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 A371024 a(n) is the least prime p such that p + 4*k*(k+1) is prime for 0 <= k <= n-1 but not for k=n. 1
 2, 3, 29, 5, 23, 269, 272879, 149, 61463, 929, 7426253, 2609, 233, 59 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(15) > 3277860277, a(16) > 3103623446, a(17) > 2853255995, a(18) = 653, a(19) > 2480173428, a(20) > 2058783580, a(21) > 1894529774, a(22) > 1896261075, a(23) > 1836831342, a(24), ..., a(100) > 15000000. Other than a(1)-a(14) and a(18), no terms < 24870000007. - Michael S. Branicky, Apr 12 2024 From David A. Corneth, Apr 12 2024: (Start) Using remainders mod q we can restrict the search. For example for a(15) a term can only be 2, 3 or 5 (mod 7). Or maybe 7 itself. If a(15) = p == 1 (mod 7) then for k = 3 we have q + 4*3*(3+1) == 0 mod 7. Similarily number 0, 4 and 6 (mod 7) produce a multiple of 7 where they should not. Doing so for various primes mod q we can reduce the number of remainders and with that the search space by combining the possible remainders using the Chinese Remainder Theorem (CRT). So the possible remainders mod 2 are 1. The possible remainders mod 3 are 2. Using the CRT, a number of the form 1 (mod 2) and 2 (mod 3) simultaneously is of the form 5 (mod 6). a(15) > 2.3*10^13 if it exists. (End) LINKS Table of n, a(n) for n=1..14. MAPLE f:= proc(p) local k; for k from 1 while isprime(p+k*(k+1)*4) do od: k end proc: A:= Vector(12): count:= 0: for i from 1 while count < 12 do v:= f(ithprime(i)); if A[v] = 0 then count:= count+1; A[v]:= ithprime(i) fi od: convert(A, list); MATHEMATICA Table[p=1; m=4; Monitor[Parallelize[While[True, If[And[MemberQ[PrimeQ[Table[p+m*k*(k+1), {k, 0, n-1}]], False]==False, PrimeQ[p+m*n*(n+1)]==False], Break[]]; p++]; p], p], {n, 1, 10}] PROG (PARI) isok(p, n) = for (k=0, n-1, if (! isprime(p + 4*k*(k+1)), return(0))); return (!isprime(p + 4*n*(n+1))); a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Mar 12 2024 (Python) from sympy import isprime, nextprime from itertools import count, islice def f(p): k = 1 while isprime(p+4*k*(k+1)): k += 1 return k def agen(verbose=False): # generator of terms adict, n, p = dict(), 1, 1 while True: p = nextprime(p) v = f(p) if v not in adict: adict[v] = p if verbose: print("FOUND", v, p) while n in adict: yield adict[n]; n += 1 print(list(islice(agen(), 14))) # Michael S. Branicky, Apr 12 2024 CROSSREFS Cf. A164926, A370387. Sequence in context: A206591 A003017 A096580 * A351693 A324941 A028868 Adjacent sequences: A371021 A371022 A371023 * A371025 A371026 A371027 KEYWORD nonn,more AUTHOR J.W.L. (Jan) Eerland, Mar 08 2024 STATUS approved

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Last modified June 13 22:21 EDT 2024. Contains 373391 sequences. (Running on oeis4.)