A049481 -id:A049481 - cp's OEIS frontend

A140560 Primes p such that p + 30 is not a prime.

2, 3, 5, 19, 47, 61, 89, 103, 113, 131, 139, 157, 173, 179, 191, 223, 229, 257, 269, 271, 293, 311, 313, 331, 347, 373, 383, 397, 421, 439, 443, 463, 467, 487, 499, 503, 509, 521, 523, 593, 599, 607, 619, 641, 659, 673, 677, 683, 691, 701, 719, 733, 751, 761

Offset: 1

Juri-Stepan Gerasimov, Jul 03 2008

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A049481.

[p: p in PrimesUpTo(1000)|not IsPrime(p+30)] // Vincenzo Librandi, Dec 18 2010

A140560:=n->`if`(isprime(n) and not isprime(n+30), n, NULL): seq(A140560(n), n=1..2*10^3); # Wesley Ivan Hurt, Apr 14 2017

Select[Prime[Range[200]],!PrimeQ[#+30]&] (* Harvey P. Dale, Oct 21 2013 *)

isok(n) = isprime(n) && !isprime(n+30); \\ Michel Marcus, Apr 14 2017

from sympy import isprime, primerange
print([p for p in primerange(1, 1001) if not isprime(p + 30)]) # Indranil Ghosh, Apr 14 2017

Corrected (3, 5 inserted) by R. J. Mathar, Apr 25 2010

A265651 Numbers n such that n-29, n-1, n+1 and n+29 are consecutive primes.

Original entry on oeis.org

14592, 84348, 151938, 208962, 241392, 254490, 397182, 420192, 494442, 527700, 549978, 581982, 637200, 641550, 712602, 729330, 791628, 850302, 975552, 995052, 1086558, 1107852, 1157670, 1245450, 1260798, 1286148, 1494510, 1555290, 1608912

Offset: 1

Karl V. Keller, Jr., Dec 11 2015

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 0 are divisible by 30 (cf. A249674).
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.
The numbers n-29 and n+1 belong to A252090 (p and p+28 are primes) and A124595 (p where p+28 is the next prime).
The numbers n-29 and n-1 belong to A049481 (p and p+30 are primes).

			14592 is the average of the four consecutive primes 14563, 14591, 14593, 14621.
84348 is the average of the four consecutive primes 84319, 84347, 84349, 84377.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

Select[Prime@Range@100000, NextPrime[#, {1, 2, 3}] == {28, 30, 58} + # &] + 29 (* Vincenzo Librandi, Dec 12 2015 *)
Mean/@Select[Partition[Prime[Range[125000]],4,1],Differences[#]=={28,2,28}&] (* Harvey P. Dale, May 02 2016 *)

from sympy import isprime,prevprime,nextprime
for i in range(0,1000001,6):
   if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-29 and nextprime(i+1) == i+29 :  print (i,end=', ')

A270754 Numbers n such that n - 31, n - 1, n + 1 and n + 31 are consecutive primes.

Original entry on oeis.org

90438, 258918, 293862, 385740, 426162, 532950, 1073952, 1317192, 1318410, 1401318, 1565382, 1894338, 1986168, 2174772, 2612790, 2764788, 3390900, 3450048, 3618960, 3797250, 3961722, 3973062, 4074870, 4306230, 4648068, 4917360, 5351010, 5460492

Offset: 1

Karl V. Keller, Jr., Mar 22 2016

nonn

This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 0 are divisible by 30 (cf. A249674).
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.
The numbers n - 31 and n + 1 belong to A049481 (p and p + 30 are primes) and A124596 (p where p + 30 is the next prime).
The numbers n - 31 and n - 1 belong to A049489 (p and p + 32 are primes).

			90438 is the average of the four consecutive primes 90407, 90437, 90439, 90469.
258918 is the average of the four consecutive primes 258887, 258917, 258919, 258949.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

from sympy import isprime,prevprime,nextprime
for i in range(0,1000001,6):
   if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-31 and nextprime(i+1) == i+31 :  print (i,end=', ')

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

Original entry on oeis.org

3, 5, 3, 11, 7, 5, 17, 13, 7, 3, 29, 19, 11, 5, 3, 41, 37, 13, 11, 7, 5, 59, 43, 17, 23, 13, 7, 3, 71, 67, 23, 29, 19, 11, 5, 3, 101, 79, 31, 53, 31, 17, 17, 7, 5, 107, 97, 37, 59, 37, 19, 23, 13, 11, 3, 137, 103, 41, 71, 43, 29, 29, 31, 13, 11, 7, 149, 109

Offset: 1

Felix Fröhlich, Jul 28 2019

The same as A231608 except that A231608 gives the upward antidiagonals of the array, while this sequence gives the downward antidiagonals.
Conjecture: All values are nonzero, i.e., for any even integer e there are infinitely many primes p such that p + e is also prime.
The conjecture is true if Polignac's conjecture is true.

			The array starts as follows:
3,  5, 11, 17, 29, 41, 59,  71, 101, 107, 137, 149, 179, 191
3,  7, 13, 19, 37, 43, 67,  79,  97, 103, 109, 127, 163, 193
5,  7, 11, 13, 17, 23, 31,  37,  41,  47,  53,  61,  67,  73
3,  5, 11, 23, 29, 53, 59,  71,  89, 101, 131, 149, 173, 191
3,  7, 13, 19, 31, 37, 43,  61,  73,  79,  97, 103, 127, 139
5,  7, 11, 17, 19, 29, 31,  41,  47,  59,  61,  67,  71,  89
3,  5, 17, 23, 29, 47, 53,  59,  83,  89, 113, 137, 149, 167
3,  7, 13, 31, 37, 43, 67,  73,  97, 151, 157, 163, 181, 211
5, 11, 13, 19, 23, 29, 41,  43,  53,  61,  71,  79,  83,  89
3, 11, 17, 23, 41, 47, 53,  59,  83,  89, 107, 131, 137, 173
7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 229, 241
5,  7, 13, 17, 19, 23, 29,  37,  43,  47,  59,  73,  79,  83

Wikipedia, Polignac's conjecture

Cf. A231608.

Cf. A001359 (row 1), A023200 (row 2), A023201 (row 3), A023202 (row 4), A023203 (row 5), A046133 (row 6), A153417 (row 7), A049488 (row 8), A153418 (row 9), A153419 (row 10), A242476 (row 11), A033560 (row 12), A252089 (row 13), A252090 (row 14), A049481 (row 15), A049489 (row 16), A252091 (row 17), A156104 (row 18), A271347 (row 19), A271981 (row 20), A271982 (row 21), A272176 (row 22), A062284 (row 25), A049490 (row 32), A020483 (column 1).

row(n, terms) = my(i=0); forprime(p=1, , if(i>=terms, break); if(ispseudoprime(p+2*n), print1(p, ", "); i++))
array(rows, cols) = for(x=1, rows, row(x, cols); print(""))
array(12, 14) \\ Print initial 12 rows and 14 columns of the array

cp's OEIS Frontend

A140560 Primes p such that p + 30 is not a prime.

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A265651 Numbers n such that n-29, n-1, n+1 and n+29 are consecutive primes.

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A270754 Numbers n such that n - 31, n - 1, n + 1 and n + 31 are consecutive primes.

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Python

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

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PARI