A058264 Smallest prime p of two consecutive primes, p < q, such that gcd( p-1, q-1 ) = 2n.

3, 13, 31, 89, 181, 661, 113, 2113, 523, 13421, 2311, 4177, 35543, 39901, 4831, 44417, 1327, 12853, 119321, 52321, 82657, 36389, 136897, 203713, 95651, 59281, 255259, 178697, 531919, 427621, 2640581, 1414849, 643303, 3021173, 175141, 1770337, 514967, 1004797, 1354393

Offset

1,1

Comments

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..183

Formula

a(n) = prime(A067605(n)). - Amiram Eldar, Mar 05 2025

Example

a(4) = 89 because gcd(89-1, 97-1) = gcd(8*11, 8*16) = 8 = 2*4 and these primes are the smallest with this property.
a(49) = 604073 because gcd(604073-1, 604171-1) = gcd(6164*98, 6165*98) = 98 = 2*49.

Mathematica

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}]; a
With[{tsp={#[[1]], #[[2]], GCD[#[[1]]-1, #[[2]]-1]}&/@Partition[Prime[ Range[ 300000]], 2, 1]}, Transpose[Flatten[Table[Select[tsp, Last[#]==2n&, 1], {n, 40}], 1]][[1]]] (* Harvey P. Dale, Jul 07 2013 *)

Prog

(PARI) list(len) = {my(v = vector(len), c = 0, p = 3, i); forprime(q = 5, , i = gcd(p-1, q-1)/2; if(i <= len && v[i] == 0, v[i] = p; c++; if(c == len, break)); p = q); v; } \\ Amiram Eldar, Mar 05 2025

Crossrefs

Cf. A006093, A058263, A067605.

Sequence in context: A147134 A146842 A146191 * A242905 A213970 A034050

Adjacent sequences: A058261 A058262 A058263 * A058265 A058266 A058267

Keyword

nonn

Author

Labos Elemer, Dec 06 2000

Extensions

Edited by Robert G. Wilson v, Feb 01 2002

Status

approved