A339213 Phi-base self numbers: positive numbers not of the form k + A055778(k).

1, 3, 6, 10, 12, 15, 19, 23, 26, 30, 32, 38, 41, 43, 52, 55, 59, 61, 64, 68, 72, 75, 79, 81, 86, 89, 91, 97, 101, 104, 108, 110, 115, 118, 120, 126, 131, 135, 137, 140, 144, 148, 151, 155, 157, 163, 166, 168, 177, 180, 184, 186, 189, 193, 197, 200, 204, 206, 213

Offset: 1

Author

Amiram Eldar, Nov 27 2020

Keywords

• nonn
• base

Comments

Analogous to self numbers (A003052) using base phi (A130600) instead of base 10.

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
• F. Michel Dekking, The sum of digits function of the base phi expansion of the natural numbers, arXiv:1911.10705 [math.NT], 2019.
• Ron Knott, Using Powers of Phi to represent Integers (Base Phi).
• Eric Weisstein's World of Mathematics, Self Number.
• Wikipedia, Golden ratio base.
• Wikipedia, Self number.
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A003052, A010061, A010064, A010067, A010070, A055778, A130600, A331083, A334308, A339211, A339212, A339214, A339215.

Programs

Mathematica

s[1] = 2; s[n_] := n + Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]][[1]]; m = 220; Complement[Range[m], Array[s, m]]