A394581 Smallest prime(n)-smooth k > primorial(n+1), where primorial = A002110.

8, 32, 216, 2352, 30184, 511056, 9702000, 223120128, 6469751750, 200560883544, 7420747075830, 304250397681384, 13082761799177000, 614889830506570476, 32589158927990089698, 1922760351270804488400, 117288381369590011752500, 7858321551352441818943371

Offset

1,1

Comments

Smallest k > primorial(n+1) such that rad(k) | primorial(n), where rad = A007947.

Smallest k > primorial(n+1) such that gpf(k) <= prime(n), where gpf = A006530.

Links

Table of n, a(n) for n=1..18.

Michael De Vlieger, Notes on this sequence.

Formula

a(n) <= A276939(n).

Example

Table of n, a(n), primorial(n+1) for n = 1..7:
  n                       a(n)                          primorial(n+1)
  --------------------------------------------------------------------
  1   A000079(4)     =       8 = 2^3                                6
  2   A003586(13)    =      32 = 2^5                               30
  3   A051037(47)    =     216 = 2^3 * 3^3                        210
  4   A002473(200)   =    2352 = 2^4 * 3 * 7^2                   2310
  5   A051038(789)   =   30184 = 2^3 * 7^3 * 11                 30030
  6   A080197(3288)  =  511056 = 2^4 * 3^3 * 7 * 13^2          510510
  7   A080681(13258) = 9702000 = 2^4 * 3^2 * 5^3 * 7^2 * 11   9699690

Mathematica

MapIndexed[(k = 1; p = Prime[First[#2] ]; While[FactorInteger[#1 + k][[-1, 1]] > p, k++]; #1 + k) &, Rest@ FoldList[Times, Prime@ Range[10] ] ]

Prog

(Python)
from functools import lru_cache
from sympy import prime, integer_log, primorial
from oeis_sequences.OEISsequences import bisection
def A394581(n):
    ps = tuple(prime(i) for i in range(1, n+1))
    @lru_cache(maxsize=None)
    def g(x, m): return sum(g(x//ps[m-1]**i, m-1) for i in range(integer_log(x, ps[m-1])[0]+1)) if m-1 else x.bit_length()
    k = g(p:=primorial(n+1), n)+1
    return bisection(lambda x:k+x-g(x, n), p, p) # Chai Wah Wu, Apr 05 2026

Crossrefs

Cf. A000079, A002110, A002473, A003586, A006530, A007947, A051037, A051038, A080197, A080681, A080682, A080683, A394580.

Sequence in context: A129798 A129792 A224664 * A222692 A268143 A093759

Adjacent sequences: A394578 A394579 A394580 * A394582 A394583 A394584

Keyword

nonn,hard,more

Author

Michael De Vlieger, Mar 27 2026

Extensions

a(15) corrected by Chai Wah Wu, Apr 04 2026

a(16)-a(18) from Daniel Suteu, Apr 07 2026

Status

approved