A278500 -id:A278500 - cp's OEIS frontend

A074200 a(n) = m, the smallest number such that (m+k)/k is prime for k=1, 2, ..., n.

1, 2, 12, 12720, 19440, 5516280, 5516280, 7321991040, 363500177040, 2394196081200, 3163427380990800, 22755817971366480, 3788978012188649280, 2918756139031688155200

Offset: 1

Jean-Christophe Colin (jc-colin(AT)wanadoo.fr), Sep 17 2002, May 10 2010

Computed by Jack Brennen and Phil Carmody.

			(12+k)/k is prime for k = 1,2,3. 12 is the smallest such number so a(3) = 12.

Walter Nissen, Calculation without Words : Doric Columns of Primes.
C. Rivera, Puzzle 181

Cf. A078502, A278500.

One less than A093553.

a[1] = 1; a[n_] := a[n] = For[dm = LCM @@ Range[n]; m = Quotient[a[n - 1], dm]*dm, True, m = m + dm, If[AllTrue[Range[n], PrimeQ[(m + #)/#] &], Return[m]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Dec 01 2016 *)

isok(m, n) = {for (k = 1, n, if ((m+k) % k, return (0), if (! isprime((m+k)/k), return(0)));); return (1);}
a(n) = {m = 1; while(! isok(m, n), m++); m;} \\ Michel Marcus, Aug 31 2013

from sympy import isprime, lcm
def A074200(n):
    a = lcm(range(1,n+1))
    m = a
    while True:
        for k in range(n,0,-1):
            if not isprime(m//k+1):
                break
        else:
            return m
        m += a # Chai Wah Wu, Feb 27 2019

Corrected by Vladeta Jovovic, Jan 08 2003

a(14) from Jens Kruse Andersen, Feb 15 2004

A278583 Numbers k such that k+1 is a prime, k+2 is twice a prime, and k+3 is three times a prime.

Original entry on oeis.org

12, 36, 156, 540, 876, 1200, 1380, 1620, 2016, 2556, 2856, 3060, 4356, 4440, 5076, 5580, 5700, 6336, 6636, 6660, 6996, 7416, 8220, 9180, 9660, 9900, 10836, 11496, 12456, 12600, 12720, 12756, 13680, 14436, 15240, 16920, 17076, 18216, 18300, 18396, 19440, 21000, 21576, 22620, 23556, 24480

Offset: 1

N. J. A. Sloane, Nov 30 2016

nonn

All terms are divisible by 12. - Daniel Poveda Parrilla, Dec 12 2016

R. K. Guy, Posting to Number Theory Mailing List, Nov 30 2016

Antti Karttunen and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 563 terms from Antti Karttunen)

Equals A036570(n) - 1.
Positions of terms >= 3 in A278500.
Cf. A074200.

Select[Range[12,25000,12],AllTrue[{#+1,(#+2)/2,(#+3)/3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 26 2020 *)

list(lim)=my(v=List()); forprime(p=2,lim+1, if(p%6==1 && isprime(p\2+1) && isprime(p\3+1), listput(v,p-1))); Vec(v) \\ Charles R Greathouse IV, Dec 03 2016

A278585 Numbers k such that k+1 is a prime, k+2 is twice a prime, k+3 is three times a prime, and k+4 is four times a prime.

Original entry on oeis.org

12720, 16920, 19440, 24480, 49680, 61560, 104160, 229320, 255360, 259680, 266400, 291720, 298200, 311040, 331920, 419400, 423480, 436800, 446880, 471240, 525240, 532800, 539400, 581520, 600600, 663600, 704160, 709920, 783720, 867000, 904800, 908040, 918360

Offset: 1

N. J. A. Sloane, Nov 30 2016

nonn

a(n) == 0 mod 120 (see comment in A163573). - Chai Wah Wu, Nov 30 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10001

Equals A163573(n) - 1.
Cf. A036570, A074200.
Positions of terms >= 4 in A278500, thus a subsequence of A278583, A089965 and A006093.

Select[Range[920000],AllTrue[{#+1,(#+2)/2,(#+3)/3,(#+4)/4},PrimeQ]&] (* Harvey P. Dale, Aug 08 2021 *)

is(k)=k%120==0 && isprime(k+1) && isprime(k/2+1) && isprime(k/3+1) && isprime(k/4+1) \\ Charles R Greathouse IV, Dec 03 2016

from sympy import prime, isprime
A278585_list = [4*q-4 for q in (prime(i) for i in range(1,10000)) if isprime(4*q-3) and isprime(2*q-1) and (not (4*q-1) % 3) and isprime((4*q-1)//3)] # Chai Wah Wu, Nov 30 2016

cp's OEIS Frontend

A074200 a(n) = m, the smallest number such that (m+k)/k is prime for k=1, 2, ..., n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A278583 Numbers k such that k+1 is a prime, k+2 is twice a prime, and k+3 is three times a prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

A278585 Numbers k such that k+1 is a prime, k+2 is twice a prime, k+3 is three times a prime, and k+4 is four times a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python