A051454 a(n) is the smallest prime factor of 1 + lcm(1..k) where k is the n-th prime power A000961(n).

2, 3, 7, 13, 61, 421, 29, 2521, 19, 89, 71, 1693, 232792561, 6659, 26771144401, 331, 101, 72201776446801, 1801, 173, 54941, 89, 442720643463713815201, 593, 5171, 239, 1222615931, 103, 7265496855919, 6562349363, 4447, 147099357127, 1931

Offset

1,1

Links

Sean A. Irvine, Table of n, a(n) for n = 1..82

Example

1 + lcm(1..8) = 29^2, so its smallest prime divisor is 29; it occurs as the 7th term in the sequence because 8 is the 7th prime power: A000961(7) = 8.

Mathematica

Join[{2}, With[{ppwr=Select[Range[200], PrimePowerQ]}, Table[FactorInteger[LCM@@Take[ ppwr, n]+ 1][[1, 1]], {n, 40}]]] (* Harvey P. Dale, May 28 2024 *)

Prog

(Magma) a:=[]; lcm:=1; for k in [1..83] do if (k eq 1) or IsPrimePower(k) then lcm:=Lcm(lcm, k); a:=a cat [Factorization(1+lcm)[1][1]]; end if; end for; a; // Jon E. Schoenfield, May 28 2018
(PARI) a(n) = {my(nb = 1, lc = 1, k = 2); while (nb != n, if (isprimepower(k), nb++; lc = lcm(lc, k)); k++; ); vecmin(factor(lc +1)[, 1]); } \\ Michel Marcus, May 29 2018
(Python)
from math import prod
from sympy import primepi, integer_nthroot, integer_log, primerange, primefactors
def A051454(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return min(primefactors(1+prod(p**integer_log(m, p)[0] for p in primerange(m+1)))) # Chai Wah Wu, Aug 15 2024

Crossrefs

Cf. A000961, A003418, A049536, A049537, A051301, A051342, A051452.

Sequence in context: A092969 A089136 A092965 * A051452 A058017 A049536

Adjacent sequences: A051451 A051452 A051453 * A051455 A051456 A051457

Keyword

nonn

Author

Labos Elemer

Status

approved