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

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

Author

Antti Karttunen, Feb 06 2020

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A000396 (a subsequence).

Programs

Mathematica

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 *)

PARI

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