This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; PrimeExponents[n_] := Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]]; c = 0; p = q = 2; Do[ While[p < 10^n, q = NextPrim[p]; If[ Apply[ GCD, PrimeExponents[p + q]] > 1, c++ ]; p = q]; Print[c], {n, 1, 8}]
a(4) = 283, the next prime is 293 and 283 + 293 = 576 = 24^2.
Transpose[Select[Partition[Prime[Range[20000]],2,1],IntegerQ[Sqrt[Plus@@# ]]&]][[1]] (* Harvey P. Dale, Aug 04 2009 *)
{ default(primelimit, 550655327); n=0; q=2; forprime (p=3, 550655327, if (issquare(p+q), write("b061275.txt", n++, " ", q)); q=p ) } \\ Harry J. Smith, Jul 20 2009
4093 is in the sequence because 4093 and 4099 are consecutive primes and (4093 + 4099)/2 = 4096 = 2^12.
Select[Partition[Prime[Range[2, 10^4]], 2, 1], GCD @@ FactorInteger[(First[#] + Last[#])/2][[;; , 2]] > 1 &][[;; , 1]] (* Amiram Eldar, Jul 04 2022 *)
for(i=3, 10^5, if(isprime(i), k=(i+nextprime(i+1))/2; if(ispower(k), print1(i, ", "))))
n=1: p(2)+p(1) = 3+2 = 5^1 n=2: p(3)+p(2) = 5+3 = 2^3 n=18: p(19)+p(18) = 61+67 = 2^7 n=564: p(565)+p(564) = 4099+4093 = 2^13
Do[s=Prime[n+1]+Prime[n]; If[Equal[Length[FactorInteger[s]], 1], Print[{n, Prime[n], s}]], {n, 1, 10000000}]
p = q = 2; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Do[q = NextPrim[p]; If[ Length[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[p + q]]] == 1, Print[n]]; p = q, {n, 1, 10^7}] (* Robert G. Wilson v, Jan 24 2004 *)
1723 is in the sequence because it is prime, nextprime(1723) = 1733, and average(1723,1733) = 1728 = 12^3.
Select[Partition[Prime[Range[2, 10^5]], 2, 1], IntegerQ[Surd[(First[#] + Last[#])/2, 3]] &][[;; , 1]] (* Amiram Eldar, Jul 04 2022 *)
{for(i=3,3*10^7,if(isprime(i),k=(i+nextprime(i+1))/2;if(ispower(k,3),print1(i,", "))))}
3+5=8=2^3 61+67=128=2^7
Select[Prime[Range[10^6]],Total[IntegerDigits[#+NextPrime[#],2]]==1&] (* James C. McMahon, Dec 19 2024 *)
47 + 53 = 100 = 10^2, so 100 is a member of this sequence.
Select[Total/@Partition[Prime[Range[13100]],2,1],GCD@@FactorInteger[#][[All,2]]>1&] (* Harvey P. Dale, Jan 22 2019 *)
for(n=1,10^5,q=prime(n)+prime(n+1);if(ispower(q),print1(q,", ")))
m=10^8; v=[]; forstep(b=2, sqrt(m), 2, forprime(p=2, 40, n=b^p; if(n>m,break); if(n==precprime(n/2)+nextprime(n/2+1), v=concat(v,n)))); v=vecsort(v) \\ Faster program. Jens Kruse Andersen, Jul 20 2014
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