A292715
Initial member of 8 consecutive primes a, b, c, d, e, f, g, h such that (a + h) = (b + g), (c + g) = (d + f), (a + f) = (b + e) and (a + g) = (b + f).
Original entry on oeis.org
6337, 14717, 77521, 83401, 130643, 344231, 357653, 380377, 496453, 505067, 587101, 593473, 970457, 1130251, 1515691, 1694191, 1936741, 2689997, 2773007, 2811163, 3665371, 3678887, 3713993, 3976361, 4024687, 4181579, 4629461, 4801673, 5438569, 5882197, 6016811
Offset: 1
6337 is a term because it is the first of 8 consecutive primes {6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379} = {a, b, c, d, e, f, g, h}; and (a + h) = (b + g), (c + g) = (d + f), (a + f) = (b + e) and (a + g) = (b + f).
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A292715:= proc(n)local a, b, c, d, e, f, g, h; 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); if (a + h) = (b + g) and (c + g) = (d + f) and (a + f) = (b + e) and (a + g)=(b + f) then RETURN (a); fi; end: seq(A292715(n), n=1..1000000);
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Select[Partition[Prime@ Range[10^6], 8, 1], Function[{a, b, c, d, e, f, g, h}, And[(a + h) == (b + g), (c + g) == (d + f), (a + f) == (b + e), (a + g) == (b + f)]] @@ # &][[All, 1]] (* Michael De Vlieger, Sep 21 2017 *)
Select[Partition[Prime[Range[420000]],8,1],Length[Union[Drop[Drop[ Differences[ #],{4}],{2}]]]==1&][[All,1]](* Harvey P. Dale, Nov 03 2017 *)
A292743
Initial member of 6 consecutive primes a, b, c, d, e, f such that (a + f) = (b + e), (a + e) = (b + d) and (c + f) = (d + e).
Original entry on oeis.org
6353, 14731, 19463, 71333, 77543, 78781, 83417, 104701, 105557, 130651, 185021, 202799, 214433, 218111, 344243, 351031, 357661, 358429, 380417, 408203, 443221, 466547, 496471, 505091, 587117, 593491, 634241, 652733, 702497, 746177, 778241, 807011, 886973, 949951
Offset: 1
6353 is a term because it is the initial member of 6 consecutive primes {6353, 6359, 6361, 6367, 6373, 6379} = {a, b, c, d, e, f}; and (a + f) = (b + e), (a + e) = (b + d) and (c + f) = (d + e).
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A292743:= proc(n)local a,b,c,d,e,f; a:=ithprime(n); b:=ithprime(n+1); c:=ithprime(n+2); d:=ithprime(n+3); e:=ithprime(n+4); f:=ithprime(n+5); if (a + f) = (b + e) and (a + e) = (b + d) and (c + f) = (d + e) then RETURN (ithprime(n)); fi; end: seq(A292743(n), n=1..100000);
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Select[Partition[Prime@ Range[10^5], 6, 1], Function[{a, b, c, d, e, f}, And[(a + f) == (b + e), (a + e) == (b + d), (c + f) == (d + e)]] @@ # &][[All, 1]] (* Michael De Vlieger, Sep 22 2017 *)
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).
Original entry on oeis.org
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
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.
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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
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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 *)
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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
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.
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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]]
Showing 1-4 of 4 results.
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