A076305 -id:A076305 - 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

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]

A206279 Smallest of three consecutive primes whose sum is a square.

Original entry on oeis.org

13, 37, 277, 313, 613, 7591, 8209, 12157, 14557, 15679, 16267, 23053, 32233, 42953, 44887, 55213, 81013, 94687, 105649, 106397, 135241, 203317, 221401, 225769, 235747, 245941, 258707, 287671, 306541, 331333, 342049, 346111, 347443, 393853, 479191, 488827

Offset: 1

Harvey P. Dale, Feb 05 2012

nonn

			a(4) = 313. The next two primes are 317 and 331, and 313 + 317 + 331 = 961 = 31^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A061275, A076304, A076305, A206280, A206281.

Transpose[Select[Partition[Prime[Range[50000]],3,1],IntegerQ[ Sqrt[ Total[#]]]&]][[1]]

p=2;q=3;forprime(r=5,1e9,if(issquare(p+q+r),print1(p", "));p=q;q=r) \\ Charles R Greathouse IV, Aug 28 2013

a(n) = A000040(A076305(n)). - Zak Seidov, Apr 07 2017

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 *)

A166261 Numbers k with property that the sum of 120 successive primes starting with prime(k) is a square.

Original entry on oeis.org

10917, 11527, 50923, 73894, 111468, 118436, 128662, 139123, 195234, 249281, 332863, 435489, 438080, 482557, 538373, 542299, 650254, 679958, 722145, 803501, 810871, 820409, 962582, 970711, 1003544, 1027732, 1030010, 1190134, 1204929, 1305603, 1636065, 1689410

Offset: 1

Zak Seidov, Oct 10 2009

Corresponding values of s = sqrt(Sum_{i=k..k+119} prime(i)) are A166262.

			a(1) = 10917: Sum_{i=0..119} prime(10917+i) = 3734^2 = A166262(1)^2,
a(2) = 11527: Sum_{i=0..119} prime(11527+i) = 3846^2 = A166262(2)^2.

Cf. A166262.

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

PrimePi/@Select[Partition[Prime[Range[169*10^4]],120,1],IntegerQ[ Sqrt[ Total[ #]]]&][[All,1]] (* Harvey P. Dale, Jan 22 2019 *)

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

S=vecsum(primes(119)); p=0; q=prime(120); for(n=1,oo, issquare(S+=q-p) && print1(n","); q=nextprime(q+1); p=nextprime(p+1)) \\ It is about 25% faster to avoid "nextprime(p)" at expense of keeping the last 120 primes used in a vector p, using {my(i=Mod(0,120)); ...p[lift(i)+1]... i++}. - M. F. Hasler, Jan 04 2020

a(30)-a(32) from Michel Marcus, Oct 15 2013

Edited by M. F. Hasler, Jan 04 2020

A076306 Numbers k such that k^3 is a sum of three successive primes.

Original entry on oeis.org

11, 47, 145, 223, 229, 267, 313, 353, 365, 391, 397, 409, 507, 565, 567, 571, 573, 641, 661, 723, 793, 799, 841, 887, 895, 1015, 1051, 1089, 1293, 1297, 1411, 1451, 1469, 1789, 1909, 1943, 2043, 2077, 2171, 2401, 2459, 2497, 2671, 2801, 2851, 2871, 2921, 3211

Offset: 1

Zak Seidov, Oct 05 2002, Nov 12 2009

nonn

prime(k) + prime(k+1) + prime(k+2) is a cube in A034961, k=A158796(n).

			11 is a term because 11^3 = 1331 = prime(85) + prime(86) + prime(87) = 439 + 443 + 449.
47 is a term because 47^3 = 103823 = prime(3696) + prime(3697) + prime(3698) = 34603 + 34607 + 34613.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..252 from Zak Seidov, terms 253..1000 from Donovan Johnson)

Cf. A076304, A076305, A034961, A158796, A227475.

okQ[n_]:=Module[{x=n^3,low,hi}, low=PrimePi[Round[x/3]]-4; hi=low+8; MemberQ[Total/@Partition[Prime[Range[low,hi]],3,1],x]]; Select[Range[5,3300],okQ]  (* Harvey P. Dale, Jan 27 2011 *)

{ p1=prime(1) ; p2=prime(2) ; p3=prime(3) ; n3=p1+p2+p3 ; for(i=1,100000000, if( ispower(n3,3,&n), print(n) ; ) ; n3 -= p1 ; p1=p2 ; p2=p3 ; p3=nextprime(p3+1) ; n3 += p3 ; ) ; } \\ R. J. Mathar, Jan 13 2007

n=0; forstep(j=3, 86231, 2, c=j^3; c3=c/3; f=0; if(denominator(c3)==1, if(isprime(c3), if(precprime(c3-1)+c3+nextprime(c3+1)==c, f=1))); p2=precprime(c3); p1=precprime(p2-1); p3=nextprime(c3); p4=nextprime(p3+1); if(p1+p2+p3==c, f=1); if(p2+p3+p4==c, f=1); if(f==1, n++; write("b076306.txt", n " " j))) /* Donovan Johnson, Sep 02 2013 */

from _future_ import division
from sympy import nextprime, prevprime, isprime
A070306_list, i = [], 3
while i < 10**6:
    n = i**3
    m = n//3
    pm, nm = prevprime(m), nextprime(m)
    k = n - pm - nm
    if isprime(m):
        if m == k:
            A070306_list.append(i)
    else:
        if nextprime(nm) == k or prevprime(pm) == k:
            A070306_list.append(i)
    i += 1 # Chai Wah Wu, May 30 2017

More terms from R. J. Mathar, Jan 13 2007
a(29)-a(48) from Donovan Johnson, Apr 27 2008
Edited by N. J. A. Sloane, Nov 12 2009 at the suggestion of R. J. Mathar

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

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

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

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A206279 Smallest of three consecutive primes whose sum 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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A166261 Numbers k with property that the sum of 120 successive primes starting with prime(k) is a square.

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A076306 Numbers k such that k^3 is a sum of three successive primes.

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