A153417 -id:A153417 - cp's OEIS frontend

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.

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

A365850 Numbers k for which k^2 + (k')^2 is a square, where k' is the arithmetic derivative of k (A003415).

Original entry on oeis.org

0, 1, 12, 15, 35, 81, 143, 323, 400, 441, 899, 1540, 1763, 2700, 3599, 4641, 5183, 5929, 9375, 10395, 10403, 11663, 13585, 18225, 19043, 21952, 22499, 30576, 32399, 35581, 36863, 39203, 48841, 51983, 57599, 72899, 79523, 97343, 121103, 148176, 166375, 175692, 176399

Offset: 1

Marius A. Burtea, Oct 09 2023

nonn

If p and p + 2 are twin primes (A001359) then m = p*(p + 2) is a term. Indeed, m' = p + (p + 2) = 2*p + 2 and m^2 + (m')^2 = p^2*(p + 2)^2 + (2*p + 2)^2 = (p^2 + 2*p + 2)^2.

More generally, if p and p + 2*k, k >= 1, are prime numbers, then m = p^k*(p + 2*k)^k is a term. Indeed, m' = k*p^(k - 1)*(p + 2*k)^k+ k*p^k*(p + 2*k)^(k - 1) = k*p^(k - 1)*(p + 2*k)^(k-1)*(2*p + 2*k). Thus, m^2 + (m')^2 = p^(2*k)*(p + 2*k)^(2*k) + (k^2)*p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(2*p + 2*k)^2 = p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(p^2*(p + 2*k)^2 + k^2*(2*p + 2*k)^2) = p^(2*k - 2)*(p + 2*k)^(2*k - 2)*(2*k^2 + 2*k*p + p^2)^2.

			For k = 12, k' = 16 and 12^2 + 16^2 = 144 + 256 = 400 = 20^2, so 12 is a term.
For k = 15, k' = 8 and 15^2 + 8^2 = 225 + 64 = 289 = 17^2, so 15 is a term.
For k = 143, k' = 24 and 143^2 + 24^2 = 144 + 256 = 21025 = 145^2, so 143 is a term.

Cf. A077800, A153417 - A153419, A156104, A242476, A252089 - A252091, A271347, A271981, A271982, A272176.

f:=func; [n:n in [0..200000] |IsSquare( n^2+( Floor(f(n))^2))];

ader:= proc(n) local f;   n*add(f[2]/f[1], f=ifactors(n)[2]) end proc:
select(t -> issqr(t^2 + ader(t)^2), [$0..10^6]; # Robert Israel, Oct 17 2023

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[0, 180000], IntegerQ[Sqrt[#^2 + d[#]^2]] &] (* Amiram Eldar, Oct 11 2023 *)

cp's OEIS Frontend

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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