A134735 -id:A134735 - cp's OEIS frontend

A219606 Prime gaps and primes interleaved.

1, 2, 2, 3, 2, 5, 4, 7, 2, 11, 4, 13, 2, 17, 4, 19, 6, 23, 2, 29, 6, 31, 4, 37, 2, 41, 4, 43, 6, 47, 6, 53, 2, 59, 6, 61, 4, 67, 2, 71, 6, 73, 4, 79, 6, 83, 8, 89, 4, 97, 2, 101, 4, 103, 2, 107, 4, 109, 14, 113, 4, 127, 6, 131, 2, 137, 10, 139, 2, 149, 6

Offset: 1

Reinhard Zumkeller, Dec 12 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Gaps
Wikipedia, Prime gap
Index entries for primes, gaps between

Cf. A134735.

import Data.List (transpose)
a219606 n = a219606_list !! (n-1)
a219606_list = concat $ transpose [a001223_list, a000040_list]

A224888 Primes of the form p^2 + (q-p)^2, where p and q are consecutive primes.

Original entry on oeis.org

5, 13, 29, 293, 997, 6257, 11897, 18773, 19421, 52457, 73477, 109597, 120413, 167381, 192737, 218233, 249017, 292717, 333029, 361237, 398261, 466553, 502781, 546137, 552113, 591377, 635353, 683933, 687341, 704117, 737897, 885517, 966353, 982117, 1018097, 1079621

Offset: 1

Thomas Ordowski, Jul 24 2013

nonn

Primes of the form A000040(n)^2 + A001223(n)^2.
Primes of the form A134735(2n-1)^2 + A134735(2n)^2.
Conjecture: a(n) ~ A093343(n).
There are 20421247 members of this sequence below 10^20. - Charles R Greathouse IV, Jul 29 2013

			3 and 5 are consecutive primes and 3^2 + (5-3)^2 = 9 + 4 = 13 is prime, so 13 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A093343.

Select[Table[Prime[n]^2 + (Prime[n + 1] - Prime[n])^2, {n, 200}], PrimeQ] (* Alonso del Arte, Jul 29 2013 *)

p=2;forprime(q=3,1e4,if(isprime(t=p^2+(q-p)^2),print1(t", "));p=q) \\ Charles R Greathouse IV, Jul 24 2013

c(x) is O( sqrt(x/log x) / log x ), where c(x) is the counting function, the number of terms less than x.

a(5), a(9)-a(36) from Charles R Greathouse IV, Jul 24 2013

A340202 Primes followed by the difference from the next prime calculated by adding the digits of the lexicographically earliest distinct term.

Original entry on oeis.org

2, 1, 3, 20, 5, 110, 7, 4, 11, 200, 13, 22, 17, 1001, 19, 40, 23, 6, 29, 1010, 31, 15, 37, 112, 41, 1100, 43, 121, 47, 24, 53, 33, 59, 2000, 61, 42, 67, 130, 71, 10001, 73, 51, 79, 202, 83, 60, 89, 8, 97, 220, 101, 10010, 103, 301, 107, 10100, 109, 310, 113

Offset: 1

Carole Dubois, Dec 31 2020

			Prime 2 + (1) = prime 3;
prime 3 + (2+0) = prime 5;
prime 5 + (1+1+0) = prime 7;
prime 7 + (4) = prime 11;
prime 11 + (2+0+0) = prime 13;
prime 13 + (2+2) = 17; etc.

Cf. A000040 (primes), A219606.

Differences between consecutive primes: A001223, A134735, A340063.

cp's OEIS Frontend

A219606 Prime gaps and primes interleaved.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

A224888 Primes of the form p^2 + (q-p)^2, where p and q are consecutive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A340202 Primes followed by the difference from the next prime calculated by adding the digits of the lexicographically earliest distinct term.

Views

Author

Keywords

Examples

Crossrefs