A124510 -id:A124510 - cp's OEIS frontend

A124508 a(n) = 2^BigO(n) * 3^omega(n), where BigO = A001222 and omega = A001221, the numbers of prime factors of n with and without repetitions.

1, 6, 6, 12, 6, 36, 6, 24, 12, 36, 6, 72, 6, 36, 36, 48, 6, 72, 6, 72, 36, 36, 6, 144, 12, 36, 24, 72, 6, 216, 6, 96, 36, 36, 36, 144, 6, 36, 36, 144, 6, 216, 6, 72, 72, 36, 6, 288, 12, 72, 36, 72, 6, 144, 36, 144, 36, 36, 6, 432, 6, 36, 72, 192, 36, 216, 6, 72, 36, 216, 6, 288, 6

Offset: 1

Reinhard Zumkeller, Nov 04 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.
Eric Weisstein's World of Mathematics, Smooth number.

Cf. A000079, A000244, A000400, A001221, A001222, A003586, A005117, A007283, A061142, A074816, A124509, A124510, A124511, A124512.

Table[2^PrimeOmega[n] 3^PrimeNu[n],{n,80}] (* Harvey P. Dale, Mar 26 2013 *)

a(n) = my(f = factor(n)); 2^bigomega(f) * 3^omega(f); \\ Amiram Eldar, Jul 11 2023

Multiplicative with p^e -> 3*2^e, p prime and e>0.
a(n) = A061142(n)*A074816(n) = A000079(A001222(n))*A000244(A001221(n)).
A124509 gives the range: A124509(n) = a(A124510(n)) and a(m) <> a(A124510(n)) for m < A124510(n).
For primes p, q with p <> q: a(p) = 6; a(p*q) = 36; a(p^k) = 3*2^k, k>0.
For squarefree numbers m: a(m) = 6^omega(m).
A001222(a(n)) = A001222(n)+1; A001221(a(n)) = 2 for n > 1.
A124511(n) = a(a(n)); A124512(n) = a(a(a(n))).

A124509 Range of A124508.

Original entry on oeis.org

1, 6, 12, 24, 36, 48, 72, 96, 144, 192, 216, 288, 384, 432, 576, 768, 864, 1152, 1296, 1536, 1728, 2304, 2592, 3072, 3456, 4608, 5184, 6144, 6912, 7776, 9216, 10368, 12288, 13824, 15552, 18432, 20736, 24576, 27648, 31104, 36864, 41472, 46656, 49152, 55296, 62208

Offset: 1

Reinhard Zumkeller, Nov 04 2006

1 together with numbers of the form 2^i * 3^j with i >= j >= 1. - Amiram Eldar, Jul 11 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number.

Subsequence of A003586.
A007283 is a subsequence.
Cf. A124508, A124510.

With[{max = 70000}, Join[{1}, Sort[Flatten[Table[2^i*3^j, {i, 1, Log2[max]}, {j, 1, Min[i, Log[3, max/2^i]]}]]]]] (* Amiram Eldar, Jul 11 2023 *)

a(n) = A124508(A124510(n)).

Sum_{n>=1} 1/a(n) = 7/5. - Amiram Eldar, Jul 11 2023

Missing term a(43) inserted and more terms added by Amiram Eldar, Jul 11 2023

cp's OEIS Frontend

A124508 a(n) = 2^BigO(n) * 3^omega(n), where BigO = A001222 and omega = A001221, the numbers of prime factors of n with and without repetitions.

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