A007898 -id:A007898 - cp's OEIS frontend

A007896 Psi_c(n), where Product_{k>1} 1/(1-1/k^s)^phi(k) = Sum_{k>0} psi_c(k)/k^s.

1, 1, 2, 3, 4, 4, 6, 7, 9, 8, 10, 12, 12, 12, 16, 18, 16, 19, 18, 24, 24, 20, 22, 32, 30, 24, 34, 36, 28, 40, 30, 42, 40, 32, 48, 60, 36, 36, 48, 64, 40, 60, 42, 60, 76, 44, 46, 86, 63, 66, 64, 72, 52, 82, 80, 96, 72, 56, 58, 128, 60, 60, 114, 104, 96, 100

Offset: 1

Felix Weinstein (wain(AT)ana.unibe.ch)

nonn

Phi(k) is the Euler totient function A000010.

			The left-hand side (a Dirichlet generating function) is
1/((1-1/2^s)*(1-1/3^s)^2*(1-1/4^s)^2*(1-1/5^s)^4*(1-1/6^s)^2*(1-1/7^s)^6* ...)
= 1 + 1/2^s + 2/3^s + 3/4^s + 4/5^s + 4/6^s + 6/7^s + 7/8^s + 9/9^s + ...,
whose coefficients are 1, 1, 2, 3, 4, 4, 6, 7, 9, ... . - _N. J. A. Sloane_, May 26 2014
G.f. = x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 7*x^8 + 9*x^9 + ...

Felix Weinstein, The Fibonacci Partitions, preprint, 1995

N. J. A. Sloane, Table of n, a(n) for n = 1..1000
F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2015.

Cf. A000010, A007897, A007898.

dircon[v_, w_] := Module[{lv = Length[v], lw = Length[w], fv, fw}, fv[n_] := If[n <= lv, v[[n]], 0]; fw[n_] := If[n <= lw, w[[n]], 0]; Table[ DirichletConvolve[fv[n], fw[n], n, m], {m, Min[lv, lw]}]];
a[n_] := Module[{A, v, w, m}, If[n<1, 0, v = Table[Boole[k == 1], {k, n}]; For[k = 2, k <= n, k++, m = Length[IntegerDigits[n, k]] - 1; A = (1 - x)^-EulerPhi[k] + x*O[x]^m // Normal; w = Table[0, {n}]; For[i = 0, i <= m, i++, w[[k^i]] = Coefficient[A, x, i]]; v = dircon[v, w]]; v[[n]]]];
Array[a, 66] (* Jean-François Alcover, Nov 12 2018, from PARI *)

{a(n) = my(A, v, w, m); if( n<1, 0, v = vector(n, k, k==1); for(k=2, n, m = #digits(n, k) - 1; A = (1 - x)^ -eulerphi(k) + x * O(x^m); w = vector(n); for(i=0, m, w[k^i] = polcoeff(A, i)); v = dirmul(v, w)); v[n])}; /* Michael Somos, May 26 2014 */

Definition corrected by Felix Weinstein (wain(AT)ana.unibe.ch), May 14 2014

A007897 a(n) is multiplicative with a(2) = 1; a(4) = 2; a(2^i) = 2^(i-2)+2 if i>2; a(p^i) = 1+(p-1)*p^(i-1)/2 if prime p>2 and i>0.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 4, 4, 3, 6, 4, 7, 4, 6, 6, 9, 4, 10, 6, 8, 6, 12, 8, 11, 7, 10, 8, 15, 6, 16, 10, 12, 9, 12, 8, 19, 10, 14, 12, 21, 8, 22, 12, 12, 12, 24, 12, 22, 11, 18, 14, 27, 10, 18, 16, 20, 15, 30, 12, 31, 16, 16, 18, 21, 12, 34, 18, 24, 12, 36, 16, 37, 19, 22, 20, 24, 14, 40, 18, 28

Offset: 1

Felix Weinstein (wain(AT)ana.unibe.ch), Dec 11 1999

From Jeffrey Shallit, Jun 14 2018: (Start)
Except for first term, the same as A180783.
Equal to the number of elements x relatively prime to n such that x mod n >= x^(-1) mod n. (End)

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 2*x^6 + 4*x^7 + 4*x^8 + 4*x^9 + ...

Felix Weinstein, The Fibonacci Partitions, preprint, 1995.

F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2018.

Cf. A007896, A007898, A180783.

a[ n_] := If[ n < 2, Boole[ n == 1],Times @@ Apply[ Function[ {p, e}, If[p == 2, If[e < 3, e, 2^(e - 2) + 2], 1 + p^(e - 1) (p - 1)/2]], FactorInteger @ n, 1]]; (* Michael Somos, May 26 2014 *)

ap(p, e) = if (p==2, if (e==1, 1, if (e==2, 2, 2^(e-2)+2)), 1+(p-1)*p^(e-1)/2);
a(n) = { my(f = factor(n)); prod(i=1, #f~, ap(f[i,1], f[i, 2]));} \\ Michel Marcus, Apr 19 2014

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k,1], e = A[k,2]; if( p==2, if( e<3, e, 2^(e-2) + 2), 1 + p^(e-1) * (p-1) / 2))))}; /* Michael Somos, May 26 2014 */

{a(n) = if( n<1, 0, direuler( p = 2, n, if( p>2, 1 / (1 - X) + (p - 1) / 2 * X / (1 - p*X), (1 + X^2) / (1 - X) + p * X^3 / (1 - p*X))) [n])}; /* Michael Somos, May 26 2014 */

Dirichlet g.f.: zeta(s) * zeta(s-1) * ((2 - 2^(s+2) + 2^(2*s+1) - 1/2^(2*s-2))/(2^(2*s+1) - 3*2^s - 1)) * Product_{p prime} (1 - (1/p^(s-1) + 1/p^s - 1/p^(2*s-1) + 1/p^(2*s))/2). - Amiram Eldar, Nov 09 2023

Definition corrected by Michel Marcus, Apr 19 2014

Changed name from phi(n) (which caused much confusion with the Euler phi-function) to a(n). - N. J. A. Sloane, May 26 2014

A175378 G.f. x^4(2x^2-1)/( (x^2-1)(x^2+x-1)(2x^3-2x^2+2*x-1) ).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 5, 8, 14, 26, 45, 75, 125, 212, 358, 598, 993, 1651, 2745, 4552, 7526, 12426, 20501, 33787, 55605, 91404, 150118, 246350, 403929, 661763, 1083393, 1772512, 2898182, 4735938, 7734765, 12626059, 20600733, 33597188, 54769606

Offset: 0

R. J. Mathar, Apr 24 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. V. Weinstein, Notes on Fibonacci Partitions, arXiv:math/0307150, variable h(n).
Index entries for linear recurrences with constant coefficients, signature (3, -2, -1, 3, -4, 0, 2).

Cf. A007896, A007897, A007898.

I:=[0, 0, 0, 0, 1, 3, 5]; [n le 7 select I[n] else 3*Self(n-1) - 2*Self(n-2) - Self(n-3) + 3*Self(n-4) - 4*Self(n-5) + 2*Self(n-7): n in [1..40]]; // Vincenzo Librandi, Dec 20 2012

LinearRecurrence[{3,-2,-1,3,-4,0,2},{0,0,0,0,1,3,5},40] (* Harvey P. Dale, Mar 07 2012 *)
CoefficientList[Series[x^4*(2*x^2 - 1)/((x^2 - 1)*(x^2 + x - 1)*(2*x^3 - 2*x^2 + 2*x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 20 2012 *)

a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +3*a(n-4) -4*a(n-5) +2*a(n-7).

cp's OEIS Frontend

A007896 Psi_c(n), where Product_{k>1} 1/(1-1/k^s)^phi(k) = Sum_{k>0} psi_c(k)/k^s.

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A007897 a(n) is multiplicative with a(2) = 1; a(4) = 2; a(2^i) = 2^(i-2)+2 if i>2; a(p^i) = 1+(p-1)*p^(i-1)/2 if prime p>2 and i>0.

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A175378 G.f. x^4(2x^2-1)/( (x^2-1)(x^2+x-1)(2x^3-2x^2+2*x-1) ).

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cp's OEIS Frontend

A007896 Psi_c(n), where Product_{k>1} 1/(1-1/k^s)^phi(k) = Sum_{k>0} psi_c(k)/k^s.

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Programs

Mathematica

PARI

Extensions

A007897 a(n) is multiplicative with a(2) = 1; a(4) = 2; a(2^i) = 2^(i-2)+2 if i>2; a(p^i) = 1+(p-1)*p^(i-1)/2 if prime p>2 and i>0.

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A175378 G.f. x^4*(2*x^2-1)/( (x^2-1)*(x^2+x-1)*(2*x^3-2*x^2+2*x-1) ).

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A175378 G.f. x^4(2x^2-1)/( (x^2-1)(x^2+x-1)(2x^3-2x^2+2*x-1) ).