A353138 Sum of (the number of digits in n to the power (each digit in n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 5, 6, 8, 12, 20, 36, 68, 132, 260, 516, 9, 10, 12, 16, 24, 40, 72, 136, 264, 520, 17, 18, 20, 24, 32, 48, 80, 144, 272, 528, 33, 34, 36, 40, 48, 64, 96, 160, 288, 544, 65, 66, 68, 72, 80, 96

Offset: 1

Author

William Halpin, Apr 26 2022

Keywords

• nonn
• look
• base

Comments

In base 10, 1 and 4624 are the only numbers where a(n)=n (conjectured).

a(n) < n when n > 2.1*(10^10).

Examples

			a(3) = 1^3 = 1;
a(164) = 3^1 + 3^6 + 3^4 = 813;
a(4624) = 4^4 + 4^6 + 4^2 + 4^4 = 4624.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A055642, A101337.

Programs

Maple

f:= proc(n) local m,L,t;
  L:= convert(n,base,10);
  m:= nops(L);
  add(m^t, t=L)
end proc:
map(f, [$1..100]);

PARI

a(n) = my(d=digits(n)); vecsum(vector(#d, k, #d^d[k])); \\ Michel Marcus, Apr 27 2022

Python

def a(n): s = str(n); return sum(len(s)**int(d) for d in s)
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Apr 26 2022

Formula

a(n) = Sum_{k=1..A055642(n)} (A055642(n))^(floor(n*10^(1-k)) mod 10).