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A156834 A156348 * A000010 7
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 (list; graph; refs; listen; history; text; internal format)
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 * A079024 A342421 A319631

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

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Last modified October 22 18:45 EDT 2021. Contains 348175 sequences. (Running on oeis4.)