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

%I A319688 #23 Jul 29 2023 03:17:38
%S A319688 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,
%T A319688 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,
%U A319688 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
%N A319688 Sum of digits when n is represented in phitorial (A001088) base.
%H A319688 Antti Karttunen, <a href="/A319688/b319688.txt">Table of n, a(n) for n = 0..18432</a>
%F A319688 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.
%F A319688 For all n >= 0, a(A231722(n)) = n.
%e A319688 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.
%e A319688 For n = 577, its phitorial representation is "300001" as 577 = 1*A001088(2) + 3*A001088(7) = 1*1 + 3*192, thus a(577) = 4.
%t A319688 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 *)
%o A319688 (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); };
%Y A319688 Cf. A000010, A001088 (gives the positions of ones), A231721, A231722.
%Y A319688 Cf. also A000120, A034968, A276150.
%K A319688 nonn,base
%O A319688 0,4
%A A319688 _Antti Karttunen_, Oct 02 2018