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

Author

Ilya Gutkovskiy, Sep 23 2022

Keywords

• nonn

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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