A380654 Number of positive integers less than or equal to n that have the same sum of distinct prime factors as n.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 1, 4, 1, 3, 1, 2, 1, 5, 6, 1, 3, 2, 1, 2, 1, 5, 1, 2, 1, 7, 1, 1, 1, 4, 1, 2, 1, 3, 2, 1, 1, 8, 5, 6, 1, 2, 1, 9, 2, 3, 1, 2, 1, 3, 1, 1, 4, 6, 1, 3, 1, 3, 1, 2, 1, 10, 1, 1, 3, 2, 2, 3, 1, 7, 4, 2, 1, 3, 2, 1, 1, 4, 1, 5, 2, 2, 1, 1, 1, 11, 1, 4, 3, 8

Offset

1,4

Comments

Ordinal transform of A008472.

Links

Table of n, a(n) for n=1..100.

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

Formula

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

Maple

b:= n-> add(i[1], 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

Mathematica

sopf[n_] := DivisorSum[n, # &, PrimeQ[#] &]; Table[Length[Select[Range[n], sopf[#] == sopf[n] &]], {n, 1, 100}]

Prog

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

Crossrefs

Cf. A008472, A067003, A263025, A380653.

Sequence in context: A305302 A340382 A055181 * A325568 A326195 A073811

Adjacent sequences: A380651 A380652 A380653 * A380655 A380656 A380657

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 29 2025

Status

approved