A061283 -id:A061283 - cp's OEIS frontend

A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

1, 2, 12, 120, 1680, 36960, 960960, 32672640, 1241560320, 57111774720, 3312482933760, 205373941893120, 15197671700090880, 1246209079407452160, 107173980829040885760, 10074354197929843261440, 1067881544980563385712640, 126010022307706479514091520, 15373222721540190500719165440

Offset: 0

Ilya Gutkovskiy, Apr 26 2018

nonn

For n > 0, a(n) is the smallest number m having exactly n distinct prime divisors and exactly 2*n - 1 prime divisors counted with multiplicity.

			a(1) = 2^1;
a(2) = 2^2*3;
a(3) = 2^3*3*5;
a(4) = 2^4*3*5*7;
a(5) = 2^5*3*5*7*11, etc.

Central terms of triangle A303555 (for n > 0).

Cf. A000079, A002110, A003680, A005179, A011782, A038547, A061283, A070175, A088860, A102476.

Join[{1}, Table[2^(n - 1) Product[Prime[j], {j, n}], {n, 18}]]

a(n) = A011782(n)*A002110(n).

A061236 Smallest number with prime(n)^3 divisors where prime(n) is n-th prime.

Original entry on oeis.org

24, 900, 810000, 729000000, 590490000000000, 531441000000000000, 430467210000000000000000, 387420489000000000000000000, 313810596090000000000000000000000, 228767924549610000000000000000000000000000, 205891132094649000000000000000000000000000000

Offset: 1

Labos Elemer, Jun 01 2001

nonn

			If p = 2, then d(128) = d(24) = d(30) = 8 and a(1) = 24 < 30 is the smallest.
If p = 5, then 2^124 > (2^24)*(3^4) > 30^4 = 810000 = a(3).

Amiram Eldar, Table of n, a(n) for n = 1..123

Cf. A000005, A005179, A003680, A030078, A037992, A061283, A061286, A061148, A061149, A061234.

For p = 2, 24 is the solution. If a prime p > 2, the suitable powers of 30 are the least solutions: a(n) = Min{x | d(x) = A000005(x) = p(n)^3} = 30^(prime(n)-1). d(2^(ppp-1)) = d(2^(pp-1)*3^(p-1)) = d(30^(p-1)) = p^3 and 2^(ppp-1) > 2^(pp-1)*3^(p-1) > 30^(p-1) holds if p > 2.

a(n) = A005179(A030078(n)) = A005179(prime(n)^3). - Amiram Eldar, Jan 23 2025

a(10)-a(11) from Amiram Eldar, Jan 23 2025

A137490 Numbers with 27 divisors.

Original entry on oeis.org

900, 1764, 2304, 4356, 4900, 6084, 6400, 10404, 11025, 12100, 12544, 12996, 16900, 19044, 23716, 26244, 27225, 28900, 30276, 30976, 33124, 34596, 36100, 38025, 43264, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 73984, 74529

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^26 (subset of A089081), p^2*q^2*r^2 (like 900, 1764, 4356, squares of A007304) or p^2*q^8 (like 2304, 6400, subset of the squares of A030628) where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000005, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[150000],DivisorSigma[0,#]==27&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

isA137490(n) = numdiv(n)==27 \\ Michael B. Porter, Apr 10 2010

A000005(a(n)) = 27.

Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 + P(2)*P(8) - P(10) + P(26) = 0.00453941..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

cp's OEIS Frontend

A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

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A061236 Smallest number with prime(n)^3 divisors where prime(n) is n-th prime.

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A137490 Numbers with 27 divisors.

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