A231722 Partial sums of A001088 starting from its second term; a(1)=0, a(n) = A001088(2)+...+A001088(n).

0, 1, 3, 7, 23, 55, 247, 1015, 5623, 24055, 208375, 945655, 9793015, 62877175, 487550455, 3884936695, 58243116535, 384392195575, 6255075618295, 53220543000055, 616806151581175, 6252662237392375, 130241496125238775, 1122152167228009975, 20960365589283433975

Offset: 1

Author

Antti Karttunen, Nov 27 2013

Keywords

• nonn

Comments

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).

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.

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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

One less than A231721.

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

Programs

Maple

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

Mathematica

Table[Sum[Product[EulerPhi[k], {k, i}], {i, 2, n}], {n, 20}] (* Wesley Ivan Hurt, Aug 09 2014 *)

PARI

a(n) = sum(i=2, n, prod(k=1, i, eulerphi(k))); \\ Michel Marcus, Aug 09 2014

Scheme

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

Formula

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