A368474 -id:A368474 - cp's OEIS frontend

A368473 Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).

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

Offset: 1

Amiram Eldar, Dec 26 2023

The terms of A005361 that are powers of 2 (A000079).

The first position of 2^k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.

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

Cf. A000079, A005361, A085629, A138302, A271727.
Cf. A366538, A367169.
Similar sequences: A322327, A368472, A368474.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[p == 2^IntegerExponent[p, 2], p, Nothing]]; Array[f, 150]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p >> valuation(p,2) == 1, print1(p, ", ")));}

a(n) = A005361(A138302(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=0} 2^k/p^(2^k)) = 1.504710204899266020302..., where d = A271727 is the asymptotic density of the exponentially 2^n-numbers.

A368472 Product of exponents of prime factorization of the exponentially odd numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Amiram Eldar, Dec 26 2023

The odd terms of A005361.

The first position of 2*k-1, for k = 1, 2, ..., is 1, 7, 24, 91, 154, 1444, 5777, 610, 92349, ..., which is the position of A085629(2*k-1) in A268335.

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

Cf. A005361, A013661, A065463, A085629, A098198, A268335.
Cf. A366438, A366439.
Similar sequences: A322327, A368473, A368474.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[OddQ[p], p, Nothing]]; Array[f, 150]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p%2, print1(p, ", ")));}

a(n) = A005361(A268335(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2)^2/d) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 1/p^5) = 1.38446562720473484463..., where d = A065463 is the asymptotic density of the exponentially odd numbers.

A369935 The maximal exponent in the prime factorization of the numbers whose all exponents are squares (A197680).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 06 2024

Differs from A368474 at n = 1, 834, 4154, 5822, 6417, ... .

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

Cf. A000290, A051903, A197680, A357016, A368474, A369936.

squareQ[n_] := IntegerQ[Sqrt[n]]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, squareQ], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]

lista(kmax) = {my(e, q); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; q = 1; for(i = 1, #e, if(!issquare(e[i]), q = 0; break)); if(q, print1(vecmax(e), ", ")));}

a(n) = A051903(A197680(n)).
a(n) = A369936(n)^2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} (k^2 * (d(k) - d(k-1))) / A357016 = 1.16184898017948977008..., where d(k) = Product_{p prime} (1 - 1/p^2 + Sum_{i=2..k} (1/p^(i^2)-1/p^(i^2+1))) for k >= 1, and d(0) = 0.

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A368473 Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).

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A368472 Product of exponents of prime factorization of the exponentially odd numbers.

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A369935 The maximal exponent in the prime factorization of the numbers whose all exponents are squares (A197680).

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