A332791 -id:A332791 - cp's OEIS frontend

A332792 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(d) * a(d).

1, 1, 1, 2, 1, 4, 1, 6, 3, 6, 1, 16, 1, 8, 7, 30, 1, 30, 1, 34, 9, 12, 1, 104, 5, 14, 21, 60, 1, 96, 1, 270, 13, 18, 11, 278, 1, 20, 15, 330, 1, 174, 1, 136, 81, 24, 1, 1176, 7, 130, 19, 186, 1, 588, 15, 804, 21, 30, 1, 1204, 1, 32, 135, 4590, 17, 402, 1, 310, 25, 348

Offset: 1

Ilya Gutkovskiy, Feb 24 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A006874, A008578 (positions of 1's), A038045, A057660, A332791.

a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
a[1] = 1; a[n_] := a[n] = Sum[If[GCD[n, k] > 1, a[n/GCD[n, k]], 0], {k, 1, n}]; Table[a[n], {n, 1, 70}]

up_to = 20000;
A332792list(n) = { my(v=vector(n)); v[1] = 1; for(n=2, #v, v[n] = sumdiv(n, d, if(d==n, 0, v[d]*eulerphi(d)))); (v); };
v332792 = A332792list(up_to);
A332792(n) = v332792[n]; \\ Antti Karttunen, Jan 22 2025

a(1) = 1; a(n) = Sum_{k=1..n, gcd(n, k) > 1} a(n/gcd(n, k)).

A333613 a(1) = 1; thereafter a(n) = Sum_{k = 1..n} a(k/gcd(n,k)).

Original entry on oeis.org

1, 2, 4, 7, 15, 21, 51, 78, 158, 230, 568, 661, 1797, 2595, 5117, 7789, 19095, 21702, 59892, 81801, 171329, 258028, 630942, 713093, 1887828, 2776798, 5727675, 8335692, 20702970, 21420664, 62826604, 92041835, 189376593, 281410640, 656577018, 742729123, 2087788417

Offset: 1

Ilya Gutkovskiy, Mar 28 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..3930
Vaclav Kotesovec, Plot of a(n)^(1/n) for n = 1..10000

Cf. A006874, A045545, A057661, A332791.

A333613:= proc(n) option remember;
if n<3 then n;
else add( A333613(lcm(n,j)/n), j = 1..n);
end if; end proc;
seq(A333613(n), n=1..40); # G. C. Greubel, Mar 08 2021

a[1] = 1; a[n_] := a[n] = Sum[a[k/GCD[n, k]], {k, n}]; Table[a[n], {n, 37}]
a[1] = 1; a[n_] := a[n] = Sum[Sum[If[GCD[k, d] == 1, a[k], 0], {k, d}], {d, Divisors[n]}]; Table[a[n], {n, 37}]

@CachedFunction
def A333613(n): return 1 if n==1 else sum( A333613(lcm(n, j)/n) for j in (1..n) )
[A333613(n) for n in (1..40)] # G. C. Greubel, Mar 08 2021

a(1) = 1; a(n) = Sum_{k = 1..n} a(lcm(n, k)/n).

a(1) = 1; a(n) = Sum_{d|n} Sum_{k = 1..d, gcd(d, k) = 1} a(k).

A341639 a(1) = 1; a(n+1) = Sum_{d|n} phi(d) * a(d) * a(n/d).

Original entry on oeis.org

1, 1, 2, 6, 19, 95, 291, 2037, 10203, 71429, 357240, 3929640, 19648533, 255430929, 1788018540, 16092167088, 144829514049, 2462101738833, 17234712244012, 327459532636228, 2947135794083881, 38312765323095109, 421440418557975839, 9693129626833444297, 87238166641520673597

Offset: 1

Ilya Gutkovskiy, Feb 16 2021

nonn

Cf. A000010, A038044, A038045, A332791.

a[1] = 1; a[n_] := a[n] = Sum[EulerPhi[d] a[d] a[(n - 1)/d], {d, Divisors[n - 1]}]; Table[a[n], {n, 25}]
a[1] = 1; a[n_] := a[n] = Sum[a[GCD[n - 1, k]] a[(n - 1)/GCD[n - 1, k]], {k, n - 1}]; Table[a[n], {n, 25}]

a(1) = 1; a(n+1) = Sum_{k=1..n} a(gcd(n,k)) * a(n/gcd(n,k)).

cp's OEIS Frontend

A332792 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(d) * a(d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A333613 a(1) = 1; thereafter a(n) = Sum_{k = 1..n} a(k/gcd(n,k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A341639 a(1) = 1; a(n+1) = Sum_{d|n} phi(d) * a(d) * a(n/d).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula