A166261 -id:A166261 - cp's OEIS frontend

A064397 Numbers k such that prime(k) + prime(k+1) is a square.

7, 15, 20, 61, 152, 190, 293, 377, 492, 558, 789, 919, 942, 1768, 2343, 2429, 2693, 2952, 3136, 3720, 4837, 5421, 5722, 6870, 7347, 8126, 8193, 9465, 9857, 9927, 10410, 10483, 10653, 12685, 13763, 13955, 16033, 16342, 17859, 18367, 18474

Offset: 1

Jason Earls, Sep 29 2001

nonn

			For k=15: prime(15) = 47 and prime(16) = 53, 47 + 53 = 100 = 10^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)

Cf. A061275 (the primes), A062703 (squares), A074924 (square root of sum), A000720.

Cf. A076305 (3 primes), A072849 (4 primes), A166255 (70 primes), A166261 (120 primes).

[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(n) = A000720(A061275(n)). - Amiram Eldar, Jun 28 2024

a(n) >> n^2/log^2 n. - Charles R Greathouse IV, Mar 08 2025

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

A072849 Prime(a(n)) + ... + prime(a(n)+3) is a square = A051395(n)^2.

Original entry on oeis.org

3, 21, 33, 84, 104, 199, 689, 1848, 1923, 3435, 3795, 3985, 4126, 4742, 5968, 6413, 6495, 7649, 8927, 9906, 16885, 17677, 20474, 20996, 22924, 23923, 36902, 38733, 40347, 40654, 41956, 42601, 43047, 44482, 44920, 51608, 52305, 56706, 66032

Offset: 1

Zak Seidov, Jun 21 2003

Conjecture: this sequence and A064397 are disjoint. That is to say, prime(n) + prime(n+1) and prime(n) + prime(n+1) + prime(n+2) + prime(n+3) cannot be squares at the same time. - Jianing Song, Nov 13 2022

			a(1) = 3 because prime(3) + prime(4) + prime(5) + prime(6) = 5+7+11+13 = 36 = 6*6.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Harvey P. Dale)

Cf. A051395 (square root of sums), A206280 (primes), A000720.

Cf. A064397 (2 primes), A076305 (3 primes), A166255 (70 primes), A166261 (120 primes).

Flatten[Position[Partition[Prime[Range[70000]],4,1],?(IntegerQ[ Sqrt[ Total[ #]]]&),{1},Heads->False]] (* _Harvey P. Dale, Dec 09 2015 *)

lista(m) = {for (n = 1, m, if (issquare(prime(n) + prime(n+1) + prime(n+2) + prime(n+3)), print1(n, ", ")););}  \\ Michel Marcus, Apr 19 2013

a(n) = A000720(A206280(n)). - Amiram Eldar, Jun 28 2024

Definition corrected by Zak Seidov, Dec 13 2014

A166255 Numbers k with property that the sum of 70 successive primes starting with prime(k) is a square.

Original entry on oeis.org

71, 201, 1024, 1594, 10915, 36934, 51050, 60054, 60914, 71822, 80331, 85230, 92916, 96352, 103271, 114667, 151019, 158591, 183484, 184348, 193979, 196078, 223587, 277476, 295890, 309502, 317601, 334181, 338139, 369101, 485330, 494188, 530832

Offset: 1

Zak Seidov, Oct 10 2009

Sum_{i=k..k+69} prime(i) = s^2; and the values of s are A166256.

			prime(71)+...+prime(71+69) = 200^2 = A166256(1)^2,
prime(201)+...+prime(201+69) = 322^2 = A166256(2)^2,
prime(1024)+...+prime(1024+69) = 770^2 = A166256(3)^2.

David A. Corneth, Table of n, a(n) for n = 1..693 (first 100 terms from Zak Seidov)

Cf. A166256.

Cf. A064397 (2 primes), A076305 (3 primes), A072849 (4 primes), A166261 (120 primes).

PrimePi[First[#]]&/@Select[Partition[Prime[Range[1000000]],70,1], IntegerQ[ Sqrt[ Total[#]]]&] (* Harvey P. Dale, Jun 13 2011 *)

A230327 Index of smallest prime such that the sum of n consecutive primes starting with this specific prime is a square.

Original entry on oeis.org

7, 6, 3, 42, 107, 6, 38, 1, 1631, 170, 38, 119, 5, 546, 78, 309, 85, 604, 199, 57, 270, 2, 3, 333, 45, 2, 178, 1708, 291, 2, 35, 72, 322, 19, 84, 5, 155, 346, 122, 2175, 1395, 24, 886, 2, 3108, 168, 14, 499, 340, 294, 156, 578, 325, 240, 115, 61, 283, 1035

Offset: 2

Michel Marcus, Oct 16 2013

nonn

			a(2)=7 because 17+19 (2 terms) = 36 is a square, 17 being the 7th prime.
a(3)=6 because 13+17+19 (3 terms) =49 is a square, 13 being the 6th prime.

Cf. A064397 (2 primes), A076305 (3 primes), A072849 (4 primes), A166255 (70 primes), A166261 (120 primes).

Cf. A132955 (primes themselves), A132956 (squares=sums), A132957 (square roots of sums).

a(n, lim=10^5) = {n --; pr = primes(lim); for (i = 1, lim-n, s = sum(k=i, i+n, pr[k]); if (issquare(s), return (i));); return (0);} \\ Michel Marcus, Oct 16 2013

A358156 a(n) is the smallest number k such that the sum of k consecutive prime numbers starting with the n-th prime is a square.

Original entry on oeis.org

9, 23, 4, 1862, 14, 3, 2, 211, 331, 163, 366, 3, 124, 48, 2, 449, 8403, 121, 35, 2, 4, 105, 77, 43, 190769, 1726, 234, 248, 200, 295, 293, 73, 4, 873, 32, 64, 2456139382, 8, 4519, 14, 123, 5, 9395, 296, 26, 5, 3479, 810, 9, 7091, 1669, 157, 1189, 12559, 269, 4930, 21, 376, 3

Offset: 1

Todor Szimeonov, Nov 01 2022

nonn

a(60) > 10^10 and a(68) > 10^13. - Martin Ehrenstein, Nov 09 2022

			For n=7, prime(7) = 17 and starting there 2 primes 17 + 19 = 36 which is square, so that a(7)=2.

Todor Szimeonov, Square of prime numbers

Cf. A000040, A000290, A105720, A230327 (exchanges the roles of n, k), A287027 (squares reached).

Indices of terms: A064397 (2's), A076305 (3's), A072849 (4's), A166255 (70's), A166261 (120's).

f:= proc(n) local p,s,k;
  p:= ithprime(n); s:= p;
  for k from 2 do
    p:= nextprime(p);
    s:= s+p;
    if issqr(s) then return k fi
  od
end proc:
map(f, [$1..36]); # Robert Israel, Nov 08 2022

a[n_] := Module[{p = s = Prime[n], k = 1}, While[! IntegerQ[Sqrt[s]], p = NextPrime[p]; s += p; k++]; k]; Array[a, 36] (* Amiram Eldar, Nov 08 2022 *)

a(25)-a(36) from Robert Israel, Nov 08 2022

a(37)-a(59) from Martin Ehrenstein, Nov 09 2022

A166262 Numbers n with property that n^2 is a sum of some 120 successive primes.

Original entry on oeis.org

3734, 3846, 8660, 10602, 13248, 13690, 14318, 14936, 17934, 20458, 23902, 27614, 27704, 29176, 30942, 31064, 34238, 35070, 36216, 38346, 38532, 38774, 42236, 42428, 43190, 43742, 43794, 47308, 47622, 49708, 56070, 57036, 58856, 65692, 66122, 66940, 68016

Offset: 1

Zak Seidov, Oct 10 2009

nonn

n^2=sum(prime(k),k=m,m+119); corresponding values of m: 10917, 11527, 50923, 73894, 111468, 118436, 128662, 139123, 195234 (A166261).

			a(1)=3734: 3734^2=sum[Prime[i], {i,10917,10917+119}],
a(2)=3846: 3846^2=sum[Prime[i], {i,11527,11527+119}].

Cf. A166261.

Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[5*10^6]],120,1]),IntegerQ] (* Harvey P. Dale, Jul 17 2019 *)

lista(nn) = {pr = primes(nn); for (i=1, nn-119, s = sum(k=i, i+119, pr[k]); if (issquare(s), print1(sqrtint(s), ", ")););} \\ Michel Marcus, Oct 15 2013

a(35)-a(37) from Michel Marcus, Oct 15 2013

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A064397 Numbers k such that prime(k) + prime(k+1) is a square.

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

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A072849 Prime(a(n)) + ... + prime(a(n)+3) is a square = A051395(n)^2.

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A166255 Numbers k with property that the sum of 70 successive primes starting with prime(k) is a square.

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Mathematica

A230327 Index of smallest prime such that the sum of n consecutive primes starting with this specific prime is a square.

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PARI

A358156 a(n) is the smallest number k such that the sum of k consecutive prime numbers starting with the n-th prime is a square.

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A166262 Numbers n with property that n^2 is a sum of some 120 successive primes.

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