A319688 Sum of digits when n is represented in phitorial (A001088) base.

0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6

Offset: 0

Author

Antti Karttunen, Oct 02 2018

Keywords

• nonn
• base

Examples

			For n = 9, its phitorial representation is "102" as 9 = 1*A001088(2) + 0*A001088(3) + 2*A001088(4) = 1*1 + 0*2 + 2*4. Thus a(9) = 1+0+2 = 3.
For n = 577, its phitorial representation is "300001" as 577 = 1*A001088(2) + 3*A001088(7) = 1*1 + 3*192, thus a(577) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..18432

Crossrefs

Cf. A000010, A001088 (gives the positions of ones), A231721, A231722.

Cf. also A000120, A034968, A276150.

Programs

Mathematica

With[{max = 7}, bases = EulerPhi[Range[max, 1, -1]]; nmax = Times @@ bases - 1; a[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; Array[a, nmax, 0]] (* Amiram Eldar, Jul 29 2023 *)

PARI

A319688(n) = { my(s=0, i=3, d, b); while(n, b = eulerphi(i); d = (n%b); s += d; n = (n-d)/b; i++); (s); };

Formula

Starting from i=3, compute the remainder when n is divided by phi(i), and then continue iterating with n -> floor(n/phi(i)), and i -> i+1, until n is zero. a(n) is the sum of remainders encountered in process.

For all n >= 0, a(A231722(n)) = n.