A292743 -id:A292743 - cp's OEIS frontend

A293393 Initial member of 10 consecutive primes {a, b, c, d, e, f, g, h, i, j} such that (j - e) = (i - d) = (h - c) = (g - b) = (f - a).

541, 547, 557, 1019, 4229, 4231, 35099, 59617, 91199, 105997, 708251, 998969, 1208209, 1260323, 1376461, 1435997, 1556393, 1752197, 1996217, 2092249, 2152811, 2271383, 2349917, 3011011, 3919199, 3919211, 4020167, 4020197, 4089037, 4089073, 4797503, 4897331, 5124023

Offset: 1

K. D. Bajpai, Oct 08 2017

nonn

12689273 is the smallest term such that 12689273 +- 6 are both prime.

			541 is a term because it is the initial member of 10 consecutive primes {541, 547, 557, 563, 569, 571, 577, 587, 593, 599} = {a, b, c, d, e, f, g, h, i, j}: {(j - e) = (i - d) = (h - c) = (g - b) = (f - a)} = {(599 - 569) = (593 - 563) = (587 - 557) = (577 - 547) = (571 - 541)} = 30.

Harvey P. Dale, Table of n, a(n) for n = 1..100

Cf. A000040, A292618, A292715, A292743.

A293393:= proc(n)local a, b, c, d, e, f, g, h, i, j; a:=ithprime(n); b:=ithprime(n+1); c:=ithprime(n+2); d:=ithprime(n+3); e:=ithprime(n+4); f:=ithprime(n+5); g:=ithprime(n+6); h:=ithprime(n+7); i:=ithprime(n+8); j:=ithprime(n+9); if (j - e) = (i - d) and (j - e)= (h - c) and (j - e)= (g - b) and (j - e)= (f - a)then RETURN (a); fi; end: seq(A293393(n), n=1..500000);
# Alternative:
P:= select(isprime, [seq(i,i=3..10^7,2)]):
Q:= P[6..-1]-P[1..-6]:
J:= select(t -> nops(convert(Q[t..t+4],set))=1, [$1..nops(Q)-4]):
P[J]; # Robert Israel, Oct 09 2017

Select[Partition[Prime@ Range[10^6], 10, 1], Equal[#10 - #5, #9 - #4, #8 - #3, #7 - #2, #6 - #1] & @@ # &][[All, 1]] (* Michael De Vlieger, Oct 08 2017 *)
udQ[n_]:=Length[Union[Differences[TakeDrop[n,5]][[1]]]]==1; Select[ Partition[ Prime[ Range[360000]],10,1],udQ][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 19 2018 *)

for(n = 1, 50000, a = prime(n); b = prime(n+1); c = prime(n+2); d = prime(n+3); e = prime(n+4); f = prime(n+5); g = prime(n+6); h = prime(n+7); i = prime(n+8); j = prime(n+9); if((j - e)==(i - d) && (j - e)==(h - c) && (j - e)==(g - b) && (j - e)==(f - a), print1 (a," ,")));

A293619 Initial member of 6 consecutive primes a, b, c, d, e, f such that both (f + a)/(d - c) and (e + b)/(d - c) are prime.

Original entry on oeis.org

41, 941, 2269, 2411, 5101, 7193, 7283, 12011, 13159, 18427, 19183, 19961, 25589, 27751, 28579, 31151, 35771, 37313, 41543, 47087, 47939, 50459, 52691, 57251, 58229, 58897, 64279, 64553, 65827, 67121, 67411, 67741, 70853, 78277, 81869, 86353, 88993, 90007, 91253

Offset: 1

K. D. Bajpai, Oct 13 2017

nonn

			41 is a term because it is the smallest member of 6 consecutive primes {41, 43, 47, 53, 59, 61} = {a, b, c, d, e, f} and both (f + a)/(d - c) = 17 and (e + b)/(d - c) = 17 are prime.
941 is a term because it is the smallest member of 6 consecutive primes {941, 947, 953, 967, 971, 977} = {a, b, c, d, e, f} and both (f + a)/(d - c) = 137 and (e + b)/(d - c) = 137 are prime.
7193 is a term because it is the smallest member of 6 consecutive primes {7193, 7207, 7211, 7213, 7219, 7229} = {a, b, c, d, e, f} and both (f + a)/(d - c) = 7211 and (e + b)/(d - c) = 7213 are prime.

Cf. A000040, A292618, A292715, A292743, A293395.

Select[Partition[Prime@Range[50000], 6, 1], Function[{a, b, c, d, e, f}, And[PrimeQ[(f + a)/(d - c)] && PrimeQ[(e + b)/(d - c)]]] @@ # &][[All, 1]]

cp's OEIS Frontend

A293393 Initial member of 10 consecutive primes {a, b, c, d, e, f, g, h, i, j} such that (j - e) = (i - d) = (h - c) = (g - b) = (f - a).

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