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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.

%I A230624 #48 Jan 05 2025 19:51:40
%S A230624 0,2,10,14,22,38,62,94,158,206,318,382,478,606,766,958,1022,1534,1662,
%T A230624 1726,1790,1918,1982,2238,2622,2686,3006,3262,3582,3966,4734,5118,
%U A230624 5374,5758,5886,6782,8830,9342,9470,9598,10878,12926,13182,13438,14718,18686,22526
%N 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.
%C A230624 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.
%C A230624 It is not known if this sequence is infinite.
%C A230624 The eight terms 10 through 206 are all twice primes (cf. A349820).
%H A230624 David Applegate, <a href="/A230624/b230624.txt">Table of n, a(n) for n = 1..547</a>, terms < 10^9 (first 90 terms from Lars Blomberg)
%H A230624 David Applegate, <a href="/A230624/a230624.pdf">Two graphs to accompany Comments (see next link)</a>
%H A230624 David Applegate and N. J. A. Sloane, <a href="/A230624/a230624.txt">Comments on A230624, Numbers that are generated in every base</a>
%H A230624 Santanu Bandyopadhyay, <a href="https://www.ese.iitb.ac.in/~santanu/RM8.pdf">Self-Number</a>, Indian Institute of Technology Bombay (Mumbai, India, 2020).
%H A230624 Santanu Bandyopadhyay, <a href="/A003052/a003052_3.pdf">Self-Number</a>, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]
%H A230624 Cai, Tianxin, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/34-2/cai2.pdf">On k-self-numbers and universal generated numbers</a>, Fibonacci Quart. 34 (1996), no. 2, 144--146. MR1386983 (97c:11008)
%H A230624 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%e A230624 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.
%Y A230624 Cf. A003052, A349820, A349821, A349822, A350607.
%Y A230624 For first differences see A349823.
%Y A230624 This is the limiting row of A350601.
%K A230624 nonn,base
%O A230624 1,2
%A A230624 _N. J. A. Sloane_, Oct 27 2013
%E A230624 More terms from _Lars Blomberg_, Oct 12 2015
%E A230624 More terms from _David Applegate_, Jan 02 2022