A245591 -id:A245591 - cp's OEIS frontend

A240169 Numbers n such that (6n)^3 is the sum of a twin prime pair.

1, 29, 65, 81, 99, 136, 165, 174, 176, 191, 200, 266, 295, 301, 319, 346, 351, 370, 400, 411, 431, 434, 436, 456, 491, 494, 526, 541, 599, 651, 676, 714, 746, 790, 924, 956, 991, 1011, 1131, 1161, 1194, 1259, 1274, 1280, 1304, 1374, 1550, 1641, 1644, 1649, 1714, 1715, 1739, 1804, 1811, 1814, 1830, 1879, 1941, 2000

Offset: 1

Zak Seidov, Aug 02 2014

nonn

No terms end with 2, 3, 7, 8. Minimal differences are 1; e,g., a(52) - a(51) = 1715 - 1714. There are no three consecutive terms with common difference 1.
Distribution of last digits for first 61000 terms: W(0..9) = (10190, 10162, 0, 0, 10178, 10222, 10027, 0, 0, 10221).
For "existing" digits distribution is rather uniform.

			m = 1: (6m)^3 = 216 = 107 + 109, m = 29: (6m)^3 = 5268024 = 2634011 + 2634013.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A245591.

select(n -> isprime(108 * n^3 - 1) and isprime(108 * n^3 + 1), [$1..1000]); # Robert Israel, Aug 03 2014

Select[Range[1000], PrimeQ[216#^3/2 - 1] && PrimeQ[216#^3/2 + 1] &] (* Alonso del Arte, Aug 02 2014 *)

N=2*10^3; for(k=1,N,p=216*k^3; if(isprime(p/2-1)&&isprime(1+p/2), print1(k, ", ")))

a(n) = (1/6)*A245591(n+1)^(1/3).

A119767 Perfect powers which are the sum of twin prime pairs.

Original entry on oeis.org

8, 36, 144, 216, 1764, 2304, 5184, 7056, 8100, 30276, 41616, 69696, 93636, 138384, 166464, 207936, 224676, 279936, 298116, 352836, 360000, 412164, 562500, 725904, 777924, 876096, 944784, 956484, 1077444, 1299600, 1468944, 1617984, 1920996, 2160900, 2286144, 2304324, 2509056

Offset: 1

Walter Kehowski, Jun 18 2006

Since twin primes greater than (3,5) are either occur as (5,7) mod 12 or (11,1) mod 12, all sums of such twin primes are divisible by 12. Thus all powers are divisible by 12 and are best looked at in base 12. For example, a(3) = 5E + 61 = 100, where E is eleven.

			8 = 2^3 = 3 + 5 (twin primes). Thus 8 is a member of this sequence.
36 = 6^2 = 17 + 19 (twin primes). Thus 36 is a member of this sequence.
a(3) = 71 + 73 = 144.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A037072, A245591.

egcd := proc(n::nonnegint) local L; if n=0 or n=1 then n else L:=ifactors(n)[2]; L:=map(z->z[2],L); igcd(op(L)) fi end: L:=[]: for w to 1 do for x from 1 to 2*12^2 do s:=6*x; for r from 2 to 79 do t:=s^r; if egcd(s)=1 and andmap(isprime,[(t-2)/2,(t+2)/2]) then print((t-2)/2,(t+2)/2,t)); L:=[op(L),[(t-2)/2,(t+2)/2,t]]; fi; od od od; L:=sort(L,(a,b)->a[1]z[3],L);

Lim=2600000;ts=Select[Prime[Range[PrimePi[Lim]]], PrimeQ[# + 2] &]2+2;pp=Join[{1}, Select[Range[Lim], GCD@@FactorInteger[#][[All, 2]]>1&]] ;s={};Do[ If[MemberQ[ pp,ts[[n]]],AppendTo[s,ts[[n]]]] ,{n,Length[ts]}];s (* James C. McMahon, Sep 18 2024 *)

a(N) = for(n=1,N,if(ispower(n),if(nextprime(n/2)-precprime(n/2)==2&&precprime(n/2)+nextprime(n/2)==n,print1(n,", ")))) \\ vary the program's range for any N; Derek Orr, Jul 27 2014

R. J. Mathar pointed out that 8 was missing. Once corrected, the old A245591 could be merged into this entry. - N. J. A. Sloane, Jul 30 2014

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A240169 Numbers n such that (6n)^3 is the sum of a twin prime pair.

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