A078502 -id:A078502 - 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

A093554 a(n) is the smallest number m such that (m-k+1)/k is prime for k=1,2,...,n.

Original entry on oeis.org

2, 5, 11, 11, 174599, 7224839, 10780559, 10780559, 1086338816639, 50060257410239, 7720634052774719, 227457297898150319, 7272877497848202239, 7272877497848202239

Offset: 1

Farideh Firoozbakht, Apr 14 2004

a(n) is the smallest prime number p such that floor(p/k) are also primes for all k=1,2,...,n.
This sequence is A078502 - 1. See that entry for more information and further terms. - N. J. A. Sloane, May 04 2009
It is obvious that this sequence is increasing and each term is prime. If n>4 then a(n)==9 (mod 10).
a(n) = -1 (mod 120) for n > 4, see A078502. - Jean-Christophe Hervé, Sep 15 2014

			Floor(5/2) is prime; floor(11/2) and floor(11/3) are primes; floor(11/2), floor(11/3) and floor(11/4) are primes; floor(7224839/2)...floor(7224839/5) are primes.
a(8)=10780559 because all the eight numbers 10780559, (10780559-1)/2, (10780559-2)/3, (10780559-3)/4, (10780559-4)/5, (10780559-5)/6, (10780559-6)/7 and (10780559-7)/8 are primes and 10780559 is the smallest number m such that (m-k+1)/k is prime for k=1,2,...,8.

Cf. A093553, A181680.

isokp(v) = (type(v) == "t_INT") && isprime(v);
a(n) = {if (n==1, return (2)); forprime(p=2, , nb = 0; for (k=1, n-1, if (! isokp((p-k)/(k+1)), break, nb++); ); if (nb==n-1, return(p)); ); }  \\ Michel Marcus, Sep 15 2014; Jun 22 2025

Added more terms (from A078502), Joerg Arndt, Sep 15 2014

Edited by N. J. A. Sloane, May 18 2022

cp's OEIS Frontend

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

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