A118478 a(n) is the smallest m such that m*(m+1) is divisible by the first n prime numbers.

1, 2, 5, 14, 209, 714, 714, 62985, 367080, 728364, 64822394, 1306238010, 11182598504, 715041747420, 51913478860880, 454746157008780, 9314160363311804, 261062105979210899, 261062105979210899, 696537082207206753590, 54097844397380813592485, 286495021083846822067820

Offset

1,2

Comments

a(n)*(a(n)+1)/(product of first n primes) = 1, 1, 1, 1, 19, 17, 1, 409, 604, 82, 20951, 229931, 411012, 39080794, 4382914408, ... - Robert G. Wilson v, May 13 2006 [This is now A215021. - N. J. A. Sloane, Aug 02 2012]

Links

Jinyuan Wang, Table of n, a(n) for n = 1..30 (terms 1..25 from Robert Gerbicz)

Formula

a(n) = Min_{m | m*(m+1) is divisible by A002110(n)}.

Example

a(8) = 62985 since 62985*62986 = 2*3*5*7*11*13*17*19*409, i.e., it is divisible by the first 8 prime numbers (2,3,..,19).

Mathematica

f[n_] := Block[{k = 1, p = Times @@ Prime@Range@n}, While[ !IntegerQ@ Sqrt[4k*p + 1], k++ ]; Floor@ Sqrt[k*p]]; Array[f, 15] (* Robert G. Wilson v, May 13 2006 *)

Prog

(PARI) P=primes(25); T=1; for(n=1, 25, T*=P[n]; m=T; for(k=2^(n-1), 2^n-1, u=binary(k); a=1; for(i=1, n, if(u[i], a*=P[i])); b=T/a; w=bezout(a, b); if(w[1]<=0, w[1]+=b); c=a*w[1]-1; m=min(m, c); w[1]=b-w[1]; if(w[1]<=0, w[1]+=b); c=a*w[1]; m=min(m, c)); print1(m, ", ")) \\ Robert Gerbicz, Aug 24 2006
(Haskell)
import Data.List (elemIndex); import Data.Maybe (fromJust)
a118478 n = (+ 1) . fromJust $ elemIndex 0 $
            map (flip mod (a002110 n)) $ tail a002378_list
-- Reinhard Zumkeller, Jun 14 2015
(Python 3.8+)
from itertools import combinations
from math import prod
from sympy import sieve, prime, primorial
from sympy.ntheory.modular import crt
def A118478(n): return 1 if n == 1 else int(min(min(crt((m, (k:=primorial(n))//m), (0, -1))[0], crt((k//m, m), (0, -1))[0]) for m in (prod(d) for l in range(1, n//2+1) for d in combinations(sieve.primerange(prime(n)+1), l)))) # Chai Wah Wu, May 31 2022

Crossrefs

Cf. A002110, A002378, A193314, A215021, A053670, A059958.

Sequence in context: A097595 A243979 A081483 * A179675 A193314 A270351

Adjacent sequences: A118475 A118476 A118477 * A118479 A118480 A118481

Keyword

nonn

Author

Giovanni Resta, May 05 2006

Extensions

More terms from Robert Gerbicz, Aug 24 2006

Status

approved