A341953 - OEIS

A341953

Replace each digit d in the decimal representation of n with the digital root of d^n.

1, 4, 9, 4, 2, 9, 7, 1, 9, 10, 11, 11, 19, 17, 18, 19, 14, 11, 19, 40, 81, 77, 59, 11, 25, 49, 81, 71, 59, 90, 91, 94, 99, 94, 92, 99, 97, 91, 99, 40, 71, 11, 49, 77, 18, 49, 74, 11, 49, 70, 81, 47, 29, 11, 55, 79, 81, 41, 29, 90, 91, 94, 99, 94, 92, 99, 97

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The digits 0, 1, 3, 6 and 9 will always be replaced by the same digits: 0 -> 0, 1 -> 1, 3 -> 9, 6 -> 9 and 9 -> 9.

LINKS

Table of n, a(n) for n=1..67.

Sebastian Karlsson, Generalized and plotted in arbitrary bases

EXAMPLE

a(14) = 17, since 1^14 = 1 and 4^14 = 268435456. 2 + 6 + 8 + 4 + 3 + 5 + 4 + 5 + 6 = 43 and 4 + 3 = 7. Thus, the digital root of 268435456 is 7. This means that for 14, "1" gets replaced by "1" and "4" gets replaced by "7".

MATHEMATICA

digroot[n_] := If[n == 0, 0, Mod[n - 1, 9] + 1]; a[n_] := FromDigits[digroot /@ (IntegerDigits[n]^n)]; Array[a, 100] (* Amiram Eldar, Feb 24 2021 *)

PROG

(Python)

def D(d, n):

return 0 if d == 0 else (pow(d, n, 9)-1)%9 + 1

def a(n):

return int(''.join(str(D(int(d), n)) for d in str(n)))

(PARI) r(n) = if(n, (n-1)%9+1) \\ A010888

a(n) = fromdigits(apply(x->r(x^n), digits(n))); \\ Michel Marcus, Mar 21 2021

CROSSREFS

Cf. A010888, A106649, A139337, A322131, A339023, A341767.