A231608 -id:A231608 - cp's OEIS frontend

A086505 a(n) is the n-th smallest prime p such that p+2n is also prime.

3, 7, 11, 23, 31, 29, 53, 73, 53, 89, 157, 73, 137, 199, 73, 281, 229, 127, 383, 229, 149, 389, 463, 193, 359, 547, 239, 467, 823, 197, 857, 883, 283, 809, 499, 389, 1013, 907, 421, 827, 1201, 373, 1151, 1231, 367, 1307, 1279, 577, 1229, 1009, 631, 1427, 1783

Offset: 1

Amarnath Murthy, Jul 29 2003

nonn

Note the patterns in the graph. These patterns depend on the number of prime factors of n: see color graph for different n's: n primes - black dots, n multiples of 3 - red dots, n multiples of 15 - green dots, n multiples of 105 - blue dots. - Zak Seidov, Nov 28 2013

Zak Seidov, Table of n, a(n) for n = 1..10000
Zak Seidov, Color graph for different n's

Cf. A020483, A231608.

N:= 10^4: # to get all terms before the first with a(n)+2*n > N
Primes:= select(isprime, {seq(2*i+1, i=1..N)}):
for n from 1 do
R:= Primes intersect map(`+`, Primes, -2*n);
if nops(R) < n then break fi;
A[n]:= R[n];
od:
seq(A[j],j=1..n-1); # Robert Israel, Aug 07 2014

Edited by Sam Alexander, Feb 26 2004

A246901 a(n) is the n-th smallest prime p such that p+4n is also prime.

Original entry on oeis.org

3, 5, 11, 31, 41, 23, 73, 131, 47, 97, 149, 83, 229, 167, 89, 337, 311, 167, 307, 293, 149, 499, 509, 211, 457, 509, 311, 607, 743, 211, 787, 839, 331, 877, 521, 419, 1171, 911, 421, 787, 1289, 419, 1279, 1103, 433, 1327, 1361, 619, 1123, 1103, 617, 1663, 1721, 661, 1039, 1553, 739, 2179, 2111, 599

Offset: 1

Zak Seidov, Nov 16 2014

nonn

There is an array defined by: A(n,k) is the k-th smallest prime such that p+4*n is also prime (analog of A231608). It starts
 7  13  19  37  43  67  79  97 103
 5  11  23  29  53  59  71  89 101
 7  11  17  19  29  31  41  47  59
 7  13  31  37  43  67  73  97 151
11  17  23  41  47  53  59  83  89
 7  13  17  19  23  29  37  43  47
13  19  31  43  61  73  79 103 109
11  29  41  47  71 107 131 149 167
 7  11  17  23  31  37  43  47  53
 7  13  19  31  43  61  67  73  97
a(n) = A(n,n) reads along the main diagonal of this array. - R. J. Mathar, Nov 19 2014

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A086505.

a(n) = nb=0; forprime(p=3,, if (isprime(p+4*n), nb++; if (nb==n, return (p)))); \\ Michel Marcus, Nov 16 2014

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

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A086505 a(n) is the n-th smallest prime p such that p+2n is also prime.

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A246901 a(n) is the n-th smallest prime p such that p+4n is also prime.

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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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