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A341953 Replace each digit d in the decimal representation of n with the digital root of d^n.

%I A341953 #15 Jan 16 2025 20:39:12
%S A341953 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,
%T A341953 81,71,59,90,91,94,99,94,92,99,97,91,99,40,71,11,49,77,18,49,74,11,49,
%U A341953 70,81,47,29,11,55,79,81,41,29,90,91,94,99,94,92,99,97
%N A341953 Replace each digit d in the decimal representation of n with the digital root of d^n.
%C A341953 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.
%H A341953 Robert Israel, <a href="/A341953/b341953.txt">Table of n, a(n) for n = 1..10000</a>
%H A341953 Sebastian Karlsson, <a href="/A341953/a341953.pdf">Generalized and plotted in arbitrary bases</a>
%e A341953 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".
%t A341953 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 *)
%o A341953 (Python)
%o A341953 def D(d, n):
%o A341953     return 0 if d == 0 else (pow(d, n, 9)-1)%9 + 1
%o A341953 def a(n):
%o A341953     return int(''.join(str(D(int(d), n)) for d in str(n)))
%o A341953 (PARI) r(n) = if(n, (n-1)%9+1) \\ A010888
%o A341953 a(n) = fromdigits(apply(x->r(x^n), digits(n))); \\ _Michel Marcus_, Mar 21 2021
%Y A341953 Cf. A010888, A106649, A139337, A322131, A339023, A341767.
%K A341953 nonn,base,look
%O A341953 1,2
%A A341953 _Sebastian Karlsson_, Feb 24 2021