A093553 -id:A093553 - cp's OEIS frontend

A036570 Primes p such that (p+1)/2 and (p+2)/3 are also primes.

13, 37, 157, 541, 877, 1201, 1381, 1621, 2017, 2557, 2857, 3061, 4357, 4441, 5077, 5581, 5701, 6337, 6637, 6661, 6997, 7417, 8221, 9181, 9661, 9901, 10837, 11497, 12457, 12601, 12721, 12757, 13681, 14437, 15241, 16921, 17077, 18217

Offset: 1

N. J. A. Sloane

nonn

The prime p is followed by two semiprimes; a third semiprime is not possible. - T. D. Noe, Jul 23 2008
A subsequence of A005383, which has A163573 as a subsequence. - M. F. Hasler, Feb 26 2012
Similarly, the only "prime sandwiched by semiprimes" is 5. - Zak Seidov, Aug 04 2013
For n > 1, a(n) == 1 or (7 mod 10). If a(n) == 3 (mod 10), then (a(n) + 2)/3 == 0 (mod 5) which is a composite number if a(n) > 13. - Chai Wah Wu, Nov 30 2016
All terms are congruent to 1 (mod 12). - Zak Seidov, Feb 16 2017

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A005383, A074200, A093553, A147615, A163573.
A278583 is an equivalent sequence.
See also A278585.

lst={};Do[p=Prime[n];If[PrimeQ[(p+1)/2]&&PrimeQ[(p+2)/3],AppendTo[lst,p]],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 31 2009 *)

is_A036570(n)={ !(n%3-1) & isprime(n\3+1) & isprime(n\2+1) & isprime(n) }
for(n=1,2e4,is_A036570(n) & print1(n","))  \\ M. F. Hasler, Feb 26 2012

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

Original entry on oeis.org

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

A208455 Primes p such that (p+k)/(k+1) is a prime number for k=1,...,5.

Original entry on oeis.org

5516281, 16831081, 18164161, 29743561, 51755761, 148057561, 153742681, 158918761, 175472641, 189614881, 212808961, 297279361, 298965241, 322030801, 467313841, 527428441, 661686481, 668745001, 751524481, 808214401

Offset: 1

M. F. Hasler, Feb 27 2012

nonn

Subsequence of A071368 consisting of elements ending in the digit 1. (Proof: Let n=10k+1, {n,n+1,...,n+5}={P1,2*P2,...,6*P6} with P1,...,P6 prime. Obviously n+4=10k+5=5*P5. Since n+1==n+5 (mod 4), none of these two can be 4*P4. Thus, n+3=4*P4, whence n==P4 (mod 3) and n cannot be 3*P3. Therefore n=P1 and n+2=3*P3. Then n+5 is an even multiple of 3, n+5=6*P6, and n+1=2*P2 is the only remaining choice.)

Also: The subsequence of p in A204592 such that (p+5)/6 is a prime number. All terms are congruent to 1 modulo 2520 = A003418(9) = 7!/2 = 5*7*8*9.

M. F. Hasler, Table of n, a(n) for n = 1..1000

Cf. A093553. A207825 is a subsequence.

Select[Range[1,810*10^6,2520],PrimeQ[#]&&AllTrue[Table[(#+k)/(k+1),{k,5}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 21 2015 *)

{my(p=1); until(,isprime(p+=2520) || next; for(j=2, 6, isprime(p\j+1)||next(2)); print1(p","))}

is_A208455(p,c=6)={ isprime(p) || return; for(j=2, c, isprime(p\j+1) || return); 1 }

Value of A208455(1000) = 147435621481 = 58506199*2520+1 confirmed by Zak Seidov.

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

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A036570 Primes p such that (p+1)/2 and (p+2)/3 are also primes.

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

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A208455 Primes p such that (p+k)/(k+1) is a prime number for k=1,...,5.

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

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