A082588 -id:A082588 - cp's OEIS frontend

A348955 a(1) = 1; a(n) = Sum_{d|n, d <= sqrt(n)} a(d)^2.

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 6, 1, 3, 1, 6, 2, 2, 1, 7, 2, 2, 2, 6, 1, 4, 1, 6, 2, 2, 2, 11, 1, 2, 2, 7, 1, 7, 1, 6, 3, 2, 1, 11, 2, 3, 2, 6, 1, 7, 2, 7, 2, 2, 1, 12, 1, 2, 3, 10, 2, 7, 1, 6, 2, 4, 1, 15, 1, 2, 3, 6, 2, 7, 1, 11, 6, 2, 1, 12, 2, 2, 2, 10, 1, 12

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

nonn

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

Cf. A008578 (positions of 1's), A067868, A068108, A082588, A337135, A348956.

a[1] = 1; a[n_] := a[n] = DivisorSum[n, a[#]^2 &, # <= Sqrt[n] &]; Table[a[n], {n, 90}]

A348955(n) = if(1==n,n,sumdiv(n,d,if((d*d)<=n,A348955(d)^2,0))); \\ Antti Karttunen, Nov 05 2021

G.f.: Sum_{k>=1} a(k)^2 * x^(k^2) / (1 - x^k).

a(4^n) = A067868(n).

A343511 a(n) = 1 + Sum_{d|n, d < n} a(d)^2.

Original entry on oeis.org

1, 2, 2, 6, 2, 10, 2, 42, 6, 10, 2, 146, 2, 10, 10, 1806, 2, 146, 2, 146, 10, 10, 2, 23226, 6, 10, 42, 146, 2, 314, 2, 3263442, 10, 10, 10, 42814, 2, 10, 10, 23226, 2, 314, 2, 146, 146, 10, 2, 542731938, 6, 146, 10, 146, 2, 23226, 10, 23226, 10, 10, 2, 141578, 2, 10, 146, 10650056950806, 10

Offset: 1

Ilya Gutkovskiy, Apr 17 2021

nonn

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, Apr 24 2021

Alois P. Heinz, Table of n, a(n) for n = 1..3000
Index entries for sequences related to prime signature

Cf. A025487, A006881, A007018, A007304, A067824, A082588, A333120.

a:= proc(n) option remember;
      1+add(a(d)^2, d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 17 2021

a[n_] := a[n] = 1 + Sum[If[d < n, a[d]^2, 0], {d, Divisors[n]}]; Table[a[n], {n, 65}]

lista(nn) = {my(va = vector(nn)); for (n=1, nn, va[n] = 1 + sumdiv(n, d, if (dMichel Marcus, Apr 18 2021

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A343511(n): return 1+sum(A343511(d)**2 for d in divisors(n) if d < n) # Chai Wah Wu, Apr 17 2021

G.f.: x / (1 - x) + Sum_{n>=1} a(n)^2 * x^(2*n) / (1 - x^n).
a(p^k) = A007018(k) for p prime.
From Bernard Schott, Apr 24 2021: (Start)
a(A006881(n)) = 10 for signature [1, 1].
a(A054753(n)) = 146 for signature [2, 1].
a(A007304(n)) = 314 for signature [1, 1, 1].
a(A065036(n)) = 23226 for signature [3, 1].
a(A085986(n)) = 42814 for signature [2, 2].
a(A085987(n)) = 141578 for signature [2, 1, 1]. (End)

A333120 a(1) = 1, a(n+1) = Sum_{d|n} a(d)^2.

Original entry on oeis.org

1, 1, 2, 5, 27, 730, 532906, 283988804837, 80649641272747674596596, 6504364637422885153991868441922618991334787221, 42306759336557340251452848811862268800945638555088445693345741249091605099189510505344903572

Offset: 1

Ilya Gutkovskiy, Mar 08 2020

nonn

Cf. A003238, A007018, A082588.

a[1] = 1; a[n_] := a[n] = Sum[a[d]^2, {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 11}]

a(n) ~ c^(2^n), where c = 1.10850959824748299278192262396861284785509520261602375885447280659977154... - Vaclav Kotesovec, Mar 08 2020

A332778 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * a(d)^2.

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 6, 15, 14, 24, 10, 96, 12, 48, 56, 255, 16, 344, 18, 656, 108, 120, 22, 9840, 84, 168, 434, 2448, 28, 4608, 30, 65535, 260, 288, 264, 137376, 36, 360, 360, 432512, 40, 16776, 42, 14720, 7208, 528, 46, 96974880, 258, 9464, 608, 28656, 52, 425864, 600

Offset: 1

Ilya Gutkovskiy, Feb 23 2020

nonn

Cf. A000010, A006874, A007018, A023900, A069097, A082588.

a[1] = 1; a[n_] := Sum[If[d < n, EulerPhi[n/d] a[d]^2, 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
a[1] = 1; a[n_] := Sum[a[GCD[n, k]]^2, {k, 1, n - 1}]; Table[a[n], {n, 1, 55}]

a(1) = 1; a(n) = Sum_{k=1..n-1} a(gcd(n, k))^2.

cp's OEIS Frontend

A348955 a(1) = 1; a(n) = Sum_{d|n, d <= sqrt(n)} a(d)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A343511 a(n) = 1 + Sum_{d|n, d < n} a(d)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A333120 a(1) = 1, a(n+1) = Sum_{d|n} a(d)^2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A332778 a(1) = 1; a(n) = Sum_{d|n, d < n} phi(n/d) * a(d)^2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula