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

1,1

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.

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.

Example

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.

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]

Crossrefs

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

A164318 is a subsequence.

Sequence in context: A216639 A350301 A171098 * A118845 A050717 A335034

Adjacent sequences: A349759 A349760 A349761 * A349763 A349764 A349765

Keyword

nonn

Author

Amiram Eldar, Nov 29 2021

Status

approved