A074200 -id:A074200 - 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

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

A093553 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, 3, 13, 12721, 19441, 5516281, 5516281, 7321991041, 363500177041, 2394196081201, 3163427380990801, 22755817971366481, 3788978012188649281, 2918756139031688155201

Offset: 1

Farideh Firoozbakht, Apr 14 2004

This sequence is A074200(n) + 1. See that entry for more information. - N. J. A. Sloane, May 04 2009
It is obvious that this sequence is increasing and each term is prime. If n > 3 then a(n) == 1 (mod 10).
From Jean-Christophe Hervé, Sep 14 2014: (Start)
a(n) == 1 (mod 120) for all n > 3 (see A163573).
a(4) = 12721 is a quite remarkable number: it is a palindromic prime, its 5 (prime) digits sum to 13, still a prime number (and the preceding element in this sequence, among other things), and as the fourth element of this sequence, it is the smallest prime such that (p-1)/2, (p-2)/3 and (p-3)/4 are also prime, and many other properties. (End)

			a(9)=363500177041 because all the nine numbers 363500177041,
(363500177041+1)/2, (363500177041+2)/3, (363500177041+3)/4,
(363500177041+4)/5, (363500177041+5)/6, (363500177041+6)/7,
(363500177041+7)/8 and (363500177041+8)/9 are primes and
363500177041 is the smallest number m such that (m+k-1)/k is prime for k=1,2,...,9.

Walter Nissen, Calculation without Words : Doric Columns of Primes, Up for the Count !

Cf. A072875.

A278500 a(n) = largest k such that n+1 = a prime, n+2 = 2 * a prime, ..., n+k is k times a prime, a(n) = 0 if n+1 is not a prime.

Original entry on oeis.org

1, 2, 0, 2, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 30 2016

nonn

First 4 occurs at n=12720, first 5 occurs at n=19440. See A074200.

			a(12) = 3 as 13 = 1*prime, 14 = 2*prime, 15 = 3*prime.

Antti Karttunen, Table of n, a(n) for n = 1..21000

Cf. A072668 (positions of zeros), A006093 (nonzeros), A089965 (positions of terms >= 2), A278583 (of terms >= 3), A278585 (of terms >= 4).

Cf. A074200 (position of the first term >= n).

Table[If[CompositeQ[n + 1], 0, k = 1; While[Times @@ Boole@ Map[PrimeQ, MapIndexed[#1/First@ #2 &, (n + Range@ k)]] == 1, k++]; k - 1], {n, 120}] (* Michael De Vlieger, Dec 01 2016 *)

A278500(n) = { my(k=1); while((!((n+k)%k) && isprime((n+k)/k)), k = k+1); (k-1); }
for(n=1, 2^20, write("b278500.txt", n, " ", A278500(n)));

(define (A278500 n) (let loop ((k 1)) (let ((h (/ (+ n k) k))) (if (or (not (integer? h)) (zero? (A010051 h))) (- k 1) (loop (+ 1 k))))))

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

A078502 a(n) = smallest positive integer N such that (N - k)/k is prime for k = 1, 2, ..., n.

Original entry on oeis.org

3, 6, 12, 12, 174600, 7224840, 10780560, 10780560, 1086338816640, 50060257410240, 7720634052774720, 227457297898150320, 7272877497848202240, 7272877497848202240

Offset: 1

Joseph L. Pe, Jan 05 2003

The idea for the sequence and first eleven terms are from Ken Wilke.

a(n) == 0 (mod 120) for n > 4: because a(n)/2, a(n)/3, a(n)/4 and a(n)/5 must be integer, a(n) == 0 (mod 60); and if a(n) == 60 (mod 120), (a(n)-4)/4 == 14 (mod 120) would not be prime; thus a(n) == 0 (mod 120). A more general result is a(n) == 0 (mod lcm(1,2,...,n)) for all n >= 1, and a(n) == 0 (mod 2*lcm(1,2,...,n)) for n > 4. - Jean-Christophe Hervé, Sep 15 2014

			(12-k)/k is prime for k = 1,2,3,4 and 12 is the smallest positive integer satisfying this property. Hence a(4) = 12.

Carlos Rivera, Puzzle 206. (N-k)/k Primes, Prime Puzzles and Problems Connection.
Carlos Rivera, Puzzle 977. A special set of primes, Prime Puzzles and Problems Connection.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 7272877497848202240

See A093554 for another version.

Cf. A074200 (equivalent sequence for (N+k)/k prime).

a(n)=k=1; while(k,c=0; for(i=1,n,if(k%i==0&&isprime(k/i-1),c++)); if(c==n,return(k));k++)
n=1;while(n<10,print1(a(n),", ");n++) \\ Derek Orr, Sep 15 2014

a(n) == 0 (mod A003418(n)) because of the divisibility condition (A003418(n) = lcm(1,2,...,n)). - Jean-Christophe Hervé, Sep 15 2014

Corrected and extended by Jens Kruse Andersen, Jan 10 2003

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

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

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A093553 a(n) is the smallest number m such that (m+k-1)/k is prime for k=1,2,...,n.

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A278500 a(n) = largest k such that n+1 = a prime, n+2 = 2 * a prime, ..., n+k is k times a prime, a(n) = 0 if n+1 is not a prime.

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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.

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A078502 a(n) = smallest positive integer N such that (N - k)/k is prime for k = 1, 2, ..., n.

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