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A375552 Yet unseen terms in the enumeration of A375553, prepended by [2, 5].

%I A375552 #38 Jun 21 2025 07:54:22
%S A375552 2,5,3,7,19,31,11,17,13,29,23,41,47,139,61,37,53,67,59,43,89,103,109,
%T A375552 83,73,97,79,101,71,131,167,137,107,199,163,151,191,233,113,127,227,
%U A375552 211,179,173,239,157,193,149,181,277,313,197,223,307,251,271,331,263,241,229,349
%N A375552 Yet unseen terms in the enumeration of A375553, prepended by [2, 5].
%C A375552 Conjecture: This is a permutation of the prime numbers.
%H A375552 Chai Wah Wu, <a href="/A375552/b375552.txt">Table of n, a(n) for n = 1..242</a>
%p A375552 aList := proc(upto) local P, p, q, Y; Y := 2,5;
%p A375552    P := select(isprime, [seq(2..upto)]):
%p A375552    for p in P do for q in P do
%p A375552       if isprime(q+(p+q)*10^(1+ilog10(q))) then break fi od:
%p A375552    if not member(q, [Y]) then Y := Y,q fi od;
%p A375552 Y end: aList(100000);
%t A375552 spq[p_] := Module[{k = 2}, While[!PrimeQ[(p + k)*10^IntegerLength[k] + k], k = NextPrime[k]]; k];
%t A375552 Join[{2, 5}, DeleteDuplicates @ Table[spq[p], {p, Prime[Range[30000]]}]]
%t A375552 (* _Jean-François Alcover_, Oct 01 2024, after _Harvey P. Dale_ in A375553 *)
%o A375552 (SageMath)
%o A375552 from more_itertools import unique_everseen
%o A375552 def f(p):
%o A375552     for q in Primes():
%o A375552         if is_prime(q + (p + q)*10^(1 + int(log(q, 10)))): return q
%o A375552 a = lambda n: unique_everseen((f(p) for p in prime_range(n)))
%o A375552 print([2, 5] + list(a(999)))
%o A375552 (PARI)
%o A375552 f(n) = my(k=2); while (!isprime(eval(concat(Str(prime(n)+k), Str(k)))), k = nextprime(k+1)); k; \\ A375553
%o A375552 lista(nn) = my(list=List()); listput(list, 2); listput(list, 5); for (n=1, nn, my(k=f(n)); if (#select(x->(x==k), Vec(list)) == 0, listput(list, k));); Vec(list); \\ _Michel Marcus_, Sep 17 2024
%o A375552 (Python)
%o A375552 from itertools import count, islice
%o A375552 from sympy import isprime, nextprime
%o A375552 def A375552_gen(): # generator of terms
%o A375552     p, a = 2, set()
%o A375552     yield from (2,5)
%o A375552     while True:
%o A375552         q, m = 2, 10
%o A375552         for l in count(1):
%o A375552             while q<m:
%o A375552                 if isprime(m*(p+q)+q):
%o A375552                     if q not in a:
%o A375552                         yield q
%o A375552                         a.add(q)
%o A375552                     break
%o A375552                 q = nextprime(q)
%o A375552             else:
%o A375552                 m *= 10
%o A375552                 continue
%o A375552             break
%o A375552         p = nextprime(p)
%o A375552 A375552_list = print(list(islice(A375552_gen(),61))) # _Chai Wah Wu_, Sep 18 2024
%Y A375552 Cf. A000040, A375553, A174414, A055642.
%K A375552 nonn
%O A375552 1,1
%A A375552 _Peter Luschny_, Sep 17 2024