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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.
0
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
1,1
COMMENTS
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.
EXAMPLE
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
PROG
(PARI) 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
CROSSREFS
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).
Sequence in context: A242034 A336863 A205708 * A089730 A351368 A105445
KEYWORD
nonn,tabl
AUTHOR
Felix Fröhlich, Jul 28 2019
STATUS
approved