A132349 -id:A132349 - cp's OEIS frontend

A132350 If n > 1 is a k-th power with k >= 2 then a(n) = 0, otherwise a(n) = 1.

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

N. J. A. Sloane, Nov 11 2007

nonn

			a(4) = 0 because 4 = 2^2.
a(8) = 0 because 8 = 2^3.
a(12) = 1 because 12 is not a perfect power (though it is divisible by a perfect power).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A132349, A132351, A132352, A075802, A001597.

a132350 1 = 1
a132350 n = 1 - a075802 n  -- Reinhard Zumkeller, Jun 14 2013

Table[Boole[GCD@@FactorInteger[n][[All, 2]] == 1], {n, 100}] (* Alonso del Arte, May 28 2018 *)

(a(n)=!ispower(n)); (r(nMax) = for(j=1,nMax,print1(!ispower(j)","))); r(100)

a(n) = 1 - A075802(n) for n >= 2. - R. J. Mathar, Nov 12 2007

Given the Möbius function mu(n) = A008683(n), a(n) = abs(mu(n)) unless n is in A303946. - Alonso del Arte, May 28 2018

Edited by M. F. Hasler, Jun 01 2018

A375341 The maximum exponent in the prime factorization of the numbers that have exactly one non-unitary prime factor.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 3, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 3, 2, 3, 6, 2, 2, 2, 4, 2, 2, 5, 2, 3, 2, 2, 4, 2, 5, 2, 2, 3, 3, 8, 2, 2, 3, 2, 3, 4, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3

Offset: 1

Amiram Eldar, Aug 12 2024

The positive terms in A375339.

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

Cf. A005361, A048109, A051903, A060687, A082293, A154945, A190641, A375339, A375340.

s[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], Nothing]]; Array[s, 300]

lista(kmax) = {my(e); for(k = 1, kmax, e = select(x -> x > 1, factor(k)[,2]); if(#e == 1, print1(e[1], ", ")));}

a(n) = A051903(A190641(n)).
a(n) = A005361(A190641(n)).
a(n) = A375339(A190641(n)).
a(n) = A132349(A057521(A190641(n))).
a(n) = 2 if and only if A190641(n) is in A060687.
a(n) = 3 if and only if A190641(n) is in A048109.
a(n) <= 3 if and only if A190641(n) is in A082293.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2*p-1)/((p-1)*(p^2-1)) / Sum_{p prime} 1/(p^2-1) = A375340 / A154945 = 2.74622231282166656595... .
Asymptotic second raw moment:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)^2 = Sum_{p prime} (4*p^2-3*p+1)/((p-1)^3*(p+1)) / Sum_{p prime} 1/(p^2-1) = 9.064902009520365378603... .
Asymptotic second central moment, or variance, is  - ^2 = 1.52316501808078192104... and the asymptotic standard deviation is sqrt( - ^2) = 1.23416571743051667098... .

A132352 Partial sums of A132351.

Original entry on oeis.org

1, 3, 6, 9, 13, 18, 24, 30, 36, 43, 51, 60, 70, 81, 93, 105, 118, 132, 147, 163, 180, 198, 217, 237, 257, 278, 299, 321, 344, 368, 393, 418, 444, 471, 499, 527, 556, 586, 617, 649, 682, 716, 751, 787, 824, 862, 901, 941, 981, 1022, 1064, 1107, 1151, 1196, 1242, 1289, 1337

Offset: 1

N. J. A. Sloane, Nov 11 2007

nonn

Cf. A132349-A132351, A132384.

cp's OEIS Frontend

A132350 If n > 1 is a k-th power with k >= 2 then a(n) = 0, otherwise a(n) = 1.

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A375341 The maximum exponent in the prime factorization of the numbers that have exactly one non-unitary prime factor.

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A132352 Partial sums of A132351.

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