A332208 -id:A332208 - 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)));

A331752 Numbers k such that squarefree part of sigma(k) is equal to squarefree part of 2*k.

Original entry on oeis.org

6, 28, 468, 496, 775, 2268, 3780, 4655, 7448, 8128, 9000, 10880, 10976, 25137, 40131, 40176, 58752, 62775, 66960, 91000, 137541, 137940, 140800, 160930, 167400, 173600, 195938, 224450, 307125, 377055, 399360, 406224, 417477, 494832, 569184, 603288, 634725, 639158, 658368, 773175, 869022, 881280, 889056, 1005480

Offset: 1

Antti Karttunen, Feb 06 2020

nonn

Numbers k such that A007913(sigma(k)) is equal to A007913(2*k), thus numbers for which sigma(k) has the same set of distinct prime factors with an odd exponent as 2*k.
Among the first 257 terms, these four are also in A228058:
  46277101  = 61 * 13^2 * 67^2,
  49889853  = 13 * 3^2 * 653^2,
  106706925 = 13 * 3^2 * 5^2 * 191^2,
  676830973 = 37 * 7^2 * 13^2 * 47^2.

			For n = 46277101 = 61 * 13^2 * 67^2, sigma(46277101) = 51703722 = 2 * 3^2 * 7^2 * 31^2 * 61, with A007913(sigma(46277101)) = 2*61 = A007913(2*46277101), thus 46277101 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..257
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A006532, A007913, A228058, A331751, A332208.

Cf. A000396 (a subsequence).

Select[Range[10^6], SameQ @@ Map[Sqrt[#] /. (c_: 1)*a_^(b_: 0) :> (c*a^b)^2 &, {DivisorSigma[1, #], 2 #}] &] (* Michael De Vlieger, Feb 08 2020, after Bill Gosper at A007913 *)

isA331752(n) = (core(2*n)==core(sigma(n)));

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)));

cp's OEIS Frontend

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

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A331752 Numbers k such that squarefree part of sigma(k) is equal to squarefree part 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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