A215021 -id:A215021 - cp's OEIS frontend

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

Giovanni Resta, May 05 2006

nonn

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]

			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).

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

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

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

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 *)

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

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

More terms from Robert Gerbicz, Aug 24 2006

A215085 a(n) = (A214089(n)^2 - 1) divided by four times the product of the first n primes.

Original entry on oeis.org

1, 1, 1, 1, 19, 17, 1, 2567, 3350, 128928, 3706896, 1290179, 100170428, 39080794, 61998759572, 7833495265, 45119290746, 581075656330, 8672770990, 15792702394898740, 594681417768520250, 25509154378676494, 1642780344643617537867, 480931910076867717575

Offset: 1

J. Stauduhar, Aug 02 2012

nonn

When floor(A214089(n) / 2) = A118478(n), a(n) = A215021(n).

			A214089(14) = 1430083494841, n#_14 = 13082761331670030, and (1430083494841^2 - 1) / (4 * 13082761331670030) = 39080794, so a(14) = 39080794.

J. Stauduhar, Table of n, a(n) for n = 1..30

A215085 := proc(n)
        (A214089(n)^2-1)/4/A002110(n) ;
end proc: # R. J. Mathar, Aug 21 2012

from itertools import product
from sympy import sieve, prime, isprime, primorial
from sympy.ntheory.modular import crt
def A215085(n):
    return (
        1
        if n == 1
        else (
            int(
                min(
                    filter(
                        isprime,
                        (
                            crt(tuple(sieve.primerange(prime(n) + 1)), t)[0]
                            for t in product((1, -1), repeat=n)
                        ),
                    )
                )
            )
            ** 2
            - 1
        )
        // 4
        // primorial(n)
    )  # Chai Wah Wu, May 31 2022
for n in range(1, 16):
    print(A215085(n), end=", ")

a(n) = (A214089(n)^2 - 1) / (4 * A002110(n)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions