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A270696
Perfect powers that are the average of two consecutive primes.
1
4, 9, 64, 81, 144, 225, 324, 441, 625, 1089, 1681, 1728, 2601, 3600, 4096, 5184, 6084, 8464, 12544, 13689, 16641, 17576, 19044, 19600, 21952, 25281, 27225, 28224, 29584, 36864, 38025, 39204, 45369, 46656, 47524, 51984, 56169, 74529, 87025, 88804, 91809, 92416, 95481
OFFSET
1,1
COMMENTS
The corresponding lesser primes are in A242380. - Michel Marcus, Mar 23 2016
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
1728 is a term because 1728 = 12^3 = (1723 + 1733)/2.
MATHEMATICA
Select[Mean/@Partition[Prime[Range[10000]], 2, 1], GCD@@FactorInteger[#][[All, 2]]>1&] (* Harvey P. Dale, Jun 22 2022 *)
PROG
(PARI) for(n=2, 1e5, if((nextprime(n) - n) == (n - precprime(n)) && ispower(n), print1(n, ", ")));
(PARI) list(lim)=my(v=List(), t); forprime(e=2, logint(lim\=1, 2), for(m=2, sqrtnint(lim, e), t=m^e; if(t-precprime(t)==nextprime(t)-t, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Mar 21 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Altug Alkan, Mar 21 2016
STATUS
approved