A339214 Factorial-base self numbers: numbers not of the form k + A034968(k).

1, 4, 11, 18, 36, 43, 61, 68, 86, 93, 111, 118, 125, 132, 139, 157, 164, 182, 189, 207, 214, 232, 239, 246, 253, 260, 278, 285, 303, 310, 328, 335, 353, 360, 367, 374, 381, 399, 406, 424, 431, 449, 456, 474, 481, 488, 495, 502, 520

Offset: 1

Author

Amiram Eldar, Nov 27 2020

Keywords

• nonn
• base

Comments

Analogous to self numbers (A003052) using factorial base representation (A007623) instead of decimal expansion.

The numbers of terms that do not exceed 10^k, for k = 0, 1, ..., are 1, 2, 10, 90, 878, 8749, 87455, 874499, 8744934, 87449296, 874492907, ... . Apparently, the asymptotic density of this sequence exists and equals 0.08744929... . - Amiram Eldar, Aug 08 2025

References

• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Rosalind Guaraldo, On the Density of the Image Sets of Certain Arithmetic Functions - III, The Fibonacci Quarterly, Vol. 16, No. 6 (1978), pp. 481-488.
• Eric Weisstein's World of Mathematics, Self Number.
• Wikipedia, Factorial number system.
• Wikipedia, Self number.
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A003052, A007623, A010061, A010064, A010067, A010070, A034968, A118363, A339211, A339212, A339213, A339215.

Cf. also A219650, A219658, A230412.

Programs

Mathematica

max = 6; s[n_] := n + Plus @@ IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; m = max!; Complement[Range[m], Array[s, m]]