A281570
Numbers n such that (n+1)^k + (-n)^k is prime for each k = 2, 3, 4, 5, 7, and 8.
Original entry on oeis.org
1, 9387629, 18276717, 40036062, 252447645, 293291802, 319596455, 327091015, 401241904, 421675344, 471333967, 483656680, 1059439524, 1162179372, 1651177394, 2339341839, 2423329650, 2596829984, 2749510742, 2903809499, 2941064795, 2956438949
Offset: 1
9387629 is a term because 9387630^3 - 9387629^3, 9387630^5 - 9387629^5, 9387630^7 - 9387629^7 and 9387629^2 + 9387630^2, 9387629^4 + 9387630^4, 9387629^8 + 9387630^8 are prime numbers.
A307493
Primes that are both centered triangular and centered square.
Original entry on oeis.org
16381, 23199907725541, 873105326726527441, 169377932722437899461, 532026300937919058017204151243671297356368598920355705257429996547710782877327451810988538831181
Offset: 1
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Select[LinearRecurrence[{195, -195, 1}, {1, 85, 16381}, 43], PrimeQ[#] &]
A308341
Hypotenuses of primitive Pythagorean triangles two sides of which are Pythagorean primes.
Original entry on oeis.org
13, 421, 1861, 5101, 16381, 60901, 83641, 100801, 106261, 135721, 161881, 205441, 218461, 337021, 388081, 431521, 571381, 637321, 697381, 926161, 1108561, 1460341, 1515541, 1806901, 1899301, 2334961, 2574181, 2601481, 2740141, 2834581, 2853661, 3248701, 3403441, 3723721, 3889261, 4503001
Offset: 1
13 is a term because 13 and 5 are Pythagorean primes and are sides of {5,12,13}.
421 is a term because 421 and 29 are Pythagorean primes and are sides of {29,420,421}.
1861 is a term because 1861 and 61 are Pythagorean primes and are sides of {61,1860,1861}.
5101 is a term because 5101 and 101 are Pythagorean primes and are sides of {101,5100,5101}.
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hyp(n) = {return((2*((n-1)/2)^2) + (2*((n-1)/2)) + 1);}
lista(n) = forprime(p=2, n, if((p%4 == 1) && isprime(p) && isprime(hyp(p)), print1(hyp(p), ", ")));
lista(3100)
A321616
Primes p = k^2 + (k-1)^2 such that k^p - (k-1)^p is prime.
Original entry on oeis.org
5, 61, 113, 1741
Offset: 1
The prime 5 = 2^2 + 1^2 and 2^5 - 1^5 = 31 is prime.
We have 61 = 6^2 + 5^2, 113 = 8^2 + 7^2, 1741 = 30^2 + 29^2.
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f[k_]:=k^2 + (k-1)^2 ; seqQ[k_]:=Module[{p=f[k]}, PrimeQ[p] && PrimeQ[k^p - (k-1)^p ]]; f[Select[Range[30], seqQ]] (* Amiram Eldar, Nov 15 2018 *)
pQ[k_]:=Module[{c=k^2+(k-1)^2},If[AllTrue[{c,k^c-(k-1)^c},PrimeQ],c,Nothing]]; Array[pQ,30] (* Harvey P. Dale, Aug 27 2023 *)
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lista(nn) = {for (k=1, nn, if (isprime(p=k^2 + (k-1)^2) && isprime(k^p - (k-1)^p), print1(p, ", ")););} \\ Michel Marcus, Nov 18 2018
A380882
Centered square numbers which are sphenic numbers.
Original entry on oeis.org
1105, 2665, 3445, 7565, 8845, 14965, 15665, 16745, 17485, 18241, 20605, 22685, 23545, 27145, 28085, 32005, 32513, 35113, 37265, 48985, 50245, 50881, 55445, 56785, 62305, 71065, 74885, 78013, 80401, 81205, 84461, 85285, 88621, 89465, 109045, 111865, 113765, 116645, 118585, 119561
Offset: 1
1105 = 5 * 13* 17 which is the 24-th centered square number.
18241 = 17 * 29 * 37 which is the 96-th centered square number.
32513 = 13 * 41 * 61 which is 128-th centered square number.
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Select[Table[2*n*(n+1) + 1, {n, 0, 250}], FactorInteger[#][[;; , 2]] == {1, 1, 1} &] (* Amiram Eldar, Feb 07 2025 *)
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