A333753 -id:A333753 - cp's OEIS frontend

A333750 Number of prime power divisors of n that are <= sqrt(n).

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 3, 0, 2, 1, 1, 1, 3, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 3, 1, 2, 1, 2, 0, 2, 1, 3, 1, 1, 0, 4, 0, 1, 2, 3, 1, 2, 0, 2, 1, 3, 0, 4, 0, 1, 2, 2, 1, 2, 0, 4, 2, 1, 0, 4, 1, 1, 1, 3, 0, 4, 1, 2, 1, 1, 1, 4, 0, 2, 2, 3

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

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

Cf. A001222, A038548, A063962, A069288, A069291, A073093, A246655, A333748, A333749, A333753.

f:= proc(n) local p;
  add(min(padic:-ordp(n,p),floor(1/2*log[p](n))),p=numtheory:-factorset(n))
end proc:
map(f, [$1..200]); # Robert Israel, Apr 22 2020

Table[DivisorSum[n, 1 &, # <= Sqrt[n] && PrimePowerQ[#] &], {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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

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

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)).

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

A333750 Number of prime power divisors of n that are <= sqrt(n).

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

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

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