A156834 - OEIS

%I #18 Apr 22 2021 09:05:58

%S 1,2,3,5,5,12,7,17,19,30,11,63,13,56,99,89,17,154,19,269,237,132,23,

%T 509,301,182,379,783,29,1230,31,881,813,306,2125,2431,37,380,1299,

%U 4157,41,4822,43,3695,6175,552,47,8529,5587,6266,2787

%N A156348 * A000010

%C Conjecture: for n>1, a(n) = n iff n is prime. Companion to A156833.

%H Seiichi Manyama, <a href="/A156834/b156834.txt">Table of n, a(n) for n = 1..10000</a>

%F Equals A156348 * A054525 * [1, 2, 3,...]; where A054525 = the inverse Mobius transform.

%F a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-2, d-1). - _Seiichi Manyama_, Apr 22 2021

%F G.f.: Sum_{k >= 1} phi(k) * (x/(1 - x^k))^k. - _Seiichi Manyama_, Apr 22 2021

%e a(4) = 5 = (1, 2, 0, 1) dot (1, 1, 2, 2) = (1 + 2 + 0 + 2), where row 4 of A156348 = (1, 2, 0, 1) and (1, 1, 2, 2) = the first 4 terms of Euler's phi function.

%p A156834 := proc(n)

%p         add(A156348(n,k)*numtheory[phi](k),k=1..n) ;

%p end proc: # _R. J. Mathar_, Mar 03 2013

%t a[n_] := DivisorSum[n, EulerPhi[#] * Binomial[# + n/# - 2, #-1] &]; Array[a, 100] (* _Amiram Eldar_, Apr 22 2021 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-2, d-1)); \\ _Seiichi Manyama_, Apr 22 2021

%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*(x/(1-x^k))^k)) \\ _Seiichi Manyama_, Apr 22 2021

%Y Cf. A156348, A000010, A156833, A157020, A343553.

%Y Equals row sums of triangle A157030. [_Gary W. Adamson_, Feb 21 2009]

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Feb 16 2009

%E Extended beyond a(14) by _R. J. Mathar_, Mar 03 2013