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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, 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
1,3
LINKS
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
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 *)
PROG
(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
CROSSREFS
Cf. A000079 (positions of 1's), A051903, A058933, A289023.
Sequence in context: A332723 A210040 A349385 * A285329 A289023 A085985
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 23 2022
STATUS
approved