A344513 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.

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

Offset

1,1

Comments

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.

References

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

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

Links

Table of n, a(n) for n=1..10.

Eric Weisstein's World of Mathematics, Self Number.

Wikipedia, Self number.

Index entries for Colombian or self numbers and related sequences

Example

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

Mathematica

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]

Crossrefs

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

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

Sequence in context: A284700 A050629 A348140 * A219707 A219822 A048558

Adjacent sequences: A344510 A344511 A344512 * A344514 A344515 A344516

Keyword

nonn,base,more

Author

Amiram Eldar, May 21 2021

Status

approved