A274034 -id:A274034 - cp's OEIS frontend

A366762 Numbers whose canonical prime factorization contains only exponents which are congruent to 1 modulo 3.

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, Oct 21 2023

First differs from A274034 at n = 42, and from A197680 and A361177 at n = 84.

The asymptotic density of this sequence is zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 = A253905 * A065465 = 0.644177671086029533405... .

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

Similar sequences with exponents of a given form: A000290 (2*k), A268335 (2*k+1), A000578 (3*k), A182120 (3*k+2).
Cf. A002117, A065465, A253905, A330523.
Cf. A197680, A274034, A361177, A366761.

q[n_] := AllTrue[FactorInteger[n][[;; , 2]], Mod[#, 3] == 1 &]; Select[Range[120], q]

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

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

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... .

A273487 Density of numbers without prime exponents in their factorization.

Original entry on oeis.org

6, 5, 0, 4, 4, 5, 6, 0, 8, 4, 2, 1, 9, 1, 2, 6, 9, 1, 3, 9, 0, 4, 4, 4, 3, 6, 1, 1, 0, 4, 6, 5, 9, 6, 4, 5, 5, 7, 7, 0, 1, 0, 2, 9, 6, 9, 2, 2, 0, 5, 4, 9, 7, 6, 0, 2, 0, 1, 9, 3, 5, 8, 8, 5, 5, 5, 2, 3, 4, 2, 8, 6, 9, 1, 6, 8, 2, 1, 3, 6, 7, 7, 4, 9, 3

Offset: 0

Charles R Greathouse IV, Jul 01 2016

			0.6504456084219126913904443611046...

Density of A274034.

eser := 1-x^2+x^4 ;
for pidx from 3 to 100 do
    p := ithprime(pidx) ;
    eser := eser -x^p+x^(p+1) ;
end do:
eser := taylor(eser,x=0,p) ;
gfun[seriestolist](eser) ;
subsop(1=NULL,%) ;
L := EULERi(%) ;
Digits := 180 ;
x := 1.0 ;
for i from 2 to nops(L) do
    if op(i,L) <> 0 then
        x := x*evalf(Zeta(i)^op(i,L)) ;
        printf("%.70f\n",x) ;
    fi ;
end do; # R. J. Mathar, Jul 11 2016

leps=log(2)*(1-bitprecision(1.))
f(x)=my(s=0.);forprime(p=2,1-leps/log(x),s+=x^-p);s
6/Pi^2*prodeuler(p=2,1e6,(1-(1-1/p)*f(p))/(1-1/p^2))

Prod_{p prime} 1 - (1 - 1/p)*Sum_{q prime} p^-q.

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A366762 Numbers whose canonical prime factorization contains only exponents which are congruent to 1 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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A273487 Density of numbers without prime exponents in their factorization.

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