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A374029 Lexicographically earliest strictly increasing sequence of prime numbers whose partial products, starting from the second, are all Fermat pseudoprimes to base 2 (A001567).

%I A374029 #13 Jun 30 2024 06:34:04
%S A374029 11,31,41,61,181,54001,54721,61561,123121,225721,246241,430921,523261,
%T A374029 800281,2400841,9603361,28810081,76826881,96033601,15909022209601,
%U A374029 133133396385601,5791302742773601,15443473980729601,61773895922918401,386086849518240001,13706083157897520001
%N A374029 Lexicographically earliest strictly increasing sequence of prime numbers whose partial products, starting from the second, are all Fermat pseudoprimes to base 2 (A001567).
%C A374029 Are all the terms of the form 10*k+1?
%H A374029 Daniel Suteu, <a href="/A374029/b374029.txt">Table of n, a(n) for n = 1..28</a>
%H A374029 <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>.
%e A374029 The partial products begin with 11, 11 * 31 = 341 = A001567(1), 11 * 31 * 41 = 13981 = A001567(29), 11 * 31 * 41 * 61 = 852841 = A001567(234), 11 * 31 * 41 * 61 * 181 = 154364221 = A001567(2509), ... .
%t A374029 pspQ[n_] := PowerMod[2, n - 1, n] == 1; a[1] = 11; a[n_] := a[n] = Module[{p = NextPrime[a[n-1]], r = Product[a[i], {i, 1, n - 1}]}, While[! pspQ[p*r], p = NextPrime[p]]; p]; Array[a, 10]
%o A374029 (PARI) ispsp(n) = Mod(2, n)^(n-1) == 1;
%o A374029 lista(len) = {my(prd = 1, c = 0, k = 11); while(c < len, while(!ispsp(prd * k), k = nextprime(k+1)); prd *= k; print1(k,", "); c++);}
%o A374029 (PARI) my(P=List(11), base=2); my(m = vecprod(Vec(P))); my(L = znorder(Mod(base, m))); print1(P[1], ", "); while(1, forstep(p=lift(1/Mod(m, L)), oo, L, if(p > P[#P] && isprime(p) && base % p != 0, if((m*p-1) % znorder(Mod(base, p)) == 0, print1(p, ", "); listput(P, p); L = lcm(L, znorder(Mod(base, p))); m *= p; break)))); \\ _Daniel Suteu_, Jun 30 2024
%Y A374029 Cf. A001567, A374027, A374028.
%K A374029 nonn
%O A374029 1,1
%A A374029 _Amiram Eldar_, Jun 26 2024
%E A374029 a(20)-a(26) from _Daniel Suteu_, Jun 30 2024