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
Examples
100 is included, as its canonical prime factorization (2^2)*(5^2) contains only exponents which are congruent to 2 modulo 3.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Douglas Latimer)
Programs
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Mathematica
Join[{1},Select[Range[5000],Union[Mod[Transpose[FactorInteger[#]][[2]],3]] == {2}&]] (* Harvey P. Dale, Aug 18 2014 *) -
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
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
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