A353538 Numbers k such that k and k+1 both have an abundancy index that is larger than Pi^2/6.

44, 63, 75, 104, 135, 164, 188, 189, 195, 224, 230, 231, 255, 284, 285, 315, 344, 374, 375, 404, 405, 434, 435, 440, 441, 464, 494, 495, 524, 567, 584, 675, 692, 735, 764, 819, 824, 825, 854, 855, 944, 945, 975, 1034, 1035, 1070, 1071, 1124, 1155, 1196, 1215, 1274

Offset: 1

Author

Amiram Eldar, Apr 25 2022

Keywords

• nonn

Comments

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 3, 43, 399, 4292, 41306, 413310, 4155925, 414963651, 4149603572, ...

Apparently, the asymptotic density of this sequence is 0.04149...

Examples

			44 is a term since sigma(44)/44 = 84/44 = 1.909... and sigma(45)/45 = 78/45 = 1.733... are both larger than Pi^2/6 = 1.644... .
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A013661.

Subsequence of A353537.

Subsequences: A096399, A353539, A353540, A353541.

Programs

Mathematica

q[n_] := DivisorSigma[-1, n] > Pi^2/6; Select[Range[1300], q[#] && q[# + 1] &]

PARI

isok(k) = ((sigma(k)/k) > Pi^2/6) && ((sigma(k+1)/(k+1)) > Pi^2/6); \\ Michel Marcus, Apr 25 2022