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

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

1,2

Comments

By convention 1 is included as the first term.

Links

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

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

A062503 is a subsequence.

Subsequence of A001694.

Cf. A002117, A366762.

Sequence in context: A232241 A163836 A175085 * A099998 A360902 A365003

Adjacent sequences: A182117 A182118 A182119 * A182121 A182122 A182123

Keyword

nonn,easy

Author

Douglas Latimer, Apr 12 2012

Status

approved