A349762 Numbers k such that phi(k) = A000010(k) is an abundant number (A005101) and d(k) = A000005(k) is a deficient number (A005100).

13, 19, 21, 25, 26, 27, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 49, 54, 55, 56, 57, 61, 62, 65, 66, 67, 70, 71, 73, 74, 77, 78, 79, 81, 82, 86, 87, 88, 89, 91, 93, 95, 97, 100, 101, 103, 104, 105, 109, 110, 111, 112, 113, 114, 115, 119, 122, 123, 125, 127, 129

Offset: 1

Author

Amiram Eldar, Nov 29 2021

Keywords

• nonn

Comments

Sándor (2005) proved that this sequence is infinite by showing that it includes all the numbers of the form 3^(p^2-1) where p is a prime.

Examples

			13 is a term since phi(13) = 12 is an abundant number, sigma(12) = 28 > 2*12 = 24, and d(13) = 2 is a deficient number, sigma(2) = 3 < 2*2 = 4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.

Crossrefs

Cf. A000005, A000010, A005100, A005101, A349758, A349759, A349761.

A164318 is a subsequence.

Programs

Mathematica

abQ[n_] := DivisorSigma[1, n] > 2*n; defQ[n_] := DivisorSigma[1, n] < 2*n; q[n_] := abQ[EulerPhi[n]] && defQ[DivisorSigma[0, n]]; Select[Range[150], q]