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.

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A120851 Numbers n such that n is prime and is equal to the product of the first k primes minus the sum of the first k primes, for some k.

Original entry on oeis.org

193, 29989, 13082761331669749
Offset: 1

Views

Author

Carlos Alves, Jul 08 2006

Keywords

Comments

It is in the spirit of A096345 (only for 2 consecutive primes).
The corresponding values of k are 4, 6, 14, 548, 1190, ... a(4) = 2.452... * 10^1691, a(5) = 1.263... x 10^4142. - Amiram Eldar, Dec 19 2018

Examples

			193 = -(2+3+5+7)+(2*3*5*7) and 193 is prime.

Crossrefs

Cf. A096345, A096342, A120850.

Programs

  • Mathematica
    tb = {}; Do[pq = -Plus @@ Prime[Range[1, k]] + Times @@ Prime[Range[1, k]]; If[PrimeQ[pq], AppendTo[tb, pq]], {k, 1, 200}]; tb

Extensions

The next term is too large to include.

A162358 The larger member q in a pair of successive primes (p,q) such that p*q+p+q is prime.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 23, 29, 43, 47, 53, 61, 83, 89, 97, 103, 113, 127, 139, 167, 173, 179, 227, 233, 263, 313, 389, 397, 419, 421, 443, 449, 487, 523, 557, 563, 587, 599, 617, 647, 653, 691, 787, 809, 811, 821, 857, 967, 991, 1021, 1051, 1103, 1117, 1153, 1181
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 01 2009

Keywords

Examples

			q=3 is the larger member in (2,3) where 2*3+2+3=11 is prime.
q=5 is the larger member in (3,5) where 3*5+3+5=23 is prime.

Links

Crossrefs

Cf. A096342, A126148

Programs

  • Mathematica
    lst={};Do[p=Prime[n]*Prime[n+1]+Prime[n]+Prime[n+1];If[PrimeQ[p],AppendTo[lst, Prime[n+1]]],{n,6!}];lst
Transpose[Select[Partition[Prime[Range[200]],2,1],PrimeQ[ Times@@#+ Total[ #]]&]] [[2]] (* Harvey P. Dale, Jan 19 2016 *)

Formula

a(n) = A151800(A126148(n)).

Extensions

Definition rephrased by R. J. Mathar, Sep 11 2009

A284550 Integers n such that prime(n) + prime(n+1) + prime(n+2) + prime(n+3) + prime(n)*prime(n+1)*prime(n+2)*prime(n+3) is prime.

Original entry on oeis.org

1, 2, 9, 35, 45, 61, 80, 84, 97, 98, 124, 130, 140, 142, 175, 179, 185, 213, 241, 249, 287, 300, 324, 344, 346, 352, 366, 368, 369, 384, 389, 398, 400, 409, 431, 436, 437, 462, 515, 520, 525, 530, 544, 565, 592, 594, 595, 614, 615, 627, 628, 682, 719, 745, 778, 798, 835, 852, 861
Offset: 1

Views

Author

Zak Seidov, Mar 29 2017

Keywords

Examples

			n=1: 2+3+5+7+2*3*5*7=227=A000040(49),
n=2: 3+5+7+11+3*5*7*11=1181=A000040(194).

Links

Crossrefs

Cf. A000040, A096342, A126199.

Programs

A291464 Primes p such that p^3*q^3 + p^2 + q^2 is prime, where q is next prime after p.

Original entry on oeis.org

2, 11, 13, 41, 97, 277, 389, 1093, 1229, 1409, 1429, 1627, 1823, 1931, 1979, 2437, 2521, 2549, 2657, 2689, 2719, 2729, 2731, 2969, 3019, 3413, 3539, 3593, 3613, 3623, 3697, 4003, 4027, 4289, 4327, 4583, 4751, 5051, 5323, 5503, 5657, 5783, 6143, 6221, 6299, 6329
Offset: 1

Views

Author

K. D. Bajpai, Aug 24 2017

Keywords

Examples

			a(1) = 2 is prime; 3 is the next prime: 2^3*3^3 + 2^2 + 3^2 = 8*27 + 4 + 9 = 229 that is a prime.
a(2) = 11 is prime; 13 is the next prime: 11^3*13^3 + 11^2 + 13^2 = 1331*2197 + 121 + 169 = 2924497 that is a prime.

Links

Crossrefs

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241, A291339, A291374.

Programs

  • Magma
    [p: p in PrimesUpTo(5000) | IsPrime(p^3*q^3 + p^2 + q^2) where q is NextPrime(p)];
  • Maple
    select(p -> andmap(isprime,[p,(p^3*nextprime(p)^3+p^2+nextprime(p)^2)]), [seq(p, p=1..10^4)]);
  • Mathematica
    Select[Prime[Range[5000]], PrimeQ[#^3*NextPrime[#]^3 + #^2 + NextPrime[#]^2] &]
Select[Partition[Prime[Range[1000]],2,1],PrimeQ[#[[1]]^3 #[[2]]^3+#[[1]]^2+#[[2]]^2]&][[;;,1]] (* Harvey P. Dale, Sep 11 2023 *)
  • PARI
    forprime(p=1, 5000, q=nextprime(p+1); p3=p^3; p2=p^2; q3=q^3; q2=q^2; if(ispseudoprime(p3*q3 + p2 + q2), print1(p, ", ")));

A349985 Primes of the form (product of 4 consecutive primes) + (sum of the same 4 consecutive primes).

Original entry on oeis.org

227, 1181, 765169, 575772529, 1844619589, 7916858557, 31095441001, 37809636673, 75033373321, 80635078873, 234564891361, 302257557157, 443314943881, 463236781489, 1215371749321, 1347613229509, 1534404944209, 2967342092629, 5573043569437, 6390859845289
Offset: 1

Views

Author

Lyle Blosser, Jan 08 2022

Keywords

Comments

It is conjectured that this sequence is infinite, and that similar lists of primes can be generated by using any even number of consecutive primes. Specifying 2 consecutive primes results in A096342. However, it should be noted that the percentage of resulting primes (as compared to all numbers generated in this manner) decreases as the number of consecutive primes to consider increases.

Examples

			227 is a term since 227 is prime and is generated by (2*3*5*7) + (2+3+5+7).
1181 is a term since 1181 is prime and is generated by (3*5*7*11) + (3+5+7+11).

Links

Crossrefs

Cf. A096342.

Programs

  • Mathematica
    Select[Table[s=NextPrime[p,Range@4-1];Total@s+Times@@s,{p,Prime@Range@300}],PrimeQ] (* Giorgos Kalogeropoulos, Jan 09 2022 *)

A249444 Primes representable as p^q + p + q, where p and q are primes.

Original entry on oeis.org

13, 137, 251, 353, 2213, 4933, 24421, 78137, 148933, 205441, 371311, 493121, 524309, 571873, 912773, 1225153, 1594339, 4330913, 7189253, 13652161, 18191713, 21254213, 28629187, 31855333, 42508901, 49431233, 73560481, 81183173, 99253313, 178454113, 184220581, 192100613
Offset: 1

Views

Author

Alex Ratushnyak, Jan 12 2015

Keywords

Crossrefs

Cf. A053810, A065017, A066938, A096342, A253776.
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