A340995 -id:A340995 - cp's OEIS frontend

A159929 INVERT transform of phi(n), A000010.

1, 1, 2, 5, 11, 26, 57, 131, 296, 669, 1515, 3430, 7765, 17577, 39790, 90069, 203897, 461562, 1044847, 2365239, 5354224, 12120455, 27437267, 62110208, 140599921, 318278385, 720492104, 1630990029, 3692099407, 8357867190, 18919843773, 42829166807, 96953101328, 219474357191, 496827773575

Offset: 0

Gary W. Adamson, Apr 26 2009

nonn

Number of compositions of n into parts where there are phi(k) sorts of part k. - Joerg Arndt, Sep 30 2012

			a(6) = 57 = (1, 1, 2, 2, 4, 2) dot (26, 11, 5, 2, 1, 1) = (26 + 11 + 10 + 4 + 4 + 2).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000010.

Row sums of A340995.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*numtheory[phi](i), i=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Sep 22 2017

a[n_] := a[n] = If[n == 0, 1, Sum[a[n-i] EulerPhi[i], {i, 1, n}]];
a /@ Range[0, 35] (* Jean-François Alcover, Oct 31 2020, after Maple *)

N=66;  x='x+O('x^N);
Vec( 1/( 1 - sum(k=1,N, eulerphi(k)*x^k ) ) - 1 )
/* Joerg Arndt, Sep 30 2012 */

INVERT transform of A000010.
G.f.: 1/( 1 - Sum_{k>=1} phi(k) * x^k ) where phi = A000010. Joerg Arndt, Sep 30 2012
a(n) ~ c * d^n, where d = 2.26371672715382105671101924573765243871241560288177676216035633730282369149... is the root of the equation Sum_{k>=1} phi(k)/d^k = 1 and c = 0.42880036544961338799475947921442516792321060146527623589359809145075482942... - Vaclav Kotesovec, Aug 18 2021

a(0)=1 prepended by Alois P. Heinz, Sep 22 2017

A065093 Convolution of A000010 with itself.

Original entry on oeis.org

1, 2, 5, 8, 16, 20, 36, 44, 68, 76, 120, 124, 188, 196, 276, 272, 404, 380, 544, 532, 716, 668, 968, 860, 1184, 1120, 1472, 1332, 1896, 1624, 2204, 2036, 2656, 2352, 3284, 2752, 3684, 3356, 4324, 3744, 5192, 4312, 5720, 5180, 6540, 5628, 7768, 6388, 8476

Offset: 1

Vladeta Jovovic, Nov 11 2001

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.

Cf. A000010, A000385, A055507, A065474, A330319.

Column k=2 of A340995.

Table[Sum[EulerPhi[j]*EulerPhi[n-j], {j, 1, n-1}], {n, 2, 50}] (* Vaclav Kotesovec, Aug 18 2021 *)

{ for (n=1, 1000, a=sum(k=1, n, eulerphi(k)*eulerphi(n+1-k)); write("b065093.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 06 2009

a(n) = Sum_{k=1..n} phi(k)*phi(n+1-k), where phi is Euler totient function (A000010).
G.f.: (1/x)*(Sum_{k>=1} mu(k)*x^k/(1 - x^k)^2)^2. - Ilya Gutkovskiy, Jan 31 2017
a(n) ~ (n^3/6) * c * Product_{primes p|n+1} ((p^3-2*p+1)/(p*(p^2-2))), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474) (Ingham, 1927). - Amiram Eldar, Jul 13 2024

A340994 a(n) is the (2n)-th term of the n-fold self-convolution of the Euler totient function phi.

Original entry on oeis.org

1, 1, 5, 19, 89, 391, 1817, 8429, 39697, 187849, 894965, 4282191, 20572961, 99158645, 479294877, 2322365959, 11276837761, 54860498415, 267336028565, 1304677123305, 6375749480369, 31195075605755, 152798541606529, 749184538847397, 3676699991008897

Offset: 0

Alois P. Heinz, Feb 01 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1431

Cf. A000010, A008683, A340995.

b:= proc(n, k) option remember; `if`(k=0, 1,
      `if`(k=1, numtheory[phi](n+1), (q->
       add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..29);

a(n) = [x^(2n)] (Sum_{j>=1} mu(j)*x^j/(1-x^j)^2)^n.
a(n) = A340995(2n,n).
a(n) ~ c * d^n / sqrt(n), where d = 5.0117569538757703168577972551675369123003378927616324330274512382246419... and c = 0.287455327702489527773675891801880332800309441856159133456758815116... - Vaclav Kotesovec, Aug 18 2021

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A159929 INVERT transform of phi(n), A000010.

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A065093 Convolution of A000010 with itself.

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A340994 a(n) is the (2n)-th term of the n-fold self-convolution of the Euler totient function phi.

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