A110358 Beginning with 3, the least prime which is the product of one or more previous terms + 2.

3, 5, 7, 17, 19, 23, 37, 53, 59, 61, 71, 73, 97, 107, 109, 113, 163, 179, 181, 257, 293, 307, 347, 349, 359, 367, 373, 401, 439, 487, 491, 499, 547, 557, 631, 751, 773, 797, 853, 881, 883, 887, 907, 971, 1009, 1039, 1049, 1051, 1097, 1103, 1123, 1283, 1297

Offset: 1

Author

Amarnath Murthy, Jul 23 2005

Keywords

• nonn

Comments

Conjecture: The sequence is infinite.

Subbarao & Yip prove that if there is an integer m such that the equation Phi_2(x) = m has a unique solution, where Phi_2 is the 2nd Schemmel totient function (A058026), then x == 0 (mod a(n)^2) for each term in this sequence. They conjectured an analog to Carmichael's conjecture, that this equation has no unique solution to any integer m, and prove that any counterexample to this conjecture is > 10^120000, a bound calculated from the first 10000 terms of this sequence. A proof that this sequence is infinite would prove the conjecture. - Amiram Eldar, Mar 25 2017

Examples

			After 3, 5 and 7 the next term is 3*5 + 2 = 17, then 17 + 2 = 19, then 3*7 + 2 = 23, then 5*7 + 2 = 37, etc.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• M. V. Subbarao and L. W. Yip, Carmichael's conjecture and some analogues, Théorie des nombres/Number Theory: Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, Jean M. de Koninck and Claude Levesque, eds., Walter de Gruyter, 1989, pp. 928-941.

Crossrefs

Cf. A058026.

Programs

Mathematica

L={3}; p=3; While[Length[L] < 100, p = NextPrime@p; If[SquareFreeQ[p - 2] && SubsetQ[L, First /@ FactorInteger[p - 2]], AppendTo[L, p]]]; L (* Giovanni Resta, Mar 25 2017 *)

Extensions

More terms from John Pammer (jcp5027(AT)psu.edu), Oct 10 2005

Corrected and extended by Joshua Zucker, May 08 2006