cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A240715 Primes p such that p*q*r + 6 and p*q*r - 6 are primes where q and r are the next two primes after p.

Original entry on oeis.org

569, 1531, 1549, 7103, 7451, 9013, 10627, 10853, 11779, 11783, 12671, 12941, 14821, 14851, 17489, 18493, 20717, 20959, 25237, 26309, 27739, 29669, 29873, 34549, 35977, 36251, 37591, 38351, 38639, 39551, 40129, 45589, 46957, 47317, 48781, 55163, 55259
Offset: 1

Views

Author

K. D. Bajpai, Apr 10 2014

Keywords

Examples

			569 is in the sequence because 569*571*577 + 6 = 187466729 and 569*571*577 - 6 = 187466717 are both prime where 571 and 577 are the next two primes after 569.
1531 is in the sequence because 1531*1543*1549 + 6 = 3659253823 and 1531*1543*1549 - 6 = 3659253811 are both prime where 1543 and 1549 are the next two primes after 1531.

Links

Crossrefs

Cf. A006094, A048880, A051507, A240596.

Programs

  • Magma
    [p: p in PrimesUpTo(10^5) | IsPrime(t-6) and IsPrime(t+6) where t is p*NextPrime(p)*NextPrime(NextPrime(p))]; // Bruno Berselli, Apr 11 2014
  • Maple
    KD := proc(n) local a,b,d; a:=ithprime(n)*ithprime(n+1)*ithprime(n+2); b:=a+6; d:=a-6; if  isprime(b) and isprime(d) then RETURN (ithprime(n)); fi; end: seq(KD(n), n=1..10000);
  • Mathematica
    c = 0; Do[If[PrimeQ[Prime[n]*Prime[n+1]*Prime[n+2] +6] && PrimeQ[Prime[n]*Prime[n+1]*Prime[n+2] -6],c=c+1;Print[c, " ", Prime[n]]],{n,1,500000}];
KD={};   f=Prime[n+1]*Prime[n+2];  Do[p=Prime[n]; If[ PrimeQ[p*f+6] && PrimeQ[p*f-6], AppendTo[KD,p]], {n,10000}]; KD
Select[Partition[Prime[Range[6000]],3,1],AllTrue[Times@@#+{6,-6},PrimeQ]&][[All,1]] (* Harvey P. Dale, Oct 29 2022 *)

A240618 Primes p such that p*q*r + 2 is prime, where q and r are the next two primes after p.

Original entry on oeis.org

3, 13, 257, 307, 409, 431, 587, 719, 733, 1031, 1109, 1123, 1129, 1237, 1657, 1987, 1999, 2143, 2179, 2239, 2411, 2467, 2843, 3041, 3191, 3433, 3691, 3943, 4051, 4219, 4289, 4561, 4567, 4721, 4817, 4831, 4943, 4993, 5521, 5563, 5623, 5689, 5813, 6257, 6983, 7043
Offset: 1

Views

Author

K. D. Bajpai, Apr 09 2014

Keywords

Comments

All the terms in the sequence, except a(1), are congruent to 1 mod 6.

Links

Crossrefs

Cf. A000040, A048880, A051507, A240596.

Programs

  • Maple
    KD := proc(n) local a,b; a:=ithprime(n)*ithprime(n+1)*ithprime(n+2); b:=a+2; if  isprime(b) then RETURN (ithprime(n)); fi; end: seq(KD(n), n=1..2000);
  • Mathematica
    KD={}; Do[p=Prime[n]; If[PrimeQ[p*Prime[n+1]*Prime[n+2] + 2], AppendTo[KD,p]], {n,1000}]; KD

A240621 Primes p such that p*q + 6 and p*q - 6 are primes where p and q are consecutive primes.

Original entry on oeis.org

5, 7, 11, 19, 23, 37, 79, 97, 307, 349, 439, 479, 503, 719, 907, 983, 991, 1061, 1069, 1109, 1597, 1609, 1621, 1867, 2111, 2609, 3301, 3371, 3851, 4129, 4211, 4639, 4999, 5119, 5471, 5683, 5779, 5867, 5939, 6563, 7951, 9337, 9461, 9551, 10061, 10181, 10273, 12251
Offset: 1

Views

Author

K. D. Bajpai, Apr 09 2014

Keywords

Examples

			7 is in the sequence because 7*11 + 6 = 83 and 7*11 - 6 = 71 are both prime where 7 and 11 are consecutive primes.
37 is in the sequence because 37*41 + 6 = 1523 and 37*41 - 6 = 1511 are both prime where 37 and 41 are consecutive primes.

Links

Crossrefs

Cf. A006094, A048880, A051507, A240596.

Programs

  • Maple
    KD := proc(n) local a,b,d; a:=ithprime(n)*ithprime(n+1); b:=a+6; d:=a-6; if  isprime(b) and isprime(d) then RETURN (ithprime(n)); fi; end: seq(KD(n), n=1..5000);
  • Mathematica
    Select[Partition[Prime[Range[1500]],2,1],AllTrue[Times@@#+{6,-6},PrimeQ]&][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 21 2017 *)
  • PARI
    isok(p) = isprime(p) && (q = nextprime(p+1)) && isprime(p*q-6) && isprime(p*q+6); \\ Michel Marcus, Apr 12 2014

A240725 Primes p such that p^2*q^2*r^2 + 12 and p^2*q^2*r^2 - 12 are primes where q and r are next two primes after p.

Original entry on oeis.org

23, 31, 43, 521, 1061, 2153, 3457, 4019, 4943, 5477, 6991, 7577, 8291, 8539, 10993, 11953, 14767, 17957, 18439, 26321, 40993, 41047, 53269, 57917, 71347, 79979, 80989, 88997, 91499, 92269, 94561, 108457, 109111, 112019, 117671, 121763, 133103, 140407, 147073
Offset: 1

Views

Author

K. D. Bajpai, Apr 11 2014

Keywords

Comments

In the expression prime(n)^2 * prime(n+1)^2 * prime(n+2)^2 +/- c, c = 12 is the smallest integer that yields a sequence of such primes. That means for c = 1...11 no such sequence with a large number of primes is obtained.

Examples

			23 is prime and appears in the sequence because 23^2 * 29^2 * 31^2 + 12 = 427538341 and 23^2 * 29^2 * 31^2 - 12 = 427538317 are both prime where 29 and 31 are the next two primes after 23.
31 is prime and appears in the sequence because 31^2 * 37^2 * 41^2 + 12 = 2211538741 and 31^2 * 37^2 * 41^2 - 12 = 2211538717 are both prime where 37 and 41 are the next two primes after 31.

Links

Crossrefs

Cf. A006094, A048880, A051507, A240596, A240715.

Programs

  • Maple
    KD := proc(n) local a, b, d; a:=ithprime(n)^2*ithprime(n+1)^2*ithprime(n+2)^2; b:=a+12; d:=a-12; if  isprime(b) and isprime(d) then RETURN (ithprime(n)); fi; end: seq(KD(n), n=1..20000);
  • Mathematica
    c=0;Do[If[PrimeQ[Prime[n]^2*Prime[n+1]^2*Prime[n+2]^2+12]&&PrimeQ[Prime[n]^2*Prime[n+1]^2*Prime[n+2]^2-12],c=c+1;Print[c," ", Prime[n]]], {n,1,1000000}];
KD = {}; Do[p = Prime[n]; If[PrimeQ[Prime[n]^2*Prime[n + 1]^2*Prime[n + 2]^2 + 12] && PrimeQ[Prime[n]^2*Prime[n + 1]^2*Prime[n + 2]^2 - 12], AppendTo[KD, p]], {n, 10000}]; KD
Showing 1-4 of 4 results.

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