A156834 A156348 * A000010

1, 2, 3, 5, 5, 12, 7, 17, 19, 30, 11, 63, 13, 56, 99, 89, 17, 154, 19, 269, 237, 132, 23, 509, 301, 182, 379, 783, 29, 1230, 31, 881, 813, 306, 2125, 2431, 37, 380, 1299, 4157, 41, 4822, 43, 3695, 6175, 552, 47, 8529, 5587, 6266, 2787

Offset

1,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

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

a(n) = Sum_{d|n} phi(d) * binomial(d+n/d-2, d-1). - Seiichi Manyama, Apr 22 2021

G.f.: Sum_{k >= 1} phi(k) * (x/(1 - x^k))^k. - Seiichi Manyama, Apr 22 2021

Example

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.

Maple

A156834 := proc(n)
        add(A156348(n, k)*numtheory[phi](k), k=1..n) ;
end proc: # R. J. Mathar, Mar 03 2013

Mathematica

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

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(d)*binomial(d+n/d-2, d-1)); \\ Seiichi Manyama, Apr 22 2021
(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

Crossrefs

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

Equals row sums of triangle A157030. [Gary W. Adamson, Feb 21 2009]

Sequence in context: A133278 A343395 A050368 * A376017 A079024 A357000

Adjacent sequences: A156831 A156832 A156833 * A156835 A156836 A156837

Keyword

nonn,easy

Author

Gary W. Adamson, Feb 16 2009

Extensions

Extended beyond a(14) by R. J. Mathar, Mar 03 2013

Status

approved