A322001 Digits of n interpreted in factorial base: a(Sum d_k*10^k) = Sum d_k*k!

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 6

Offset: 0

Author

M. F. Hasler, Nov 27 2018

Keywords

• nonn
• base

Comments

More terms than usual are given to distinguish the sequence from A081594, A028897 and A244158, which agree up to a(99). The last two correspond to k! replaced by 2^k resp. Catalan(k).

This is a left inverse to A007623 (factorial base representation of n): A322001(A007623(n)) = n for all n >= 0. One could imagine variants which have a(n) = 0 or a(n) = -1 if n is not a term of A007623. Restricted to the range of A007623, it is also a right inverse to A007623, at least up to the 10 digit terms, beyond which A007623 becomes non-injective.

Crossrefs

Cf. A007623 (right inverse), A081594, A028897, A244158.

Programs

Mathematica

a[n_] := Module[{d=Reverse@IntegerDigits[n]}, Sum[d[[i]]*i!, {i,1,Length[d]}]]; Array[a, 100, 0] (* Amiram Eldar, Nov 28 2018 *)

PARI

A322001(n)=sum(i=1,#n=Vecrev(digits(n)),n[i]*i!) \\ M. F. Hasler, Nov 27 2018