A231722 - OEIS

%I #21 Aug 10 2014 07:06:22

%S 0,1,3,7,23,55,247,1015,5623,24055,208375,945655,9793015,62877175,

%T 487550455,3884936695,58243116535,384392195575,6255075618295,

%U 53220543000055,616806151581175,6252662237392375,130241496125238775,1122152167228009975,20960365589283433975

%N Partial sums of A001088 starting from its second term; a(1)=0, a(n) = A001088(2)+...+A001088(n).

%C a(n+1) gives the index to the last term in each row of A231716. Specifically, for all n>=1, A231716(a(n+1)) = A033312(n+1).

%C a(n) = natural number which is written as the n-th repunit in "totient phi number system": 0, 1, 10, 11, 100, 101, 110, 111, 200, 201, 210, 211, 300, 301, 310, 311, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 1200, ..., and so on. Note how the 1st, the 3rd, the 7th and 23rd terms of this list are 1, 11, 111, and 1111.

%C In this number system the i-th digit from right (the least significant digit = digit_0) may contain integers in range 0..A000010(i+3)-1, and the value of the number is obtained as sum_{i=0..} digit_i * A001088(i+2).

%H Vincenzo Librandi, <a href="/A231722/b231722.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = A231721(n)-1. [The terms are one less than the partial sums of "phitorials", A001088, cumulatively summed from their first term]

%p with(numtheory): A231722:=n->add(product(phi(k), k=1..i), i=2..n): seq(A231722(n), n=1..20); # _Wesley Ivan Hurt_, Aug 09 2014

%t Table[Sum[Product[EulerPhi[k], {k, i}], {i, 2, n}], {n, 20}] (* _Wesley Ivan Hurt_, Aug 09 2014 *)

%o (Scheme)

%o (define (A231722 n) (- (A231721 n) 1))

%o (PARI) a(n) = sum(i=2, n, prod(k=1, i, eulerphi(k))); \\ _Michel Marcus_, Aug 09 2014

%Y One less than A231721.

%Y Cf. A000010 (Euler's totient function phi), A001088 (its partial products, "phitorials"), A231716, A033312.

%K nonn

%O 1,3

%A _Antti Karttunen_, Nov 27 2013