A058933 -id:A058933 - cp's OEIS frontend

A357310 a(n) is the number of j in the range 1 <= j <= n with the same maximal exponent in prime factorization as n.

1, 1, 2, 1, 3, 4, 5, 1, 2, 6, 7, 3, 8, 9, 10, 1, 11, 4, 12, 5, 13, 14, 15, 2, 6, 16, 3, 7, 17, 18, 19, 1, 20, 21, 22, 8, 23, 24, 25, 4, 26, 27, 28, 9, 10, 29, 30, 2, 11, 12, 31, 13, 32, 5, 33, 6, 34, 35, 36, 14, 37, 38, 15, 1, 39, 40, 41, 16, 42, 43, 44, 7, 45, 46, 17, 18, 47, 48, 49, 3

Offset: 1

Ilya Gutkovskiy, Sep 23 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000079 (positions of 1's), A051903, A058933, A289023.

f:= proc(n) option remember; `if`(n=1, 0,
      max(map(i-> i[2], ifactors(n)[2])))
    end:
b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+x^f(n)) end:
a:= n-> coeff(b(n), x, f(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 23 2022

Table[Length[Select[Range[n], If[# == 1, 0, Max @@ Last /@ FactorInteger[#]] == If[n == 1, 0, Max @@ Last /@ FactorInteger[n]] &]], {n, 1, 80}]
seq[max_] := Module[{e = Join[{0}, Table[Max @@ FactorInteger[n][[;; , 2]], {n, 2, max}]], c = Table[0, {max}]}, Do[c[[k]] = 1 + Count[e[[1 ;; k - 1]], e[[k]]], {k, 1, max}]; c]; seq[100] (* Amiram Eldar, Jan 05 2024 *)

lista(nmax) = {my(e = vector(nmax, k, if(k==1, 0, vecmax(factor(k)[,2]))), c); for(k = 1, nmax, c =  1; for(j = 1, k-1, c += (e[j] == e[k])); print1(c, ", "));} \\ Amiram Eldar, Jan 05 2024

a(n) = |{j <= n : A051903(j) = A051903(n)}|.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1/zeta(2)^2 + Sum_{k>=3} (1/zeta(k+1) - 1/zeta(k))^2 = 0.43029326822775728041... . - Amiram Eldar, Jan 05 2024

A380653 Number of positive integers less than or equal to n that have the same sum of prime factors (with repetition) as n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 1, 4, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 1, 5, 2, 3, 1, 2, 1, 6, 2, 3, 1, 2, 1, 4, 1, 1, 4, 5, 1, 3, 1, 2, 1, 3, 1, 6, 1, 1, 5, 2, 2, 3, 1, 6, 7, 2, 1, 4, 2, 1, 1, 3, 1, 7, 2, 1, 1, 1, 1, 8, 1, 4, 4, 5

Offset: 1

Ilya Gutkovskiy, Jan 29 2025

nonn

Ordinal transform of A001414.

Eric Weisstein's World of Mathematics, Sum of Prime Factors.

Cf. A001414, A058933, A263025, A380654.

b:= n-> add(i[1]*i[2], i=ifactors(n)[2]):
p:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= b(n); p(t):= p(t)+1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 30 2025

sopfr[1] = 0; sopfr[n_] := Plus @@ Times @@@ FactorInteger@ n; Table[Length[Select[Range[n], sopfr[#] == sopfr[n] &]], {n, 1, 100}]

from sympy import factorint
from collections import Counter
from itertools import count, islice
def agen(): # generator of terms
    sopfrcount = Counter()
    for n in count(1):
        key = sum(p*e for p, e in factorint(n).items())
        sopfrcount[key] += 1
        yield sopfrcount[key]
print(list(islice(agen(), 100))) # Michael S. Branicky, Jan 30 2025

a(n) = |{j <= n : sopfr(j) = sopfr(n)}|.

A380049 Partial sums of A072203.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 12, 14, 15, 17, 20, 24, 27, 29, 30, 32, 35, 39, 44, 48, 51, 55, 58, 60, 61, 63, 66, 70, 75, 81, 88, 94, 99, 103, 106, 110, 113, 115, 116, 118, 121, 125, 130, 136, 141, 147, 154, 160, 167, 173, 180, 188, 195, 201, 206, 210, 213, 217, 220, 224, 227, 231, 234, 236

Offset: 1

Tsuyoshi Hanatate, Jan 10 2025

nonn

Vaclav Kotesovec, Plot of a(n) / (2*n^(3/2)/(-3*zeta(1/2))) for n = 1..1000000

Cf. A058933, A072203.

Accumulate[Accumulate[Table[-LiouvilleLambda[n], {n, 2, 100}]]] (* Vaclav Kotesovec, Jan 15 2025 *)

f(n) = 1 - sum(i=1, n, (-1)^bigomega(i)); \\ A072203
a(n) = sum(k=1, n, f(k)); \\ Michel Marcus, Feb 06 2025

a(n) = Sum_{k=1..n} A072203(k).

Conjecture: The average value of a(n) is 2*n^(3/2)/(-3*zeta(1/2)). - Vaclav Kotesovec, Jan 15 2025

cp's OEIS Frontend

A357310 a(n) is the number of j in the range 1 <= j <= n with the same maximal exponent in prime factorization as n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A380653 Number of positive integers less than or equal to n that have the same sum of prime factors (with repetition) as n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A380049 Partial sums of A072203.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula