A393537 a(0) = 1, a(n) = prime(n)^n * Product_{i=n+1..2*n-1} prime(i).

1, 2, 45, 9625, 5836831, 15553822427, 32236669290839, 244550840131742083, 1076634344588170139749, 15523512347282949461101843, 511006101928935400533016553383, 5159935278544519492336177325477489, 236913479132009500453399121439012305479, 7010308064542918968646157932339658127887807

Offset

0,2

Comments

Smallest numbers k with omega(k) = n, such that both pi(lpf(k)) = omega(k) and pi(gpf(k)) = bigomega(k), where pi = A000720, lpf = A020639, gpf = A006530, omega = A001221, and bigomega = A001222.

a(1) = 2 is the only even number and the only prime in the sequence.

For n > 1, a(n) is in A391025, which is the sequence of odd terms in A332785.

Squarefree kernel rad(a(n)) = A007947(a(n)) = Product_{i=n..2*n-1} prime(i), i.e., product of the T(n,k)-th prime for T(n,k) = n+k, n >= 1, k = 0..n-1, which is A094727.

Prime signature of a(n) is n followed by n-1 copies of 1. For instance, prime signature of a(3) = 9625 is {3,1,1}, since 9625 = 5^3 * 7^1 * 11^1.

Least number m with prime signature of a(n) is A303557(n).

Links

Michael De Vlieger, Table of n, a(n) for n = 0..162

Index to sequences related to prime signature

Formula

Also a(0) = 1, a(n) = prime(n)^(n-1) * Product_{i=n..2*n-1} prime(i).

Example

Table of n, a(n) for n = 0..7:
  n                  a(n)
  -----------------------------------------------------------
  0                    1
  1                    2 =  2
  2                   45 =  3^2 * 5
  3                 9625 =  5^3 * 7 * 11
  4              5836831 =  7^4 * 11 * 13 * 17
  5          15553822427 = 11^5 * 13 * 17 * 19 * 23
  6       32236669290839 = 13^6 * 17 * 19 * 23 * 29 * 31
  7   244550840131742083 = 17^7 * 19 * 23 * 29 * 31 * 37 * 41

Maple

A393537 := proc(n) local i; ithprime(n)^n * mul(ithprime(i), i = n + 1 .. 2*n - 1) end proc: seq(A393537(n), n = 1 .. 13); # Felix Huber, Feb 24 2026

Mathematica

{1}~Join~Array[Prime[#]^#*Product[Prime[i], {i, # + 1, 2*# - 1}] &, 13]

Prog

(PARI) a(n) = if (n==0, 1, prime(n)^n * prod(i=n+1, 2*n-1, prime(i)));

Crossrefs

Cf. A000720, A001221, A001222, A006530, A020639, A094727, A303557, A332785, A391025, A392970, A393102.

Sequence in context: A392154 A100101 A332244 * A090601 A266016 A071777

Adjacent sequences: A393534 A393535 A393536 * A393539 A393540 A393541

Keyword

nonn,easy

Author

Michael De Vlieger, Feb 20 2026

Status

approved