A336065 -id:A336065 - cp's OEIS frontend

A336064 Numbers divisible by the maximal exponent in their prime factorization (A051903).

2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

The asymptotic density of this sequence is A336065 = 0.848957... (Schinzel and Šalát, 1994).

			4 = 2^2 is a term since A051903(4) = 2 is a divisor of 4.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Schinzel and Tibor Šalát, Remarks on maximum and minimum exponents in factoring, Mathematica Slovaca, Vol. 44, No. 5 (1994), pp. 505-514.

Cf. A051903, A124184, A336065.
A005117 (except for 1) is subsequence.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336063.

H[1] = 0; H[n_] := Max[FactorInteger[n][[;; , 2]]]; Select[Range[2, 100], Divisible[#, H[#]] &]

isok(m) = if (m>1, (m % vecmax(factor(m)[,2])) == 0); \\ Michel Marcus, Jul 08 2020

A374586 The maximum exponent in the prime factorization of the numbers that are divisible by the maximum exponent in their prime factorization.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jul 12 2024

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

Cf. A051903, A336064, A336065, A374587.

f[n_] := Module[{e = If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]}, If[e > 0 && Divisible[n, e], e, Nothing]]; Array[f, 150]

lista(kmax) = {my(e); for(k = 2, kmax, e = vecmax(factor(k)[, 2]); if(!(k % e), print1(e, ", ")));}

a(n) = A051903(A336064(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) / Sum_{k>=1} d(k) = 1.46254458894503050564..., where d(k) = (1/zeta(k+1)) * Product_{p prime, p|k} ((p^(k-e(p,k)+1) - 1)/(p^(k+1) - 1)) - (1/zeta(k)) * Product_{p prime, p|k} ((p^(k-e(p,k)) - 1)/(p^k - 1)) for k >= 2, and d(1) = 1/zeta(2), and e(p,k) is the exponent of the largest of power of p that divides k.

A144976 Nonsquarefree numbers k such that k is divisible by the maximal exponent in the prime factorization of k.

Original entry on oeis.org

4, 12, 16, 18, 20, 24, 27, 28, 36, 44, 48, 50, 52, 54, 60, 68, 72, 76, 80, 84, 90, 92, 98, 100, 108, 112, 116, 120, 124, 126, 132, 135, 140, 144, 148, 150, 156, 160, 164, 168, 172, 176, 180, 188, 189, 192, 196, 198, 204, 208, 212, 216, 220, 228, 234, 236, 240, 242, 244, 252, 256, 260, 264, 268, 270, 272

Offset: 1

Giovanni Teofilatto, Sep 28 2008

nonn

The asymptotic density of this sequence is A336065 - A059956 = 0.24103009315... . - Amiram Eldar, Jan 05 2024

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Intersection of A013929 and A336064.

Cf. A059956, A137257, A336065.

A051903 := proc(n) local a,ifs,p,e; a := 1 ; max( seq(op(2,p),p=ifactors(n)[2]) ); end: isA013929 := proc(n) RETURN( not isprime(n) and A051903(n) > 1 ) ; end: isA144976 := proc(n) RETURN( isA013929(n) and (n mod A051903(n)) = 0 ); end: for n from 4 to 400 do if isA144976(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Oct 24 2008

Select[Range[300],!SquareFreeQ[#]&&Divisible[#,Max[FactorInteger[#][[All,2]]]]&] (* Harvey P. Dale, Jul 01 2017 *)

is(n) = {my(e = factor(n)[, 2], emax); if(n == 1, 0, emax = vecmax(e); emax > 1 && !(n % emax));} \\ Amiram Eldar, Jan 05 2024

{A013929(i): A051903(A013929(i)) | A013929(i)}. - R. J. Mathar, Oct 24 2008

Adapted definition, inserted 18, 20 and extended. - R. J. Mathar, Oct 24 2008

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A336064 Numbers divisible by the maximal exponent in their prime factorization (A051903).

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A374586 The maximum exponent in the prime factorization of the numbers that are divisible by the maximum exponent in their prime factorization.

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A144976 Nonsquarefree numbers k such that k is divisible by the maximal exponent in the prime factorization of k.

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