A067604 -id:A067604 - 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

A067603 Least k such that gcd(prime(k)+1, prime(k+1)+1) = 2n.

Original entry on oeis.org

2, 4, 9, 72, 34, 91, 62, 478, 205, 2016, 522, 909, 1440, 5375, 2149, 6610, 7604, 2976, 5229, 7488, 11251, 7499, 8805, 20179, 18526, 70885, 28193, 40985, 33847, 17625, 27069, 77199, 66156, 90764, 26186, 141235, 70317, 856719, 110769, 50523, 217229

Offset: 1

Robert G. Wilson v, Jan 31 2002

nonn

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

			a(1) = 2, the first entry in A066940,
a(2) = 4, the first entry in A066941,
a(3) = 9, the first entry in A066942,
a(4) = 72, the first entry in A066943,
a(5) = 34, the first entry in A066944.
That is to say that the first k-th prime that has gcd(prime(k+1)+1, prime(k)+1) of 2, 4, 6, 8, 10, ..., are k = 2, 4, 9, 72, 34, ..., and the prime_k = 3, 7, 23, 359, 139, 467, 293, ... (A067604).
If the floor of GCD is used, then a(0) equals 1.

Zak Seidov, Robert G. Wilson v, and Charles R Greathouse IV, Table of n, a(n) for n = 1..200 (1..100 terms from Seidov, 101..140 from Wilson, 141..200 from Greathouse)

Main entry is A067604. Cf. A066940, A066941, A066942, A066943, A066944, A000040, A001223.

P = primes(10^8);
G = gcd(P(1:end-1)+1,P(2:end)+1);
A = zeros(1,66);
for n = 1:66
    A(n) = find(G == 2*n, 1, 'first');
end
A   % Robert Israel, Aug 16 2015

t = 0*Range@ 70; p = 3; q = 5; While[p < 15*10^6, d = GCD[p + 1, q + 1]/2; If[ t[[d]] == 0, t[[d]] = PrimePi@ p]; p = q; q = NextPrime@ q]; t

a(n) = p=2; q=3; k=1; while(gcd(p+1, q+1) != 2*n, k++; p=q; q = nextprime(p+1);); k; \\ Michel Marcus, Aug 16 2015

a(n)=my(p=2,k=2*n,t); forprime(q=3,, t++; if((q-p)%k==0 && (p+1)%k==0, return(t)); p=q) \\ Charles R Greathouse IV, Aug 17 2015

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

Conjecture: a(n) = least k such that A001223(k) = 2n and A000040(k) == -1 (mod 2n). - Zak Seidov, Aug 16 2015

Edited by Robert G. Wilson v, Aug 17 2015 at the direction of Zak Seidov

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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A067603 Least k such that gcd(prime(k)+1, prime(k+1)+1) = 2n.

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