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

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

1,1

Comments

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.

Links

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

Example

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.

Maple

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]<b[1]); map(z->z[3], L);

Prog

(PARI) 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

Crossrefs

Cf. A037072, A245591.

Sequence in context: A034998 A121255 A210656 * A024208 A000427 A000428

Adjacent sequences: A119764 A119765 A119766 * A119768 A119769 A119770

Keyword

easy,nonn

Author

Walter Kehowski, Jun 18 2006

Extensions

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

Status

approved