A336064 -id:A336064 - cp's OEIS frontend

A336065 Decimal expansion of the asymptotic density of the numbers divisible by the maximal exponent in their prime factorization (A336064).

8, 4, 8, 9, 5, 7, 1, 9, 5, 0, 0, 4, 4, 9, 3, 3, 2, 8, 1, 4, 2, 7, 1, 0, 9, 7, 6, 8, 5, 4, 4, 3, 5, 2, 9, 2, 6, 7, 7, 9, 1, 4, 7, 2, 8, 9, 9, 4, 9, 1, 8, 1, 0, 0, 9, 7, 8, 8, 1, 7, 6, 4, 4, 2, 0, 5, 6, 1, 5, 7, 0, 9, 6, 6, 9, 2, 4, 6, 7, 0, 3, 0, 0, 1, 5, 8, 6

Offset: 0

Amiram Eldar, Jul 07 2020

			0.848957195004493328142710976854435292677914728994918...

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

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, A059956, A336064.

f[k_] := Module[{f = FactorInteger[k]}, p = f[[;; , 1]]; e = f[[;; , 2]]; (1/Zeta[k + 1])* Times @@ ((p^(k - e + 1) - 1)/(p^(k + 1) - 1)) - (1/Zeta[k]) * Times @@ ((p^(k - e) - 1)/(p^k - 1))]; RealDigits[1/Zeta[2] + Sum[f[k], {k, 2, 1000}], 10, 100][[1]]

Equals 1/zeta(2) + Sum_{k>=2} ((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))), where e(p,k) is the largest exponent of p dividing k.

A368714 Numbers whose maximal exponent in their prime factorization is even.

Original entry on oeis.org

1, 4, 9, 12, 16, 18, 20, 25, 28, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 100, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198, 204, 207, 208

Offset: 1

Amiram Eldar, Jan 04 2024

First differs from A240112 at n = 30.
Numbers k such that A051903(k) is even.
The asymptotic density of this sequence is Sum_{k>=2} (-1)^k * (1 - 1/zeta(k)) = 0.27591672059822700769... .

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

Cf. A051903, A240112.
Subsequences: A000290, A000302, A000583, A001014, A001016, A001025, A008454, A062503, A067259.
Similar sequences: A060476, A074661, A096432, A336064, A368715.

Select[Range[210], # == 1 || EvenQ[Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 1, kmax, if(k == 1 || !(vecmax(factor(k)[,2])%2), print1(k, ", ")));

A368715 Numbers that are not coprime to the maximal exponent in their prime factorization.

Original entry on oeis.org

4, 12, 16, 18, 20, 24, 27, 28, 36, 44, 48, 50, 52, 54, 60, 64, 68, 72, 76, 80, 84, 90, 92, 98, 100, 108, 112, 116, 120, 124, 126, 132, 135, 140, 144, 148, 150, 156, 160, 162, 164, 168, 172, 176, 180, 188, 189, 192, 196, 198, 204, 208, 212, 216, 220, 228, 234, 236, 240, 242, 244

Offset: 1

Amiram Eldar, Jan 04 2024

Subsequence of A137257 and first differs from it at n = 51.
Numbers k such that gcd(k, A051903(k)) > 1.
Includes all the nonsquarefree terms of A336064.
The asymptotic density of this sequence is 1 - 1/zeta(2) - Sum_{k>=2} (1/(f(k+1, k) * zeta(k+1)) - 1/(f(k, k) * zeta(k))) = 0.24998449199080279703..., where f(e, m) = Product_{primes p|m} ((1-1/p^e)/(1-1/p)).

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

Cf. A051903.
Subsequence of A013929 and A137257.
Similar sequences: A060476, A074661, A096432, A336064, A368714.

Select[Range[210], !CoprimeQ[#, Max[FactorInteger[#][[;;, 2]]]] &]

lista(kmax) = for(k = 2, kmax, if(gcd(k, vecmax(factor(k)[,2])) > 1, print1(k, ", ")));

A381952 a(n) is the greatest common divisor of n and the maximum exponent in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 11 2025

The asymptotic density d(k) of the occurrences of each integer k >= 1 in this sequence can be calculated. E.g., d(1) = 0.75001550800919720296... (1 - the density of A368715), and d(2) = 0.16205634516436945215... (see A381953). From these densities the asymptotic mean of this sequence can be evaluated by Sum_{k>=1} k*d(k), but it seems that the expressions for d(k) for large values of k may be complicated.

The sums of the first 10^k terms, for k = 1, 2, ..., are 11, 135, 1396, 14014, 140241, 1402521, 14025251, 140252636, 1402526282, 14025262924, ... . Apparently, the asymptotic mean of this sequence equals 1.402526... .

			a(1) = gcd(1, A051903(1)) = gcd(1, 0) = 1.
a(4) = gcd(4, A051903(4)) = gcd(4, 2) = 2.
a(16) = gcd(16, A051903(16)) = gcd(16, 4) = 4.

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

Cf. A051903, A336064, A368715, A381953.

a[n_] := GCD[n, If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]]; Array[a, 100]

a(n) = gcd(n, if(n > 1, vecmax(factor(n)[, 2]), 0));

a(n) = gcd(n, A051903(n)).
a(n) >= 2 if and only if n is in A368715.
a(A381953(n)) = 2.
a(A336064(n)) = A051903(A336064(n)).

A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

Original entry on oeis.org

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, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

The asymptotic density of this sequence is 1 (Schinzel and Šalát, 1994).

			4 = 2^2 is a term since A051904(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. A051904, A124184.
A005117 (except for 1) is subsequence.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336064.

h[1] = 0; h[n_] := Min[FactorInteger[n][[;; , 2]]]; Select[Range[2, 100], Divisible[#, h[#]] &]
Select[Range[2,100],Divisible[#,Min[FactorInteger[#][[All,2]]]]&] (* Harvey P. Dale, Aug 31 2020 *)

isok(m) = if (m>1, (m % vecmin(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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A336065 Decimal expansion of the asymptotic density of the numbers divisible by the maximal exponent in their prime factorization (A336064).

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A368714 Numbers whose maximal exponent in their prime factorization is even.

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A368715 Numbers that are not coprime to the maximal exponent in their prime factorization.

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A381952 a(n) is the greatest common divisor of n and the maximum exponent in the prime factorization of n.

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A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

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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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