A333808 -id:A333808 - cp's OEIS frontend

A333806 Number of distinct prime divisors of n that are < sqrt(n).

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

nonn

a(n) = 0 if and only if n = p^k where p is prime and k is 0, 1, or 2. - Charles R Greathouse IV, Apr 07 2020

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

Cf. A001221, A056924, A063962, A333805, A333808.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
p:= 1:
do
  p:= nextprime(p);
  if p^2 >= N then break fi;
  L:= [seq(p*k,k=p+1..N/p)];
  V[L]:= V[L]+~1
od:
convert(V,list); # Robert Israel, Apr 07 2020

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

a(n)=my(f=factor(n)[,1]); sum(i=1,#f, f[i]^2Charles R Greathouse IV, Apr 07 2020

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

a(k*n) >= a(n) for k > 0. a(n^e) = A001221(n) for e > 2. - Richard Peterson, Dec 19 2024

A333807 Sum of odd divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 8, 4, 1, 1, 9, 1, 1, 11, 1, 6, 4, 1, 1, 4, 13, 1, 4, 1, 1, 9, 1, 8, 4, 1, 6, 4, 1, 1, 11, 6, 1, 4, 1, 1, 18

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

nonn

Cf. A000593, A069289, A070039, A333805, A333808, A333810.

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

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

A347157 Sum of cubes of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 0, 8, 0, 8, 0, 35, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 160, 0, 8, 27, 8, 125, 35, 0, 8, 27, 133, 0, 35, 0, 8, 152, 8, 0, 35, 0, 133, 27, 8, 0, 35, 125, 351, 27, 8, 0, 160, 0, 8, 370, 8, 125, 35, 0, 8, 27, 476, 0, 35, 0, 8, 152

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001158, A005064, A005067, A333806, A333808, A339354, A347156, A347158.

Table[DivisorSum[n, #^3 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[Prime[k]^3 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^3 * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

A347158 Sum of 4th powers of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 16, 0, 16, 0, 16, 0, 97, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 97, 0, 16, 81, 16, 0, 722, 0, 16, 81, 16, 625, 97, 0, 16, 81, 641, 0, 97, 0, 16, 706, 16, 0, 97, 0, 641, 81, 16, 0, 97, 625, 2417, 81, 16, 0, 722, 0, 16, 2482, 16, 625, 97, 0, 16, 81, 3042

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001159, A005065, A005068, A333806, A333808, A347142, A347156, A347157.

Table[DivisorSum[n, #^4 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k]^4 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^4 * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

A347156 Sum of squares of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 4, 0, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 53, 9, 4, 0, 38, 0, 4, 58, 4, 25, 13, 0, 4, 9, 78, 0, 13, 0, 4, 34, 4, 49, 13, 0, 29

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001157, A005063, A005066, A098002, A333806, A333808, A339353, A347157, A347158.

Table[DivisorSum[n, #^2 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[Prime[k]^2 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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

cp's OEIS Frontend

A333806 Number of distinct prime divisors of n that are < sqrt(n).

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A333807 Sum of odd divisors of n that are < sqrt(n).

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A347157 Sum of cubes of distinct prime divisors of n that are < sqrt(n).

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A347158 Sum of 4th powers of distinct prime divisors of n that are < sqrt(n).

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A347156 Sum of squares of distinct prime divisors of n that are < sqrt(n).

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