A230624 Numbers k with property that for every base b >= 2, there is a number m such that m+s(m) = k, where s(m) = sum of digits in the base-b expansion of m.

0, 2, 10, 14, 22, 38, 62, 94, 158, 206, 318, 382, 478, 606, 766, 958, 1022, 1534, 1662, 1726, 1790, 1918, 1982, 2238, 2622, 2686, 3006, 3262, 3582, 3966, 4734, 5118, 5374, 5758, 5886, 6782, 8830, 9342, 9470, 9598, 10878, 12926, 13182, 13438, 14718, 18686, 22526

Offset: 1

Author

N. J. A. Sloane, Oct 27 2013

Keywords

• nonn
• base

Comments

If k is a positive term then k is even (or else k has no generator in base k+1) but not a multiple of 4 (or else k has no generator in base k/2). - David Applegate, Jan 09 2022. See A349821 and A350607 for the k/2 and (k-2)/4 sequences.

It is not known if this sequence is infinite.

The eight terms 10 through 206 are all twice primes (cf. A349820).

Examples

			10 is a member because in base 2, 7=111, 7+3=10; in base 3, 7=21, 7+3=10; in base 4, 8=20, 8+2=10; in base 5, 7=12, 7+3=10; and in bases b >= 6, 5+5=10.
		

Links

• David Applegate, Table of n, a(n) for n = 1..547, terms < 10^9 (first 90 terms from Lars Blomberg)
• David Applegate, Two graphs to accompany Comments (see next link)
• David Applegate and N. J. A. Sloane, Comments on A230624, Numbers that are generated in every base
• Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020).
• Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]
• Cai, Tianxin, On k-self-numbers and universal generated numbers, Fibonacci Quart. 34 (1996), no. 2, 144--146. MR1386983 (97c:11008)
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A003052, A349820, A349821, A349822, A350607.

For first differences see A349823.

This is the limiting row of A350601.

Extensions

More terms from Lars Blomberg, Oct 12 2015

More terms from David Applegate, Jan 02 2022