 120, 135, 270, 496, 672, 891, 1080, 1638, 1782, 3780, 8128, 18600, 20580, 24948, 26208, 30240, 32640, 32760, 35640, 41850, 44226, 55860, 66960, 164640, 167400, 185220, 199584, 200655, 273000, 293760, 307125, 401310, 441936, 446880, 502740, 523776, 544635, 614250, 707616, 802620, 819000, 884520

Offset: 1

Antti Karttunen, Feb 07 2020

nonn

Numbers k such that sigma(k) has the same set of distinct prime factors as 2*k.
Numbers k such that A007947(sigma(k)) is equal to A007947(2*k), or equally, that A087207(sigma(k)) is equal to A087207(2*k).
Of the first 256 terms 44 are odd, and none occurs in A228058. Compare also to A331752.

Antti Karttunen, Table of n, a(n) for n = 1..256
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A007947, A080398, A087207, A228058, A331751, A331752.

Subsequences: A000396, A027687.

Select[Range[10^6], SameQ @@ Map[Times @@ FactorInteger[#][[All, 1]] &, {DivisorSigma[1, #], 2 #}] &] (* Michael De Vlieger, Feb 08 2020 *)

A007947(n) = factorback(factorint(n)[, 1]);
isA332208(n) = (A007947(sigma(n)) == A007947(2*n));

{n: A080398(n) == A007947(2n)}.

A332446 Numbers k for which A087808(sigma(k)) is equal to A087808(2*k).

Original entry on oeis.org

3, 6, 11, 19, 28, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 216, 227, 251, 267, 283, 286, 307, 331, 347, 379, 419, 443, 467, 491, 496, 499, 523, 547, 563, 571, 587, 598, 619, 643, 659, 683, 691, 726, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259, 1283, 1291, 1307, 1427, 1451

Offset: 1

Antti Karttunen, Feb 14 2020

nonn

Conjecture: includes all terms of A007520. - Bill McEachen, Dec 10 2023

Cf. A000203, A007520, A087808.
Subsequences: A000396, A332445.
Cf. A331751, A331752, A332208 for similar sequences.

A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA332446(n) = (A087808(2*n)==A087808(sigma(n)));

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A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

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A332208 Numbers k such that the squarefree kernel of sigma(k) is equal to the squarefree kernel of 2*k.

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A332446 Numbers k for which A087808(sigma(k)) is equal to A087808(2*k).

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