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 1-3 of 3 results.

A242230 Primes p of the form p^2 + q + 1 where p < q are consecutive primes.

Original entry on oeis.org

61, 4561, 9511, 17299, 19471, 26737, 30109, 37447, 49957, 69439, 94561, 196699, 209311, 259603, 317539, 333517, 352249, 414097, 427069, 459013, 678157, 845491, 886429, 943819, 1027189, 1217719, 1410163, 1472587, 1647379, 2165323, 2200777, 2230549, 2603389
Offset: 1

Views

Author

K. D. Bajpai, May 08 2014

Keywords

Examples

			a(1) = 61 = 7^2 + 11 + 1: 61 is prime, 7 and 11 are consecutive primes.
a(2) = 4561 = 67^2 + 71 + 1: 4561 is prime, 67 and 71 are consecutive primes.

Links

Crossrefs

Cf. A242231, A000040, A241945, A045636, A214723, A214511.

Programs

  • Maple
    with(numtheory): A242230:= proc()local k ; k:=(ithprime(x)^2+ithprime(x+1)+1); if  isprime(k) then RETURN (k); fi;end: seq(A242230 (),x=1..500);
  • Mathematica
    A242230 = {}; Do[p = Prime[n]^2 + Prime[n + 1] + 1; If[PrimeQ[p], AppendTo[A242230, p]], {n, 500}]; A242230
Select[#[[1]]^2+#[[2]]+1&/@Partition[Prime[Range[300]],2,1],PrimeQ] (* Harvey P. Dale, Mar 28 2016 *)

A242231 Primes p of the form p^2 + q - 1 where p < q are consecutive primes.

Original entry on oeis.org

13, 31, 59, 307, 383, 557, 997, 1409, 1723, 3541, 5113, 5407, 6323, 6977, 8017, 10303, 19469, 52673, 94559, 109897, 151717, 158009, 187927, 193163, 249503, 274069, 326617, 361807, 383791, 419261, 427067, 546863, 573809, 592133, 636017, 684757, 735307, 738743
Offset: 1

Views

Author

K. D. Bajpai, May 08 2014

Keywords

Examples

			a(1) = 13 = 3^2 + 5 - 1: 13 is prime, 3 and 5 are consecutive primes.
a(2) = 31 = 5^2 + 7 - 1: 31 is prime, 5 and 7 are consecutive primes.

Links

Crossrefs

Cf. A242230, A000040, A241945, A045636, A214723, A214511.

Programs

  • Maple
    with(numtheory): A242231:= proc()local k ; k:=(ithprime(x)^2+ithprime(x+1)-1);if  isprime(k) then RETURN (k); fi;end: seq(A242231 (),x=1..500);
  • Mathematica
    A242231 = {}; Do[p = Prime[n]^2 + Prime[n + 1] - 1; If[PrimeQ[p], AppendTo[A242231, p]], {n, 500}]; A242231
Select[#[[1]]^2+#[[2]]-1&/@Partition[Prime[Range[250]],2,1],PrimeQ] (* Harvey P. Dale, Mar 05 2022 *)

A284091 Indices n where prime(n) + 2*prime(n+1) and 2*prime(n) + prime(n+1) have the same number of prime divisors counted with multiplicity.

Original entry on oeis.org

2, 3, 6, 11, 12, 15, 16, 17, 19, 20, 23, 25, 27, 30, 33, 34, 37, 38, 47, 48, 51, 53, 56, 57, 58, 60, 66, 68, 75, 76, 77, 78, 79, 86, 87, 89, 90, 93, 94, 99, 100, 101, 107, 110, 123, 124, 128, 133, 137, 138, 139, 141, 143, 145, 147, 151
Offset: 1

Views

Author

Zak Seidov, Mar 19 2017

Keywords

Examples

			n = 15, prime(n) = 47, prime(n+1) = 53, both 2*47 + 53 = 147 = 3*7^2 and 47 + 2*53 = 153 = 3^2*17 are products of 3 primes.

Links

Crossrefs

A241945 is a subsequence.
Cf. A000040, A001222.

Programs

  • Maple
    select(t -> numtheory:-bigomega(2*ithprime(t)+ithprime(t+1)) = numtheory:-bigomega(ithprime(t)+2*ithprime(t+1)), [$1..1000]); # Robert Israel, Mar 20 2017
  • Mathematica
    Select[Range[1000],PrimeOmega[{2,1}.{(p=Prime[#]),(q=Prime[#+1])}]==PrimeOmega[{1,2}.{p,q}]&]
  • PARI
    list(lim)=my(v=List(),p=2,n); forprime(q=3,, if(n++>lim, break); if(bigomega(p+2*q)==bigomega(2*p+q), listput(v,n)); p=q); Vec(v) \\ Charles R Greathouse IV, Mar 20 2017

Formula

n such that A001222(A000040(n)+2*A000040(n+1))=A001222(2*A000040(n)+A000040(n+1)). - Robert Israel, Mar 20 2017
Showing 1-3 of 3 results.

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