

A344512


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


1



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

2,1


COMMENTS

Since the sequence of baseb 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 powerfree integers and selfnumbers, 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. 384386.


LINKS



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.


KEYWORD

nonn,base,more


AUTHOR



STATUS

approved



