A344513 - OEIS

%I #6 May 22 2021 11:56:18

%S 4,13,287,294,6564,90163,1136828,3301262,276404649,5643189146

%N a(n) is the least number larger than 1 which is a self number in all the even bases b = 2*k for 1 <= k <= n.

%C Joshi (1973) proved that for all odd b the sequence of base-b self numbers is the sequence of odd numbers (A005408). Therefore, in this sequence the bases are restricted to even values. For the corresponding sequence with both odd and even bases, see A344512.

%D Vijayshankar Shivshankar Joshi, Contributions to the theory of power-free integers and self-numbers, Ph.D. dissertation, Gujarat University, Ahmedabad (India), October, 1973.

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SelfNumber.html">Self Number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Self_number">Self number</a>.

%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>

%e a(1) = 4 since the least binary self number after 1 is A010061(2) = 4.

%e a(2) = 13 since the least binary self number after 1 which is also a self number in base 2*2 = 4 is A010061(4) = A010064(4) = 13.

%t s[n_, b_] := n + Plus @@ IntegerDigits[n, b]; selfQ[n_, b_] := AllTrue[Range[n, n - (b - 1) * Ceiling @ Log[b, n], -1], s[#, b] != n &]; a[1] = 4; a[n_] := a[n] = Module[{k = a[n - 1]}, While[! AllTrue[Range[1, n], selfQ[k, 2*#] &], k++]; k]; Array[a, 7]

%Y Cf. A003052, A010061, A010064, A010067, A010070, A339211, A339212, A339213, A339214, A339215, A342729, A344512.

%Y Similar sequences: A016038, A217705, A225427, A226320, A228768, A258107.

%K nonn,base,more

%O 1,1

%A _Amiram Eldar_, May 21 2021