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a(n) is the least prime p such that the second forward difference of three consecutive primes p, q and r is n = -(p - 2q + r)/2.
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%I #15 Feb 16 2025 08:33:56

%S 3,7,23,1531,139,113,523,1069,887,6397,1129,3137,5351,2971,1327,14107,

%T 9973,19333,84871,16141,15683,73189,31907,28229,35617,35677,44293,

%U 43331,107377,34061,221327,134513,31397,480209,173359,332317,933073,736279,265621,843911,404851,155921

%N a(n) is the least prime p such that the second forward difference of three consecutive primes p, q and r is n = -(p - 2q + r)/2.

%C Inspired by A295973.

%C Except for the first three primes {2, 3, 5}, all sfds are even.

%C The only other sfd which is not covered by this sequence is when the primes are {2, 3, 5} which results in an sfd of 1.

%C Except for an sfd of 0 or 1, all values of sfd occur infinitely often.

%C As an example, sfd = -2 for p = 7, 13, 31, 37, 61, 67, 73, 97, 103, 157, 193, 223, 271, 277, 307, ..., .

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ForwardDifference.html">Forward Difference</a>.

%e a(0) = 3 since the three consecutive primes {3, 5, 7} have an sfd of 0;

%e a(1) = 7 since the three consecutive primes {7, 11, 13} have a sfd of -2;

%e a(2) = 23 since the three consecutive primes {23, 29, 31} have a sfd of -4;

%e a(3) = 1531 since the three consecutive primes {1531, 1543, 1549} have an sfd of -6;

%e a(4) = since the three consecutive primes {} have an sfd of -8; etc.

%t p = 2; q = 3; r = 5; t[_] := 0; While[p < 1000000, d = p - 2q + r; If[ t[d] == 0, t[d] = p]; p = q; q = r; r = NextPrime@ r]; Array[ t[-2#] &, 42, 0]

%o (PARI) a(n) = my(p=2, q=3); while ((p - 2*q + nextprime(q+1))/2 != -n, p=q; q=nextprime(q+1)); p; \\ _Michel Marcus_, Mar 08 2023

%Y Cf. A000040, A000230, A036263, A137501, A295746, A295973, A316791.

%K nonn,changed

%O 0,1

%A _Edward Bernstein_ and _Robert G. Wilson v_, Jul 14 2018