A270231 -id:A270231 - cp's OEIS frontend

A119768 Twin prime pairs that sum to a power.

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

Offset: 1

Walter Kehowski, Jun 18 2006

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.

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

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

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

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

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 *)

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

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

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

A330978 a(n) = (p1 + p2)/36 such that p1 >= 5 and p2 = p1 + 2 are twin primes and p1 + p2 is a k-th power with k > 1.

Original entry on oeis.org

1, 4, 6, 49, 64, 144, 196, 225, 841, 1156, 1936, 2601, 3844, 4624, 5776, 6241, 7776, 8281, 9801, 10000, 11449, 15625, 20164, 21609, 24336, 26244, 26569, 29929, 36100, 40804, 44944, 53361, 60025, 63504, 64009, 69696, 87025, 93636, 100489, 108900, 109561, 126025

Offset: 1

Hugo Pfoertner, Jan 05 2020

nonn

			a(1) = 1: p1 = 17 and p2 = 19 are the first such pair, with p1 + p2 = 36 = 6^2, (17 + 19)/36 = 1;
a(2) = 4: p1 = 71, p2 = 73; p1 + p2 = 144 = 12^2, (71 + 73)/36 = 4.

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

Cf. A119768, A270231, A330980.

isa := n -> isprime(n) and isprime(n+2) and iperfpow(2*n+2) <> FAIL:
select(isa, [$4..1000000]): map(n -> (n+1)/18, %); # Peter Luschny, Jan 05 2020

my(pp=5); forprime(p=7,130000, if(p-pp==2, if(ispower(p+pp), print1((p+pp)/36,", "))); pp=p)

A330980 a(n) = (p1 + p2)/216 such that p1 >= 5 and p2 = p1 + 2 are twin primes and p1 + p2 is a k-th power with k >= 3.

Original entry on oeis.org

1, 1296, 24389, 274625, 531441, 970299, 2343750, 2515456, 4492125, 5268024, 5451776, 6967871, 8000000, 18821096, 25672375, 27270901, 32461759, 37748736, 41421736, 43243551, 50653000, 64000000, 69426531, 80062991, 81746504, 82881856, 94818816, 100663296

Offset: 1

Hugo Pfoertner, Jan 05 2020

nonn

The values of k corresponding to the first terms are: 3, 7, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, ...

			a(1) = 1: p1 = 107 and p2 = 109 is the first pair with a sum that is a 3rd power, 216=6^3;
a(2) = 1296: p1 = 1296*108 - 1 = 139967, p2 = 1296*108 + 1 = 139969, p1 + p2 = 279936 = 6^7.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A076467, A119768, A270231, A330978.

my(pp=5,j); forprime(p=7,10000000000, if(p-pp==2, if(j=ispower(p+pp), if(j>2, print1((p+pp)/216,", ")))); pp=p)

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A119768 Twin prime pairs that sum to a power.

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A330978 a(n) = (p1 + p2)/36 such that p1 >= 5 and p2 = p1 + 2 are twin primes and p1 + p2 is a k-th power with k > 1.

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A330980 a(n) = (p1 + p2)/216 such that p1 >= 5 and p2 = p1 + 2 are twin primes and p1 + p2 is a k-th power with k >= 3.

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