A234318
Squares t^2 = (p+q+r+s)/4 which are the arithmetic mean of four consecutive primes such that p < t^2 < q < r < s.
Original entry on oeis.org
15876, 35721, 59049, 65025, 488601, 828100, 1144900, 3857296, 4822416, 4901796, 5107600, 5322249, 5856400, 6100900, 6760000, 10536516, 11716929, 12503296, 13468900, 14197824, 14638276, 15163236, 18748900, 21455424, 22127616, 22638564, 24049216, 24098281, 24108100
Offset: 1
15876 is in the sequence because 15876 = 126^2 = (15859+15877+15881+15887)/4, the arithmetic mean of four consecutive primes.
35721 is in the sequence because 35721 = 189^2 = (35677+35729+35731+35747)/4, the arithmetic mean of four consecutive primes.
Cf.
A062703 (squares: sum of two consecutive primes).
Cf.
A069495 (squares: arithmetic mean of two consecutive primes).
Cf.
A234240 (cubes: arithmetic mean of three consecutive primes).
Cf.
A234297 (squares: arithmetic mean of three consecutive primes).
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KD := proc() local a,b,d,e,f,g; a:=n^2;b:=prevprime(a); d:=nextprime(a); e:=nextprime(d); f:=nextprime(e); g:=(b+d+e+f)/4; if a=g then RETURN (a); fi; end: seq(KD(), n=2..10000);
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fcpQ[{a_,b_,c_,d_}]:=Module[{m=Mean[{a,b,c,d}]},IntegerQ[ Sqrt[ m]] && a< m< b]; Mean/@Select[Partition[Prime[Range[1600000]],4,1],fcpQ] (* Harvey P. Dale, Apr 24 2017 *)
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list(lim)=my(v=List(), p=2, q=3, r=5, t); forprime(s=7, nextprime(nextprime(lim+1)+1), t=(p+q+r+s)/4; if(denominator(t)==1 && issquare(t) && t < q, listput(v, t)); p=q; q=r; r=s); Vec(v) \\ Charles R Greathouse IV, Jan 03 2014
A245360
Perfect powers which are the sum of two consecutive primes.
Original entry on oeis.org
8, 36, 100, 128, 144, 216, 576, 1764, 2304, 3844, 5184, 7056, 8100, 8192, 12100, 14400, 14884, 21952, 30276, 41616, 43264, 48400, 53824, 57600, 69696, 74088, 93636, 106276, 112896, 138384, 148996, 166464, 168100, 197136, 206116, 207936, 219024, 220900, 224676, 272484, 279936
Offset: 1
47 + 53 = 100 = 10^2, so 100 is a member of this sequence.
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Select[Total/@Partition[Prime[Range[13100]],2,1],GCD@@FactorInteger[#][[All,2]]>1&] (* Harvey P. Dale, Jan 22 2019 *)
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for(n=1,10^5,q=prime(n)+prime(n+1);if(ispower(q),print1(q,", ")))
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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
A287027
Least sum s of consecutive prime numbers starting with prime(n) such that s is a perfect square.
Original entry on oeis.org
100, 961, 36, 14017536, 484, 49, 36, 134689, 354025, 80089, 443556, 121, 47524, 7744, 100, 700569, 344956329, 48841, 5329, 144, 324, 39601, 22801, 8649, 239438955625, 12250000, 197136, 222784, 147456, 319225, 316969, 24649, 576, 2975625, 7396, 21316, 70036245333532859364
Offset: 1
Sum of set {2,3,5,7,11,13,17,19,23} is 100 = 10^2, sum of set {3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83} is 961=31^2, sum of set {5,7,11,13}=36=6^2.
Cf.
A062703 (squares that are the sum of 2 consecutive primes),
A080665 (squares that are the sum of 3 consecutive primes),
A034707 (numbers that are sums of consecutive primes).
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f:= proc(n) local p, s;
p:= ithprime(n); s:= p;
while not issqr(s) do p:= nextprime(p); s:= s+p od:
s
end proc:
map(f, [$1..36]); # Robert Israel, May 18 2017
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Table[Set[{k, s}, {n, 0}]; While[! IntegerQ@ Sqrt[AddTo[s, Prime@ k]], k++]; s, {n, 36}] (* Michael De Vlieger, May 20 2017 *)
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a(n) = my(s=0); forprime(p=prime(n), , s=s+p; if(issquare(s), return(s))) \\ Felix Fröhlich, May 25 2017
Original entry on oeis.org
3, 5, 31, 61, 163, 193, 227, 383, 401, 521, 631, 653, 883, 1019, 1151, 1229, 1433, 1601, 1669, 1801, 1873, 2437, 2729, 3191, 3671, 4013, 4049, 4127, 4447, 5507, 5651, 5701, 5813, 6079, 6199, 6353, 6569, 6823, 6857, 7507, 7529, 7873, 7907, 8291, 8419, 8461
Offset: 1
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