A005064 -id:A005064 - cp's OEIS frontend

A347159 Sum of cubes of distinct prime divisors of n that are <= sqrt(n).

0, 0, 0, 8, 0, 8, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 35, 125, 8, 27, 8, 0, 160, 0, 8, 27, 8, 125, 35, 0, 8, 27, 133, 0, 35, 0, 8, 152, 8, 0, 35, 343, 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, A063962, A097974, A098002, A280375, A347157, A347160.

Table[DivisorSum[n, #^3 &, # <= Sqrt[n] && PrimeQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[Prime[k]^3 x^(Prime[k]^2)/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^3 * x^(prime(k)^2) / (1 - x^prime(k)).

A384815 Sum of the cubes of the exponents in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 8, 1, 2, 1, 27, 8, 2, 1, 9, 1, 2, 2, 64, 1, 9, 1, 9, 2, 2, 1, 28, 8, 2, 27, 9, 1, 3, 1, 125, 2, 2, 2, 16, 1, 2, 2, 28, 1, 3, 1, 9, 9, 2, 1, 65, 8, 9, 2, 9, 1, 28, 2, 28, 2, 2, 1, 10, 1, 2, 9, 216, 2, 3, 1, 9, 2, 3, 1, 35, 1, 2, 9, 9, 2, 3, 1, 65, 64, 2, 1, 10, 2, 2, 2, 28, 1, 10

Offset: 1

Ilya Gutkovskiy, Jun 10 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. L. Duncan, A class of additive arithmetical functions, The American Mathematical Monthly, Vol. 69, No. 1 (1962), pp. 34-36.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001222, A001620, A005064, A090885, A360970.

Join[{0}, Table[Plus @@ (#[[2]]^3 & /@ FactorInteger[n]), {n, 2, 90}]]

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2]^3); \\ Michel Marcus, Jun 10 2025

If n = Product (p_j^k_j) then a(n) = Sum (k_j^3).
From Amiram Eldar, Jul 03 2025: (Start)
Additive with a(p^e) = e^3.
Sum_{k=1..n} a(k) ~ n * log(log(n)) + B_3 * n + O(n/log(n)), where B_3 = gamma + Sum_{p prime} ((1-1/p)*Sum_{m>=1} m^3/p^m + log(1-1/p)) = 16.17021843694072992072..., and gamma is Euler's constant (A001620) (Duncan, 1962). (End)

A380098 Numbers whose sum of cubes of distinct prime factors is prime.

Original entry on oeis.org

165, 210, 390, 399, 420, 462, 495, 561, 570, 595, 615, 630, 651, 780, 798, 825, 840, 924, 957, 1050, 1085, 1140, 1170, 1173, 1197, 1218, 1235, 1245, 1260, 1302, 1386, 1435, 1470, 1482, 1485, 1495, 1554, 1560, 1596, 1615, 1680, 1683, 1705, 1710, 1767, 1771, 1815, 1845, 1848, 1885, 1890, 1938, 1950, 1953

Offset: 1

Rafik Khalfi, Jan 12 2025

nonn

			165=3*5*11 and 3^3 + 5^3 + 11^3 = 1483, which is prime. Therefore, 165 is included.

Cf. A005064, A114522, A114989.

a:=proc(n) local DPF: DPF:=factorset(n): if isprime(sum(DPF[j]^3, j=1..nops(DPF)))=true then n else fi end: seq(a(n), n=1..2000);

Select[Range[2000], PrimeQ[Total[Transpose[FactorInteger[#]][[1]]^3]]&]

from sympy import isprime, factorint
def ok(n): return isprime(sum(p**3 for p in factorint(n)))
print([k for k in range(2000) if ok(k)]) # Michael S. Branicky, Jan 12 2025

A384816 Sum of the cubes of the indices of distinct prime factors of n.

Original entry on oeis.org

0, 1, 8, 1, 27, 9, 64, 1, 8, 28, 125, 9, 216, 65, 35, 1, 343, 9, 512, 28, 72, 126, 729, 9, 27, 217, 8, 65, 1000, 36, 1331, 1, 133, 344, 91, 9, 1728, 513, 224, 28, 2197, 73, 2744, 126, 35, 730, 3375, 9, 64, 28, 351, 217, 4096, 9, 152, 65, 520, 1001, 4913, 36, 5832, 1332, 72, 1, 243, 134, 6859, 344, 737, 92

Offset: 1

Ilya Gutkovskiy, Jun 10 2025

nonn

Index entries for sequences computed from indices in prime factorization.

Cf. A000720, A005064, A066328, A332385.

Table[Plus @@ (PrimePi[#[[1]]]^3 & /@ FactorInteger[n]), {n, 70}]
nmax = 70; CoefficientList[Series[Sum[k^3 x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = my(f=factor(n)[,1]); sum(k=1, #f~, primepi(f[k])^3); \\ Michel Marcus, Jun 10 2025

If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^3), where pi = A000720.

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

cp's OEIS Frontend

A347159 Sum of cubes of distinct prime divisors of n that are <= sqrt(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A384815 Sum of the cubes of the exponents in the prime factorization of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A380098 Numbers whose sum of cubes of distinct prime factors is prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

A384816 Sum of the cubes of the indices of distinct prime factors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula