A377734 Number of integers less than n that have the same smallest prime factor as n.

0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 5, 0, 6, 2, 7, 0, 8, 0, 9, 3, 10, 0, 11, 1, 12, 4, 13, 0, 14, 0, 15, 5, 16, 2, 17, 0, 18, 6, 19, 0, 20, 0, 21, 7, 22, 0, 23, 1, 24, 8, 25, 0, 26, 3, 27, 9, 28, 0, 29, 0, 30, 10, 31, 4, 32, 0, 33, 11, 34, 0, 35, 0, 36, 12, 37, 2, 38, 0, 39

Offset

1,6

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Least Prime Factor.

Formula

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

a(n) = A078898(n) - 1.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{k>=1} (A038110(k)/A038111(k))^2 = 0.2847976823663... . - Amiram Eldar, Nov 21 2024

Mathematica

Table[Length[Select[Range[n - 1], If[# == 1, 1, FactorInteger[#][[1, 1]]] == If[n == 1, 1, FactorInteger[n][[1, 1]]] &]], {n, 80}]
seq[len_] := Module[{t = Table[FactorInteger[n][[1, 1]], {n, 1, len}], s = Table[0, {len}]}, Do[s[[i]] = Count[t[[1;; i-1]], t[[i]]], {i, 1, len}]; s]; seq[80] (* Amiram Eldar, Nov 21 2024 *)

Prog

(PARI) a(n) = if (n>1, my(p=vecmin(factor(n)[, 1])); sum(k=2, n-1, p == vecmin(factor(k)[, 1])), 0); \\ Michel Marcus, Nov 16 2024

Crossrefs

Cf. A020639, A038110, A038111, A047983, A078898, A334655, A335097, A377730.

Sequence in context: A355526 A243056 A338568 * A279119 A249738 A364020

Adjacent sequences: A377731 A377732 A377733 * A377735 A377736 A377737

Keyword

nonn,new

Author

Ilya Gutkovskiy, Nov 05 2024

Status

approved