A344512 a(n) is the least number larger than 1 which is a self number in all the bases 2 <= b <= n.

4, 13, 13, 13, 287, 287, 2971, 2971, 27163, 27163, 90163, 90163, 5940609, 5940609, 6069129, 6069129, 276404649, 276404649

Offset

2,1

Comments

Since the sequence of base-b self numbers for odd b is the sequence of the odd numbers (A005408) (Joshi, 1973), all the terms beyond a(2) are odd numbers.

For the corresponding sequence with only even bases, see A344513.

a(20) > 1.5*10^10, if it exists.

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=2..19.

Eric Weisstein's World of Mathematics, Self Number.

Wikipedia, Self number.

Index entries for Colombian or self numbers and related sequences

Formula

a(2*n+1) = a(2*n) for n >= 2.

Example

a(2) = 4 since the least binary self number after 1 is A010061(2) = 4.
a(3) = 13 since the least binary self number after 1 which is also a self number in base 3 is A010061(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[2] = 4; a[b_] := a[b] = Module[{n = a[b - 1]}, While[! AllTrue[Range[2, b], selfQ[n, #] &], n++]; n]; Array[a, 10, 2]

Crossrefs

Cf. A003052, A010061, A010064, A010067, A010070, A339211, A339212, A339213, A339214, A339215, A342729, A344513.

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

Sequence in context: A287895 A050223 A301792 * A168401 A370644 A081738

Adjacent sequences: A344509 A344510 A344511 * A344513 A344514 A344515

Keyword

nonn,base,more

Author

Amiram Eldar, May 21 2021

Status

approved