A303557 -id:A303557 - cp's OEIS frontend

A303555 Triangle read by rows: T(n,k) = 2^(n-k)*prime(k)#, 1 <= k <= n, where prime(k)# is the product of first k primes.

2, 4, 6, 8, 12, 30, 16, 24, 60, 210, 32, 48, 120, 420, 2310, 64, 96, 240, 840, 4620, 30030, 128, 192, 480, 1680, 9240, 60060, 510510, 256, 384, 960, 3360, 18480, 120120, 1021020, 9699690, 512, 768, 1920, 6720, 36960, 240240, 2042040, 19399380, 223092870, 1024, 1536, 3840, 13440, 73920, 480480, 4084080, 38798760, 446185740, 6469693230

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

T(n,k) = the smallest number m having exactly n prime divisors counted with multiplicity and exactly k distinct prime divisors.

			T(5,4) = 420 = 2^2*3*5*7, hence 420 is the smallest number m such that bigomega(m) = 5 and omega(m) = 4 (see A189982).
Triangle begins:
    2;
    4,   6;
    8,  12,  30;
   16,  24,  60,  210;
   32,  48, 120,  420, 2310;
   64,  96, 240,  840, 4620, 30030;
  128, 192, 480, 1680, 9240, 60060, 510510;
  ...

Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences related to primorial numbers
Index to sequences related to prime signature

Cf. A000079, A001221, A001222, A002110, A005179, A038547, A055079, A070175, A303557 (central terms).

Flatten[Table[2^(n - k) Product[Prime[j], {j, k}], {n, 10}, {k, n}]]

A181825 Members of A025487 whose prime signature is self-conjugate (as a partition).

Original entry on oeis.org

1, 2, 12, 36, 120, 360, 1680, 5040, 5400, 27000, 36960, 75600, 110880, 378000, 960960, 1587600, 1663200, 2882880, 7938000, 8316000, 32672640, 34927200, 43243200, 98017920, 174636000, 216216000, 277830000, 908107200, 1152597600, 1241560320, 1470268800, 1944810000

Offset: 1

Matthew Vandermast, Dec 08 2010

A025487(n) is included iff A025487(n) = A181822(n).

Closed under the binary operations of GCD and LCM, since a self-conjugate partition of Omega(a(n)) (which the prime signature of these numbers is) is the concatenation of self-conjugate hooks of decreasing size while moving downward and to the right in the Ferrers diagram, and the GCD (or LCM) of two terms a(i) and a(j) is obtained by taking the smaller (or larger, respectively) of the corresponding hooks. For example, GCD(a(8),a(11)) = GCD(5040,36960) = 1680 = a(7), and LCM(a(8),a(11)) = 110880 = a(13). The two binary operations make the set {a(n)} into a lattice order. - Richard Peterson, May 29 2020

			A025487(11) = 36 = 2^2*3^2 has a prime signature of (2,2), which is a self-conjugate partition; hence, 36 is included in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..11879 (first 578 terms from Amiram Eldar, terms <= 10^70)
David A. Corneth, PARI-program
Eric Weisstein's World of Mathematics, Self-Conjugate Partition

Cf. A025487, A181822, A303557.
Includes subsequences A006939 and A181555.
See also A181823, A181824, A181826, A181827.

\\ See Corneth link \\ David A. Corneth, Jun 03 2020

a(18)-a(32) from Amiram Eldar, Jan 19 2019

cp's OEIS Frontend

A303555 Triangle read by rows: T(n,k) = 2^(n-k)*prime(k)#, 1 <= k <= n, where prime(k)# is the product of first k primes.

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