A080665 -id:A080665 - cp's OEIS frontend

A062703 Squares that are the sum of two consecutive primes.

36, 100, 144, 576, 1764, 2304, 3844, 5184, 7056, 8100, 12100, 14400, 14884, 30276, 41616, 43264, 48400, 53824, 57600, 69696, 93636, 106276, 112896, 138384, 148996, 166464, 168100, 197136, 206116, 207936, 219024, 220900, 224676, 272484, 298116, 302500, 352836

Offset: 1

Jason Earls, Jul 11 2001

			prime(7) + prime(8) = 17 + 19 = 36 = 6^2.

Michael S. Branicky, Table of n, a(n) for n = 1..22054 (terms 1..100 from Harry J. Smith)

Squares in A001043. See A226524 for cubes.
Cf. A074924 (square roots), A061275 (lesser of the primes), A064397 (index of that prime).
Cf. A080665 (same with sum of three consecutive primes).

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

a(n) = A074924(n)^2.

a(n) = A000040(i) + A000040(i+1) with i = A064397(n) = A000720(A061275(n)). - M. F. Hasler, Jan 03 2020

Edited by Robert G. Wilson v, Mar 02 2003

Edited (crossrefs completed, obsolete PARI code deleted) by M. F. Hasler, Jan 03 2020

A076304 Numbers k such that k^2 is a sum of three successive primes.

Original entry on oeis.org

7, 11, 29, 31, 43, 151, 157, 191, 209, 217, 221, 263, 311, 359, 367, 407, 493, 533, 563, 565, 637, 781, 815, 823, 841, 859, 881, 929, 959, 997, 1013, 1019, 1021, 1087, 1199, 1211, 1297, 1353, 1471, 1573, 1613, 1683, 1685, 1733, 1735, 1739, 1751, 1761, 1769

Offset: 1

Zak Seidov, Oct 05 2002

			7 is in this sequence because 7^2 = 49 = p(6) + p(7) + p(8) = 13 + 17 + 19.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..255 from Zak Seidov)

Cf. A206279 (smallest of the 3 primes), A076305 (index of that prime), A080665 (squares = sums), A122560 (subsequence of primes).

Cf. A034961.

Select[Table[Sqrt[Sum[Prime[k], {k, n, n + 2}]], {n, 100000}], IntegerQ] (* Ray Chandler, Sep 29 2006 *)
Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[90000]],3,1]),IntegerQ]  (* Harvey P. Dale, Feb 23 2011 *)

is(n, p=precprime(n^2/3), q=nextprime(p+1), t=n^2-p-q)=isprime(t) && t==if(t>q,nextprime(q+1),precprime(p-1)) \\ Charles R Greathouse IV, May 26 2013; edited by M. F. Hasler, Jan 03 2020

A76304=[7]; apply( A076304(n)={if(n>#A76304, my(i=#A76304, N=A76304[i]); A76304=concat(A76304, vector(n-i,i, until( is(N+=2),);N))); A76304[n]}, [1..99]) \\ M. F. Hasler, Jan 03 2020

a(n) = sqrt(prime(i) + prime(i+1) + prime(i+2)) where i = A076305(n). [Corrected by M. F. Hasler, Jan 03 2020]

A076305 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is a square.

Original entry on oeis.org

6, 12, 59, 65, 112, 965, 1029, 1455, 1706, 1830, 1890, 2573, 3457, 4490, 4664, 5609, 7927, 9130, 10078, 10143, 12597, 18248, 19727, 20086, 20887, 21708, 22739, 25041, 26536, 28511, 29346, 29664, 29774, 33387, 39945, 40677, 46136, 49869, 58135

Offset: 1

Zak Seidov, Oct 05 2002

nonn

See A076304 for the square roots of the sums of the three primes.

			6 is a term because prime(6) + prime(7) + prime(8) = 13 + 17 + 19 = 49 = 7^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..600 from Harvey P. Dale)

Cf. A076304 (square roots of sums), A080665 (squares = sums), A206279 (lesser of the primes).

Cf. A064397 (same for 2 primes), A072849 (4 primes), A166255 (70 primes), A166261 (120 primes).

[k:k in [1..60000]| IsSquare(&+[NthPrime(k+m):m in [0,1,2]])]; // Marius A. Burtea, Jan 04 2020

Select[Range[60000], IntegerQ[Sqrt[Sum[Prime[k], {k, #, # + 2}]]] &] (* Ray Chandler, Sep 26 2006 *)
Position[Partition[Prime[Range[60000]],3,1],?(IntegerQ[Sqrt[ Total[ #]]]&), 1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 28 2018 *)

n=0; p=2; q=3; forprime(r=5, 1e9, n++; if(issquare(p+q+r), print1(n", ")); p=q; q=r) \\ Charles R Greathouse IV, Apr 07 2017

a(n) = A000720(A206279(n)). - M. F. Hasler, Jan 03 2020

Corrected by Ray Chandler, Sep 26 2006

A188651 Products of two primes (i.e., "semiprimes") that are the sum of three consecutive primes.

Original entry on oeis.org

10, 15, 49, 121, 143, 159, 187, 235, 287, 301, 319, 329, 371, 395, 407, 471, 519, 533, 551, 565, 581, 589, 633, 679, 689, 713, 731, 749, 771, 789, 803, 817, 841, 961, 985, 1079, 1099, 1119, 1135, 1169, 1207, 1271, 1285, 1315, 1349, 1391, 1457, 1477, 1585

Offset: 1

Zak Seidov, Apr 16 2011

nonn

Or, semiprimes in A034961 (Sums of three consecutive primes).
Subsequence of square semiprimes: {49, 121, 841, 961, 1849, 22801, 24649, 36481, 69169, ...} = {7, 11, 29, 31, 43, 151, 157, 191, 263, ...}^2 that is also a subsequence of A080665 (Squares in A034961). Cf. also A034962 (Primes A034961).
Somewhat surprisingly, the sum of two consecutive primes is never a semiprime. This follows from that fact that if p+q = 2r for primes p,q,r, then r must between p and q. So if p and q are consecutive, then r does not exist.

			a(1) = 10 = 2*5 = A034961(1) = prime(1) + prime(2) + prime(3) = 2 + 3 + 5,
a(2) = 15 = 3*5 = A034961(2) = prime(2) + prime(3) + prime(4) = 3 + 5 + 7,
a(3) = 49 = 7*7 = A080665(1) = A034961(6) = prime(6) + prime(7) + prime(8) = 13 + 17 + 19.

Zak Seidov, Table of n, a(n) for n = 1..1000

semiPrimeQ[n_Integer] := Total[FactorInteger[n]][[2]] == 2; Select[Total /@ Partition[Prime[Range[100]], 3, 1], semiPrimeQ] (* T. D. Noe, Apr 20 2011 *)

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

Zak Seidov, May 18 2017

nonn

Squares that are the sum of 4 consecutive primes: 36, 324, 576, 1764, 2304, 4900, 20736, 63504, 66564, 128164, 142884, 150544, 156816, 183184, 236196, 256036, 260100, 311364, 369664, 414736.
Squares that are the sum of 5 consecutive primes: 961, 1089, 1681, 17689, 18769, 21025, 23409, 45369, 76729, 80089, 97969, 124609, 218089, 235225, 290521, 421201, 434281.
Squares that are the sum of 6 consecutive primes: 3600, 24336, 25600, 47524, 66564, 98596, 129600, 138384, 228484, 236196, 331776, 379456, 404496, 490000, 559504.
Squares that are the sum of 7 consecutive primes: 169, 625, 2209, 10201, 25921, 235225, 342225, 361201, 380689, 383161, 426409, 508369, 531441, 537289, 543169, 564001, 603729.
Note that A007504(m) - A007504(n) ~ m^2 log(m)/2 as m -> infinity.  Heuristically this has probability ~ 1/(m sqrt(2 log(m))) of being a square.  Since the sum of these probabilities diverges, on the basis of the second Borel-Cantelli lemma we should expect a(n) to exist.  Of course, this is not a proof.  Moreover, since the sum diverges very slowly, we might expect some very large values of a(n). - Robert Israel, May 18 2017

			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).

Cf. A007504.

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

Table[Set[{k, s}, {n, 0}]; While[! IntegerQ@ Sqrt[AddTo[s, Prime@ k]], k++]; s, {n, 36}] (* Michael De Vlieger, May 20 2017 *)

a(n) = my(s=0); forprime(p=prime(n), , s=s+p; if(issquare(s), return(s))) \\ Felix Fröhlich, May 25 2017

Missing a(25) and a(37) from Giovanni Resta, May 18 2017

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A062703 Squares that are the sum of two consecutive primes.

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A076304 Numbers k such that k^2 is a sum of three successive primes.

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A076305 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is a square.

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A188651 Products of two primes (i.e., "semiprimes") that are the sum of three consecutive primes.

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A287027 Least sum s of consecutive prime numbers starting with prime(n) such that s is a perfect square.

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