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A119768 Twin prime pairs that sum to a power. 4
3, 5, 17, 19, 71, 73, 107, 109, 881, 883, 1151, 1153, 2591, 2593, 3527, 3529, 4049, 4051, 15137, 15139, 20807, 20809, 34847, 34849, 46817, 46819, 69191, 69193, 83231, 83233, 103967, 103969, 112337, 112339, 139967, 139969, 149057, 149059, 176417 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Since twin prime pairs greater than (3,5) occur as either (5,7) mod 12 or (11,1) mod 12, all sums of such twin primes are always divisible by 12. Thus all powers are divisible by 12. The first few terms in base 12 are: 15, 17, 5E, 61, 8E, 91, 615, 617, 7EE, 801, 15EE, 1601 and the corresponding powers are 30, 100, 160, 1030, 1400, 3000.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

If a(n) is the above sequence of twin primes, then a(2n-1),a(2n) is a twin prime pair and a(2n-1)+a(2n) is a power.

a(2*n-1) = A270231(n), a(2*n) = A270231(n) + 2. - Amiram Eldar, Jan 05 2020

EXAMPLE

a(5) + a(6) = 71 + 73 = 144 = 12^2.

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->op(z[1..2]), L);

MATHEMATICA

powQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; aQ[n_] := PrimeQ[n] && PrimeQ[n + 2] && powQ[2 n + 2]; s = Select[Range[10^4], aQ]; Union @ Join[s, s + 2] (* Amiram Eldar, Jan 05 2020 *)

PROG

(PARI) my(pp=3); forprime(p=5, 180000, if(p-pp==2, if(ispower(p+pp), print1(pp, ", ", p, ", "))); pp=p) \\ Hugo Pfoertner, Jan 05 2020

CROSSREFS

Cf. A001097, A001359, A006512, A069496, A270231, A330978, A330980.

Sequence in context: A045416 A038891 A287638 * A020592 A295387 A263258

Adjacent sequences:  A119765 A119766 A119767 * A119769 A119770 A119771

KEYWORD

easy,nonn,tabf

AUTHOR

Walter Kehowski, Jun 18 2006

EXTENSIONS

a(1)-a(2) inserted by Amiram Eldar, Jan 05 2020

STATUS

approved

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Last modified August 3 20:08 EDT 2020. Contains 336201 sequences. (Running on oeis4.)