A366762 -id:A366762 - cp's OEIS frontend

A369937 Numbers whose maximal exponent in their prime factorization is square.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from A366762 at n = 84, and from A197680, A361177 and A369210 at n = 95.
Numbers k such that A051903(k) is square.
The asymptotic density of this sequence is 1/zeta(2) + Sum_{k>=2} (1/zeta(k^2+1) - 1/zeta(k^2)) = 0.64939447949574562687... .

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

Cf. A000290, A013661, A051903.
Subsequences: A002416, A005117, A030514, A030635, A113849, A178739, A179665, A179692, A197680.
Similar sequences: A368714, A369938, A369939.
Cf. A361177, A366762, A369210.

Select[Range[100], IntegerQ@ Sqrt[Max[FactorInteger[#][[;; , 2]]]] &]

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

A366761 Numbers that have at least one exponent in their canonical prime factorization that is divisible by 3.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 64, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 192, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 320, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 432, 440, 448, 456, 459, 472, 488, 500, 504, 512, 513, 520

Offset: 1

Amiram Eldar, Oct 21 2023

Each term has a unique representation of as product of two numbers: one is a cube (A000578) and the second is a number that is not in this sequence.
The asymptotic density of this sequence is 1 - zeta(3) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.10483363599014046584... .
From Amiram Eldar, Jan 22 2024: (Start)
The complement of this sequence is the sequence of numbers called "unitarily 3-free", or "3-skew", by Cohen (1961).
He proved that the asymptotic density of unitarily k-free, i.e., numbers whose prime factorization contain no exponent that is divisible by k, is zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(k+1)) (see p. 228, eq. 3.18). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.

A000578 is a subsequence.

Cf. A002117, A182120, A366762.

q[n_] := ! AllTrue[FactorInteger[n][[;; , 2]], ! Divisible[#, 3] &]; Select[Range[500], q]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(!(f[i, 2]%3), return(1))); 0;}

Sum_{n>=1} 1/a(n)^s = zeta(s) * (1 - zeta(3*s) * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s))), for s > 1.

A182120 Numbers for which the canonical prime factorization contains only exponents which are congruent to 2 modulo 3.

Original entry on oeis.org

1, 4, 9, 25, 32, 36, 49, 100, 121, 169, 196, 225, 243, 256, 288, 289, 361, 441, 484, 529, 676, 800, 841, 900, 961, 972, 1089, 1156, 1225, 1369, 1444, 1521, 1568, 1681, 1764, 1849, 2048, 2116, 2209, 2304, 2601, 2809, 3025, 3125, 3249, 3364, 3481, 3721, 3844

Offset: 1

Douglas Latimer, Apr 12 2012

By convention 1 is included as the first term.

			100 is included, as its canonical prime factorization (2^2)*(5^2) contains only exponents which are congruent to 2 modulo 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Douglas Latimer)

A062503 is a subsequence.
Subsequence of A001694.
Cf. A002117, A366762.

Join[{1},Select[Range[5000],Union[Mod[Transpose[FactorInteger[#]][[2]],3]] == {2}&]] (* Harvey P. Dale, Aug 18 2014 *)

{plnt=1; k=1; print1(k, ", "); plnt++;
mxind=76 ; mxind++ ; for(k=2, 2*10^6,
M=factor(k);passes=1;
sz = matsize(M)[1];
for(k=1,sz,  if( M[k,2] % 3 != 2, passes=0));
if( passes == 1 ,
print1(k, ", "); plnt++) ; if(mxind ==  plnt, break() ))}

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2]%3 != 2, return(0))); 1;} \\ Amiram Eldar, Oct 21 2023

Sum_{n>=1} 1/a(n) = zeta(3) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.56984817927051410948... . - Amiram Eldar, Oct 21 2023

A369210 Numbers k such that the number of divisors of k^2 is a power of 3.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Amiram Eldar, Jan 16 2024

First differs from A197680 at n = 331, from A274034 at n = 42, from A361177 at n = 167, and from A366762 at n = 84.
Equivalently, square roots of the numbers whose number of divisors is a power of 3.
The asymptotic density of this sequence is Product_{p prime} ((1 - 1/p) * Sum_{k>=0} 1/p^((3^k-1)/2)) = 0.64033435998103973346... .

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

Cf. A000005, A000244, A000290.

Cf. A197680, A274034, A361177, A366762.

pow3q[n_] := n == 3^IntegerExponent[n, 3]; Select[Range[100], pow3q[DivisorSigma[0, #^2]] &]

ispow3(n) = n == 3^valuation(n, 3);
is(n) = ispow3(numdiv(n^2));

Sum_{n>=1} 1/a(n)^2 = Product_{p prime} Sum_{k>=0} 1/p^(3^k-1) = 1.52478035628964060288... .

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A369937 Numbers whose maximal exponent in their prime factorization is square.

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A366761 Numbers that have at least one exponent in their canonical prime factorization that is divisible by 3.

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A182120 Numbers for which the canonical prime factorization contains only exponents which are congruent to 2 modulo 3.

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A369210 Numbers k such that the number of divisors of k^2 is a power of 3.

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