

A110358


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


1



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
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OFFSET

1,1


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


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, ThÃ©orie des nombres/Number Theory: Proceedings of the International Number Theory Conference held at UniversitÃ© Laval, July 518, 1987, Jean M. de Koninck and Claude Levesque, eds., Walter de Gruyter, 1989, pp. 928941.


EXAMPLE

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.


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 *)


CROSSREFS

Cf. A058026.
Sequence in context: A276044 A114265 A258195 * A038971 A210479 A045400
Adjacent sequences: A110355 A110356 A110357 * A110359 A110360 A110361


KEYWORD

nonn


AUTHOR

Amarnath Murthy, Jul 23 2005


EXTENSIONS

More terms from John Pammer (jcp5027(AT)psu.edu), Oct 10 2005
Corrected and extended by Joshua Zucker, May 08 2006


STATUS

approved



