A069293 -id:A069293 - cp's OEIS frontend

A069291 Number of square divisors of n <= sqrt(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2

Offset: 1

Reinhard Zumkeller, Mar 14 2002

nonn

Terms 1, 2, 3, ... occurs for the first time at 1, 16, 108, 288, 1296, 3600, 10368, 14400, ... - Antti Karttunen, Nov 20 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000188, A035316, A046951, A069290, A069292, A069293.

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, # <= Sqrt@ n] &], {n, 120}] (* Michael De Vlieger, Nov 20 2017 *)

A069291(n) = sumdiv(n, d, (issquare(d)&&((d^2)<=n))); \\ Antti Karttunen, Nov 20 2017

G.f.: Sum_{k>=1} x^(k^4) / (1 - x^(k^2)). - Ilya Gutkovskiy, Apr 04 2020

More terms from Antti Karttunen, Nov 20 2017

A333751 Sum of nonprime divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 11, 1, 1, 1, 5, 1, 7, 1, 5, 1, 1, 1, 11, 1, 1, 1, 5, 1, 7, 1, 5, 1, 1, 1, 11, 1, 1, 1, 13, 1, 7, 1, 5, 1, 1, 1, 19, 1, 1, 1, 5, 1, 7, 1, 13, 10, 1, 1, 11, 1, 1, 1, 13, 1, 16

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A018252, A023890, A066839, A069289, A069293, A097974, A333748, A333752, A333753.

f:= proc(n) convert(select(t -> not isprime(t) and t^2 <= n, numtheory:-divisors(n)),`+`) end proc:
map(f, [$1..100]); # Robert Israel, Sep 12 2024

Table[DivisorSum[n, # &, # <= Sqrt[n] && !PrimeQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] k x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d^2<=n) && !isprime(d), d)); \\ Michel Marcus, Apr 03 2020

G.f.: Sum_{k>=1} A018252(k) * x^(A018252(k)^2) / (1 - x^A018252(k)).

A333753 Sum of prime power divisors of n that are <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 5, 0, 2, 3, 6, 0, 5, 0, 6, 3, 2, 0, 9, 5, 2, 3, 6, 0, 10, 0, 6, 3, 2, 5, 9, 0, 2, 3, 11, 0, 5, 0, 6, 8, 2, 0, 9, 7, 7, 3, 6, 0, 5, 5, 13, 3, 2, 0, 14, 0, 2, 10, 14, 5, 5, 0, 6, 3, 14, 0, 17, 0, 2, 8, 6, 7, 5, 0, 19, 12, 2, 0, 16, 5, 2, 3, 14, 0, 19

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A023889, A066839, A069289, A069293, A097974, A246655, A333750, A333751, A333752.

f:= proc(n) local F,i,j,t;
  F:= ifactors(n)[2];
  t:= 0;
  for i from 1 to nops(F) do
    j:= min(F[i,2],ilog[F[i,1]^2](n));
    t:= t + (F[i,1]^j-1)*F[i,1]/(F[i,1]-1)
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Feb 15 2023

Table[DivisorSum[n, # &, # <= Sqrt[n] && PrimePowerQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] k x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d^2<=n) && isprimepower(d), d)); \\ Michel Marcus, Apr 03 2020

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

A069292 Sum of square roots of square divisors of n <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1

Offset: 1

Reinhard Zumkeller, Mar 14 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A035316, A046951, A000188, A069290, A069291, A069293.

Table[DivisorSum[n, Sqrt@ # &, And[IntegerQ@ Sqrt@ #, # <= Sqrt@ n] &], {n, 105}] (* Michael De Vlieger, Nov 20 2017 *)

A069292(n) = { my(r="NA"); sumdiv(n, d, (issquare(d,&r)&&((d^2)<=n))*r); } \\ Antti Karttunen, Nov 20 2017

G.f.: Sum_{k>=1} k * x^(k^4) / (1 - x^(k^2)). - Ilya Gutkovskiy, Aug 19 2021

More terms from Antti Karttunen, Nov 20 2017

A333752 Sum of squarefree divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 6, 3, 4, 3, 1, 11, 1, 3, 4, 3, 6, 12, 1, 3, 4, 8, 1, 12, 1, 3, 9, 3, 1, 12, 8, 8, 4, 3, 1, 12, 6, 10, 4, 3, 1, 17, 1, 3, 11, 3, 6, 12, 1, 3, 4, 15, 1, 12, 1, 3, 9, 3, 8, 12, 1, 8, 4, 3, 1, 19, 6, 3, 4, 3, 1, 17

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

Cf. A005117, A048250, A066839, A069289, A069293, A097974, A333749, A333751, A333753.

Table[DivisorSum[n, # &, # <= Sqrt[n] && SquareFreeQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[MoebiusMu[k]^2 k x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d^2<=n) && issquarefree(d), d)); \\ Michel Marcus, Apr 03 2020

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

cp's OEIS Frontend

A069291 Number of square divisors of n <= sqrt(n).

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A333751 Sum of nonprime divisors of n that are <= sqrt(n).

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A333753 Sum of prime power divisors of n that are <= sqrt(n).

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A069292 Sum of square roots of square divisors of n <= sqrt(n).

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A333752 Sum of squarefree divisors of n that are <= sqrt(n).

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