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%I #47 Dec 15 2021 18:36:53
%S 0,1,1,2,1,2,3,6,1,1,8,2,1,2,583,1,5,4,2,3,2,2,1,1,2,3,16,3,6,1,2,1,3,
%T 2,3,4,8,2,7,1,1,4,1,2,15,2,20,8,11,6,1,1,36,1,279,29,3,4,2,1,30,1,2,
%U 9,4,7,4,4,3,10,21,1,12,2,14,6393,11,4,3,2,1,4,1,2,6,1,3,8,5,6,19,3,2,1,2,5
%N Least m such that 1 + prime(n)*2^m is a prime, or -1 if no such m exists.
%C Primes p such that p * 2^m + 1 is composite for all m are called Sierpiński numbers. The smallest known prime Sierpiński number is 271129. Currently, 10223 is the smallest prime whose status is unknown.
%C For 0 < k < a(n), prime(n)*2^k is a nontotient. See A005277. - _T. D. Noe_, Sep 13 2007
%C With the discovery of the primality of 10223 * 2^31172165 + 1 on November 6, 2016, we now know that 10223 is not a Sierpiński number. The smallest prime of unknown status is thus now 21181. The smallest confirmed instance of a(n) = -1 is for n = 78557. - _Alonso del Arte_, Dec 16 2016 [Since we only care about prime Sierpiński numbers in this sequence, 78557 should be replaced by primepi(271129) = 23738. - _Jianing Song_, Dec 15 2021]
%C Aguirre conjectured that, for every n > 1, a(n) is even if and only if prime(n) mod 3 = 1 (see the MathStackExchange link below). - _Lorenzo Sauras Altuzarra_, Feb 12 2021
%C If prime(n) is not a Fermat prime, then a(n) is also the least m such that prime(n)*2^m is a totient number, or -1 if no such m exists. If prime(n) = 2^2^e + 1 is a Fermat prime, then the least m such that prime(n)*2^m is a totient number is min{2^e, a(n)} if a(n) != -1 or 2^e if a(n) = -1, since 2^2^e * (2^2^e + 1) = phi((2^2^e+1)^2) is a totient number. For example, the least m such that 257*2^m is a totient number is m = 8, rather than a(primepi(257)) = 279; the least m such that 65537*2^m is a totient number is m = 16, rather than a(primepi(65537)) = 287. - _Jianing Song_, Dec 15 2021
%D See A046067.
%H T. D. Noe, <a href="/A057192/b057192.txt">Table of n, a(n) for n = 1..1000</a>
%H Ray Ballinger and Wilfrid Keller, <a href="http://www.prothsearch.com/sierp.html">Sierpiński Problem</a>
%H Timothy Revell, <a href="https://www.newscientist.com/article/2113283-crowdsourced-prime-number-could-help-solve-a-50-year-old-problem/">"Crowdsourced prime number could help solve a 50-year-old problem"</a>, New Scientist, November 18, 2016.
%H Tejas R. Rao, <a href="https://arxiv.org/abs/1811.06070">Effective primality test for p*2^n+1, p prime, n>1</a>, arXiv:1811.06070 [math.NT], 2018.
%H Seventeen or Bust, <a href="http://www.seventeenorbust.com/">A Distributed Attack on the Sierpinski problem</a>
%H StackExchange, <a href="https://math.stackexchange.com/questions/153488">If p is prime, does there exist a positive integer k such that 2^k*p + 1 is also prime?</a>
%e a(8) = 6 because prime(8) = 19 and the first prime in the sequence 1 + 19 * {2, 4, 8,1 6, 32, 64} = {39, 77, 153, 305, 609, 1217} is 1217 = 1 + 19 * 2^6.
%p a := proc(n)
%p local m:
%p m := 0:
%p while not isprime(1+ithprime(n)*2^m) do m := m+1: od:
%p m:
%p end: # _Lorenzo Sauras Altuzarra_, Feb 12 2021
%t Table[p = Prime[n]; k = 0; While[Not[PrimeQ[1 + p * 2^k]], k++]; k, {n, 100}] (* _T. D. Noe_ *)
%o (PARI) a(n) = my(m=0, p=prime(n)); while (!isprime(1+p*2^m), m++); m; \\ _Michel Marcus_, Feb 12 2021
%Y Cf. A058887, A058811, A058812, A002202, A014197, A005277, A005384, A005385, A051686.
%Y Cf. A046067 (least k such that (2n - 1) * 2^k + 1 is prime).
%Y a(n) = -1 if and only if n is in A076336.
%K sign
%O 1,4
%A _Labos Elemer_, Jan 10 2001
%E Corrected by _T. D. Noe_, Aug 03 2005
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