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

0,3

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.

Links

Table of n, a(n) for n=0..100.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A156230 A028897 A244158 * A081594 A038506 A091047

Adjacent sequences: A321998 A321999 A322000 * A322002 A322003 A322004

Keyword

nonn,base

Author

M. F. Hasler, Nov 27 2018

Status

approved