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.
prime(7) + prime(8) = 17 + 19 = 36 = 6^2.
PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[n_] := Block[{m = Floor[n/2]}, s = PrevPrim[m] + NextPrim[m]; If[s == n, True, False]]; Select[ Range[550], f[ #^2] &]^2
t := Table[Prime[n] + Prime[n + 1], {n, 15000}]; Select[t, IntegerQ[Sqrt[#]] &] (* Carlos Eduardo Olivieri, Feb 25 2015 *)
{for(n=1,100,(p=precprime(n^2/2))+nextprime(p+2) == n^2 && print1(n^2", "))} \\ Zak Seidov, Feb 17 2011
A062703(n)=A074924(n)^2 \\ M. F. Hasler, Jan 03 2020
from itertools import count, islice
from sympy import nextprime, prevprime
def agen(): # generator of terms
for k in count(4, step=2):
kk = k*k
if prevprime(kk//2+1) + nextprime(kk//2-1) == kk:
yield kk
print(list(islice(agen(), 37))) # Michael S. Branicky, May 24 2022
For k=15: prime(15) = 47 and prime(16) = 53, 47 + 53 = 100 = 10^2.
[n: n in [0..50000]| IsSquare(NthPrime(n) +NthPrime(n+1))]; // Vincenzo Librandi, Apr 06 2011
lst={};Do[p1=Prime[n];p2=Prime[n+1];q=(p1+p2)^0.5;If[q==IntegerPart[q], AppendTo[lst, n]], {n, 1, 9!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
j=[]; for(n=1,30000,x=prime(n)+prime(n+1); if(issquare(x),j=concat(j,n))); j
{ n=0; default(primelimit, 8500000); for (m=1, 10^9, if (issquare(prime(m) + prime(m + 1)), write("b064397.txt", n++, " ", m); if (n==175, break)) ) } \\ Harry J. Smith, Sep 13 2009
a(0)=5 because 5=2+3 a(1)=3 because 30=13+17 a(2)=1 because 100=47+53 a(3)=6 because 6000=2999+3001.
Join[{5,3},Reap[Do[Do[n=10^m k; If[n==PreviousPrime[n/2]+NextPrime[n/2],Sow[k];Break[]],{k,2000}],{m,2,50}]][[2,1]]]
f[n_] := Block[{k = 1, tn = 10^n}, While[h = k*tn/2; NextPrime[h, -1] + NextPrime@h != k*tn, k++ ]; k]; f[1] = 3; Array[f, 70, 0] (* Robert G. Wilson v, Aug 11 2010 *)
6^3 = 216 = 107 + 109.
Surd[#,3]&/@Select[Total/@Partition[Prime[Range[150*10^6]],2,1], IntegerQ[ Surd[#,3]]&] (* Harvey P. Dale, Jun 05 2018 *)
from sympy import nextprime, prevprime A074925_list = [i for i in range(2,10**4,2) if prevprime(i**3//2) + nextprime(i**3//2) == i**3] # Chai Wah Wu, Feb 22 2017
28 is in the list because prime(28)+prime(29) = 107+109 =216 = 6^3. n=1291597: prime(1291597)+prime(1291598) = 344*344*344.
PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Do[ If[ n^3 == PrevPrim[Floor[(n^3)/2]] + NextPrim[Floor[(n^3)/2]], Print[ PrimePi[ Floor[(n^3)/2]]]], {n, 2, 10^4}]
Flatten[Position[Total/@Partition[Prime[Range[20000000]],2,1],?(IntegerQ[ Surd[ #,3]]&)]] (* _Harvey P. Dale, May 28 2014 *)
from _future_ import division from sympy import isprime, prevprime, nextprime, primepi A071220_list, i = [], 2 while i < 10**6: n = i**3 m = n//2 if not isprime(m) and prevprime(m) + nextprime(m) == n: A071220_list.append(primepi(m)) i += 1 # Chai Wah Wu, May 31 2017
6 is a term since (6^2)/2 = 18 = mean(17, 19). 12 is a term since (12^2)/2 = 72 = mean(71,73). 42 is a term since (42^2)/2 = 882 = mean(881,883).
[k:k in [2..2800 by 2]| IsPrime(k*k div 2 -1) and IsPrime(k*k div 2 +1)]; // Marius A. Burtea, Jan 01 2020
isa := n -> isprime(n) and isprime(n+2) and issqr(2*n+2): select(isa, [$4..1000000]): map(n -> sqrt(2*n+2), %); # Peter Luschny, Jan 05 2020
lst={};Do[p1=Prime[n];p2=Prime[n+1];If[p2-p1==2,e=(2*(p1+1))^(1/2);i=Floor[e]; If[e==i,AppendTo[lst,i]]],{n,3*9!}];lst
(* Second program: *)
Select[Map[Sqrt[2 #] &, Mean /@ Select[Partition[Prime@ Range[10^6], 2, 1], Subtract @@ # == -2 &]], IntegerQ] (* Michael De Vlieger, Feb 18 2018 *)
forstep(n=6,1e3,6,if(isprime(n^2/2-1)&&isprime(n^2/2+1),print1(n", "))) \\ Charles R Greathouse IV, Feb 01 2013
60^2 = 3600 = prime(107) + ... + prime(112) = 587 + 593 + 599 + 601 + 607 + 613, 156^2 = 24336 = prime(557) + ... + prime(562) = 4027 + 4049 + 4051 + 4057 + 4073 + 4079.
Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[150000]],6,1]),IntegerQ] (* Harvey P. Dale, Aug 02 2021 *)
6^2=36=17+19=5+7+11+13, 18^2=324=157+163=73+79+83+89.
Module[{nn=10^7,p2,p4},p2=Total/@Partition[Prime[Range[nn]],2,1];p4=Total/@Partition[ Prime[Range[nn]],4,1];Select[Sqrt[Intersection[p2,p4]],IntegerQ]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, May 03 2024 *)
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