A067603 -id:A067603 - cp's OEIS frontend

A066940 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 2.

2, 3, 5, 6, 7, 10, 11, 12, 13, 17, 18, 20, 21, 24, 25, 26, 28, 29, 30, 33, 35, 36, 37, 41, 42, 43, 44, 45, 49, 50, 52, 53, 57, 58, 59, 60, 61, 64, 65, 66, 67, 68, 69, 70, 73, 74, 77, 78, 79, 81, 82, 83, 84, 87, 88, 89, 98, 99, 100, 101, 104, 105, 106, 109, 110, 111, 112, 113

Offset: 1

Benoit Cloitre, Jan 24 2002

nonn

Numbers k such that A063086(k) = 2. - Andrew Howroyd, Dec 10 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A063086, A066941, A066942, A066943, A066944, A067603, A067604.

Select[ Range[120], GCD[ Prime[ # + 1] + 1, Prime[ # ] + 1] == 2 & ]

isok(k) = { gcd(prime(k+1) + 1, prime(k) + 1) == 2 } \\ Harry J. Smith, Apr 09 2010

Edited by Robert G. Wilson v, Feb 01 2002

A066941 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 4.

Original entry on oeis.org

4, 8, 14, 19, 22, 27, 31, 38, 46, 47, 48, 63, 75, 85, 90, 93, 94, 95, 114, 117, 124, 131, 143, 149, 153, 154, 155, 163, 181, 192, 207, 213, 224, 229, 232, 235, 241, 242, 247, 248, 249, 261, 276, 285, 299, 303, 304, 314, 327, 328, 333, 334, 335, 348, 364, 370

Offset: 1

Benoit Cloitre, Jan 24 2002

nonn

Numbers k such that A063086(k) = 4. - Andrew Howroyd, Dec 10 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A063086, A066940, A066942, A066943, A066944, A067603, A067604.

Select[ Range[400], GCD[ Prime[ # + 1] + 1, Prime[ # ] + 1] == 4 & ]

isok(k) = { gcd(prime(k+1) + 1, prime(k) + 1) == 4 } \\ Harry J. Smith, Apr 10 2010

Edited by Robert G. Wilson v, Feb 01 2002

A066942 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 6.

Original entry on oeis.org

9, 15, 16, 23, 32, 39, 40, 51, 54, 55, 56, 71, 76, 86, 96, 97, 102, 103, 107, 108, 118, 119, 123, 139, 160, 161, 164, 165, 170, 184, 185, 194, 195, 199, 200, 208, 218, 219, 227, 238, 245, 252, 255, 267, 290, 291, 292, 293, 298, 311, 312, 329, 342, 345, 349

Offset: 1

Benoit Cloitre, Jan 24 2002

nonn

Numbers k such that A063086(k) = 6. - Andrew Howroyd, Dec 10 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A063086, A066940, A066941, A066943, A066944, A067603, A067604.

Select[ Range[400], GCD[ Prime[ # + 1] + 1, Prime[ # ] + 1] == 6 & ]
Drop[Position[Partition[Prime[Range[400]],2,1],?(GCD@@(#+1)==6&)],2]//Flatten (* _Harvey P. Dale, Nov 08 2022 *)

isok(k) = { gcd(prime(k+1) + 1, prime(k) + 1) == 6 } \\ Harry J. Smith, Apr 10 2010

Edited by Robert G. Wilson v, Feb 01 2002

A066943 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 8.

Original entry on oeis.org

72, 92, 128, 132, 156, 166, 228, 246, 281, 282, 386, 417, 507, 519, 619, 620, 640, 641, 661, 712, 738, 739, 759, 801, 853, 898, 915, 1000, 1077, 1152, 1241, 1246, 1273, 1289, 1297, 1364, 1389, 1421, 1489, 1493, 1525, 1543, 1564, 1632, 1691, 1699, 1729

Offset: 1

Benoit Cloitre, Jan 24 2002

nonn

Numbers k such that A063086(k) = 8. - Andrew Howroyd, Dec 10 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A063086, A066940, A066941, A066942, A066944, A067603, A067604.

Select[ Range[120], GCD[ Prime[ # + 1] + 1, Prime[ # ] + 1] == 8 & ]

isok(k) = { gcd(prime(k+1) + 1, prime(k) + 1) == 8 } \\ Harry J. Smith, Apr 10 2010

Edited by Robert G. Wilson v, Feb 01 2002

A066944 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 10.

Original entry on oeis.org

34, 80, 127, 145, 157, 175, 204, 222, 266, 289, 308, 316, 397, 442, 443, 518, 525, 578, 593, 656, 690, 746, 757, 773, 793, 832, 861, 866, 892, 908, 923, 949, 958, 971, 985, 1013, 1029, 1051, 1071, 1102, 1125, 1195, 1236, 1314, 1329, 1340, 1350, 1403, 1434

Offset: 1

Benoit Cloitre, Jan 24 2002

nonn

Numbers k such that A063086(k) = 10. - Andrew Howroyd, Dec 10 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A063086, A066940, A066941, A066942, A066943, A067603, A067604.

Select[ Range[1500], GCD[ Prime[ # + 1] + 1, Prime[ # ] + 1] == 10 & ]
PrimePi/@Select[Partition[Prime[Range[1500]],2,1],GCD@@(#+1)==10&][[All,1]] (* Harvey P. Dale, May 05 2018 *)

isok(k) = { gcd(prime(k+1) + 1, prime(k) + 1) == 10 } \\ Harry J. Smith, Apr 10 2010

Edited by Robert G. Wilson v, Feb 01 2002

A067604 Smallest prime p of two consecutive primes, p < q, such that gcd(p+1, q+1) = 2n.

Original entry on oeis.org

3, 7, 23, 359, 139, 467, 293, 3391, 1259, 17519, 3739, 7079, 12011, 52639, 18869, 66239, 77383, 27143, 51071, 76039, 119447, 76163, 91033, 226943, 206699, 894451, 327347, 492911, 399793, 195599, 313409, 981823, 829883, 1169939, 302329

Offset: 1

Robert G. Wilson v, Jan 31 2002

nonn

Since all consecutive primes, p < q and p greater than 2, are odd, therefore gcd(p+1, q+1) must be even.

			a(1) = 3, the 3rd prime being the first entry in A066940;
a(2) = 7, the 4th prime being the first entry in A066941;
a(3) = 23, the 9th prime being of the first entry in A066942;
a(4) = 359, the 72nd prime being the first entry in A066943;
a(5) = 139, the 34th prime being the first entry in A066944.

Charles R Greathouse IV, Table of n, a(n) for n = 1..200

Cf. A066940, A066941, A066942, A066944, A067603.

a = Table[0, {100}]; p = 3; q = 5; Do[q = Prime[n + 1]; d = GCD[p + 1, q + 1]/2; If[d < 101 && a[[d]] == 0, a[[d]] = n]; b = c, {n, 2, 10^7}]; Prime[a]

a(n)=my(k=2*n); forstep(p=k-1,oo,k, if(isprime(p) && (nextprime(p+1)-p)%k==0, return(p))) \\ Charles R Greathouse IV, Aug 17 2015

cp's OEIS Frontend

A066940 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 2.

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A066941 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 4.

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A066942 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 6.

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A066943 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 8.

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A066944 Numbers k such that gcd(prime(k+1) + 1, prime(k) + 1) = 10.

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A067604 Smallest prime p of two consecutive primes, p < q, such that gcd(p+1, q+1) = 2n.

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