A206280 -id:A206280 - cp's OEIS frontend

A061275 Smaller of two consecutive primes whose sum is a square.

17, 47, 71, 283, 881, 1151, 1913, 2591, 3527, 4049, 6047, 7193, 7433, 15137, 20807, 21617, 24197, 26903, 28793, 34847, 46817, 53129, 56443, 69191, 74489, 83231, 84047, 98563, 103049, 103967, 109507, 110441, 112337, 136237, 149057, 151247

Offset: 1

Amarnath Murthy, Apr 25 2001

			a(4) = 283, the next prime is 293 and 283 + 293 = 576 = 24^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Subset of A091624.

Cf. A000040, A064397, A206279, A206280, A206281.

Transpose[Select[Partition[Prime[Range[20000]],2,1],IntegerQ[Sqrt[Plus@@# ]]&]][[1]] (* Harvey P. Dale, Aug 04 2009 *)

{ default(primelimit, 550655327); n=0; q=2; forprime (p=3, 550655327, if (issquare(p+q), write("b061275.txt", n++, " ", q)); q=p ) } \\ Harry J. Smith, Jul 20 2009

a(n) = A000040(A064397(n)). - Amiram Eldar, Jun 28 2024

More terms from Larry Reeves (larryr(AT)acm.org) and Asher Auel, May 15 2001

Offset changed from 0 to 1 by Harry J. Smith, Jul 20 2009

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

A051395 Numbers whose square is a sum of 4 consecutive primes.

Original entry on oeis.org

6, 18, 24, 42, 48, 70, 144, 252, 258, 358, 378, 388, 396, 428, 486, 506, 510, 558, 608, 644, 864, 886, 960, 974, 1022, 1046, 1326, 1362, 1392, 1398, 1422, 1434, 1442, 1468, 1476, 1592, 1604, 1676, 1820, 1950, 2016, 2068, 2140, 2288, 2430, 2460

Offset: 1

Zak Seidov, Jun 21 2003

First of four consecutive primes in A206280.

			6 is a term because 6*6 = 5 + 7 + 11 + 13;
18 is a term because 18*18 = 324 = 73 + 79 + 83 + 89.

Charles R Greathouse IV and Zak Seidov, Table of n, a(n) for n = 1..10783 (First 3400 terms from Charles R Greathouse IV)

Cf. A072849, A206280.

lista(nn) =  {pr = primes(nn); for (i = 1, nn - 3, s = pr[i] + pr[i+1] + pr[i+2] + pr[i+3]; if (issquare(s), print1(sqrtint(s), ", ")););} \\ Michel Marcus, Oct 02 2013

is(n)=n*=n; my(p=precprime(n\4),q=nextprime(n\4+1),r,s); if(n < 3*q+p+8, r=precprime(p-1); s=n-p-q-r; ispseudoprime(s) && (s == precprime(r-1) || s == nextprime(q+1)), r=nextprime(q+1); s=n-p-q-r; ispseudoprime(s) && (s == precprime(p-1) || s == nextprime(r+1))) \\ Charles R Greathouse IV, Oct 02 2013

Numbers m such that m^2 = Sum_{i=k..k+3} prime(i) for some k.

Corrected and extended by Don Reble, Nov 20 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

A245577 Numbers k such that k^4 is a sum of 4 consecutive primes.

Original entry on oeis.org

12, 90, 208, 212, 234, 242, 314, 366, 404, 410, 416, 486, 540, 590, 750, 888, 908, 1152, 1418, 1444, 1500, 1524, 1658, 1666, 1736, 1798, 1814, 1874, 1940, 1942, 2094, 2138, 2266, 2496, 2584, 3058, 3062, 3206, 3660, 4034, 4080, 4208, 4368, 4422, 4606, 4872

Offset: 1

Zak Seidov, Nov 29 2014

nonn

			12^4 = 20736 = prime(689) + prime(689 + 1) + prime(689 + 2) + prime(689 + 3) = 5171 + 5179 + 5189 + 5197.

Robert G. Wilson v, Table of n, a(n) for n = 1..5100 (first 1189 terms from Zak Seidov)

Cf. A034963, A051395, A206280.

fQ[n_] := MemberQ[ Total@# & /@ Partition[ Table[ NextPrime[n^4/4, i], {i, {-3, -2, -1, 1, 2, 3}}], 4, 1], n^4]; Select[ Range@ 5000, fQ] (* Robert G. Wilson v, Dec 03 2014 *)

isscpn(n) = {np = n^4; p = precprime(np\4); for (i=1, 3, p = precprime(p-1);); while(1, q = nextprime(p+1); r = nextprime(q+1); s = nextprime(r+1); if ((v=p+q+r+s) == np, return (1)); if (v > np, return (0)); p = q;);} \\ Michel Marcus, Nov 30 2014

A206281 Smallest of five consecutive primes whose sum is a square.

Original entry on oeis.org

181, 199, 317, 3529, 3733, 4177, 4663, 9049, 15329, 15991, 19577, 24907, 43607, 47017, 58073, 84223, 86843, 146191, 152417, 156623, 175543, 217559, 227671, 288461, 308999, 323077, 331249, 333323, 354301, 390289, 397037, 407249, 474923, 476137, 491059, 520339

Offset: 1

Harvey P. Dale, Feb 05 2012

nonn

			a(4) = 3529. The next four primes are 3533, 3539, 3541, and 3547, and the sum of all five primes = 17689 = 133^2.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A061275, A206279, A206280.

count:= 0: Res:= NULL:
for y from 10 while count < 100 do
  target:= y^2;
  t:= prevprime(ceil(target/5));
  s:= prevprime(t);
  r:= prevprime(s);
  q:= prevprime(r);
  p:= prevprime(q);
  u:= p+q+r+s+t;
  while u < target do
    p:= q; q:= r; r:= s; s:= t; t:= nextprime(t);
    u:= p+q+r+s+t;
  od;
  if u = target then
     count:= count+1; Res:= Res, p;
  fi
od:
Res; # Robert Israel, Oct 20 2020

Transpose[Select[Partition[Prime[Range[80000]],5,1],IntegerQ[Sqrt[ Total[#]]]&]][[1]]

A226151 Numbers n such that triangular(n) is a sum of 4 consecutive primes.

Original entry on oeis.org

8, 15, 39, 56, 60, 144, 155, 203, 212, 216, 263, 388, 451, 464, 480, 555, 619, 644, 680, 723, 736, 788, 791, 799, 876, 903, 1012, 1056, 1143, 1239, 1284, 1368, 1479, 1547, 1611, 1684, 1695, 1703, 1827, 1859, 1908, 1939, 2100, 2108, 2135, 2148, 2152, 2187, 2199, 2216

Offset: 1

Alex Ratushnyak, May 28 2013

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A000217, A034963, A051395, A206280, A226154.

#include 
#include 
#include 
#define TOP (1ULL<<30)
int main() {
  unsigned long long i, j, p1, p2, p3, r, s;
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (i=3; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0 /*&& i<(1ULL<<32)*/)
        for (j=i*i>>1; j>3] |= 1 << (j&7);
  for (p3=2, p2=3, p1=5, i=7; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      s = p3 + p2 + p1 + i;
      r = sqrt(s*2);
      if (r*(r+1)==s*2) printf("%llu, ", r);
      p3 = p2, p2 = p1, p1 = i;
    }
  return 0;
}

istriangular:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n = t1*(t1+1)/2 then return t1 ; else return -1; end if; end;
A034963 := proc(n)
    add(ithprime(i),i=n..n+3) ;
end proc:
for n from 1 to 90000 do
    ist := istriangular(A034963(n)) ;
    if ist >= 0 then
        printf("%d,",ist) ;
    end if;
end do: # R. J. Mathar, Jun 04 2013

(Sqrt[8#+1]-1)/2&/@Select[Total/@Partition[Prime[Range[ 60000]],4,1], OddQ[ Sqrt[8#+1]]&] (* Harvey P. Dale, Apr 06 2016 *)

A247258 The first of four consecutive primes that add up to a fourth power.

Original entry on oeis.org

5171, 16402481, 467943389, 504990743, 749554829, 857435471, 2430292727, 4486052477, 6659865623, 7064402441, 7487094743, 13947137557, 21257639959, 30293402473, 79101562439, 155450409929, 169935221737, 440301256697, 1010752751017, 1086948034597

Offset: 1

Zak Seidov, Nov 30 2014

nonn

			5171 is in the sequence because the next three primes are 5179, 5189, 5197, and 5171 + 5179 + 5189 + 5197 = 20736 = 12^4.

Zak Seidov, Table of n, a(n) for n = 1..1189

Cf. A245577 (base of 4th power), A247259, A247265, A247266 (members of quadruple), subsequence of A206280.

A226152 Numbers n such that n^2 is an average of 4 consecutive primes.

Original entry on oeis.org

3, 9, 12, 21, 24, 35, 72, 126, 129, 179, 189, 194, 198, 214, 243, 253, 255, 279, 304, 322, 432, 443, 480, 487, 511, 523, 663, 681, 696, 699, 711, 717, 721, 734, 738, 796, 802, 838, 910, 975, 1008, 1034, 1070, 1144, 1215, 1230, 1237, 1265, 1276, 1370, 1375, 1386, 1469

Offset: 1

Alex Ratushnyak, May 28 2013

nonn

Integers of the form sqrt(A102655(k)) for any k. - R. J. Mathar, Jun 06 2013

Cf. A102655, A051395, A206280.

#include 
#include 
#include 
#define TOP (1ULL<<30)
int main() {
  unsigned long long i, j, p1, p2, p3, r, s;
  unsigned char *c = (unsigned char *)malloc(TOP/8);
  memset(c, 0, TOP/8);
  for (i=3; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0 /*&& i<(1ULL<<32)*/)
        for (j=i*i>>1; j>3] |= 1 << (j&7);
  for (p3=2, p2=3, p1=5, i=7; i < TOP; i+=2)
    if ((c[i>>4] & (1<<((i>>1) & 7)))==0) {
      s = p3 + p2 + p1 + i;
      if (s%4==0) {
        s/=4;
        r = sqrt(s);
        if (r*r==s) printf("%llu, ", r);
      }
      p3 = p2, p2 = p1, p1 = i;
    }
  return 0;
}

A034963 := proc(n)
    add(ithprime(i), i=n..n+3) ;
end proc:
for n from 1 to 90000 do
    s := A034963(n)/4 ;
    if type(s,'integer') then
    if issqr(s) then
        printf("%d, ", sqrt(s)) ;
    end if;
    end if;
end do: # R. J. Mathar, Jun 06 2013

a(n) = A051395(n)/2.

cp's OEIS Frontend

A061275 Smaller of two consecutive primes whose sum is a square.

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

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A051395 Numbers whose square is a sum of 4 consecutive primes.

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

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A245577 Numbers k such that k^4 is a sum of 4 consecutive primes.

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Mathematica

PARI

A206281 Smallest of five consecutive primes whose sum is a square.

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Maple

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A226151 Numbers n such that triangular(n) is a sum of 4 consecutive primes.

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C

Maple

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