A349019 Modified e-perfect numbers: numbers k such that A348963(k) | k.

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101

Offset: 1

Author

Amiram Eldar, Nov 06 2021

Keywords

• nonn

Comments

First differs from A225354 at n = 25.

Not to be confused with modified exponential perfect numbers (A323757).

Sándor (2006) showed that the exponential harmonic numbers of type 2 (A348964) are terms in this sequence.

All the squarefree numbers are terms (A005117), since A348963(k) = 1 if k is squarefree.

Examples

			12 is a term since A348963(12) = 3 is a divisor of 12.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
• József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

Crossrefs

A005117, A348964 and A349020 are subsequences.

Cf. A225354, A323757, A348963.

Programs

Mathematica

f[p_, e_] := p^e/DivisorSum[e, p^(e - #) &]; modEPerfQ[1] = True; modEPerfQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], modEPerfQ]