A051395 -id:A051395 - cp's OEIS frontend

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

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

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 *)

A252018 Numbers n such that n^2 is a sum of 6 consecutive primes.

Original entry on oeis.org

60, 156, 160, 218, 258, 314, 360, 372, 478, 486, 576, 616, 636, 700, 748, 832, 1070, 1108, 1152, 1250, 1564, 1614, 1636, 1644, 1686, 1710, 1738, 1846, 1862, 1878, 1924, 2010, 2060, 2062, 2156, 2182, 2376, 2490, 2530, 2748, 2754, 2774, 2824, 2826, 2834, 2860, 2896, 2902

Offset: 1

Zak Seidov, Dec 14 2014

nonn

			60^2 = 3600 = prime(107) + ... + prime(112) = 587 + 593 + 599 + 601 + 607 + 613,
156^2 = 24336 = prime(557) + ... + prime(562) = 4027 + 4049 + 4051 + 4057 + 4073 + 4079.

Zak Seidov, Table of n, a(n) for n = 1..5000

Cf. A051395, A074924, A252066.

Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[150000]],6,1]),IntegerQ] (* Harvey P. Dale, Aug 02 2021 *)

A252066 Numbers n such that n^2 is a sum of 2 and also of 4 consecutive primes.

Original entry on oeis.org

6, 24, 42, 48, 1326, 1676, 2772, 4428, 4820, 4940, 5328, 5472, 6318, 9950, 10074, 12942, 13724, 14372, 16290, 18428, 22776, 22818, 23800, 23952, 25134, 28614, 28800, 31212, 31394, 32060, 33716, 36526, 37320, 39228, 39446, 39528, 43670, 43798, 44490, 45570, 47700, 48000

Offset: 1

Zak Seidov, Dec 13 2014

nonn

			6^2=36=17+19=5+7+11+13, 18^2=324=157+163=73+79+83+89.

Zak Seidov, Table of n, a(n) for n = 1..507

Intersection of A051395 and A074924.

Module[{nn=10^7,p2,p4},p2=Total/@Partition[Prime[Range[nn]],2,1];p4=Total/@Partition[ Prime[Range[nn]],4,1];Select[Sqrt[Intersection[p2,p4]],IntegerQ]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, May 03 2024 *)

A252019 Numbers n such that n^2 is a sum of 2, 4 and 6 consecutive primes.

Original entry on oeis.org

4820, 69636, 97058, 405888, 454086, 585828, 656490, 711282, 717486, 1161132, 1348582, 1560352, 1564810, 1625370, 1811262, 1838510, 1926224, 2446248, 2601094, 2670318, 2699918, 2961770, 2966112, 3234498, 3372694, 3387258, 3705880, 3860232, 3980524, 4104264, 4147330

Offset: 1

Zak Seidov, Dec 14 2014

nonn

All terms are even. - Michel Marcus, Jul 07 2015

Zak Seidov, Table of n, a(n) for n = 1..673

Intersection of A051395, A074924 and A252018.

Also, intersection of A252066 and A252018.

A251056 Numbers n such that n^2 is a sum of 8 consecutive primes.

Original entry on oeis.org

38, 414, 466, 514, 714, 844, 850, 1076, 1136, 1186, 1370, 1512, 1544, 1580, 1600, 1700, 1844, 1900, 1918, 2028, 2114, 2250, 2304, 2320, 2330, 2364, 2396, 2404, 2450, 2674, 2846, 2894, 3076, 3314, 3346, 3506, 3612, 3622, 3676, 3718, 3774, 3866, 3912, 3966, 4012, 4126, 4506, 4700

Offset: 1

Zak Seidov, Dec 14 2014

nonn

			38^2 = 1444 = prime(38) + ... + prime(45) = 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197,
414^2 = 171396 = prime(2401) + ... + prime(2408) = 21391 + 21397 + 21401 + 21407 + 21419 + 21433 + 21467 + 21481.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A074924, A051395, A252018, A252019, A252066.

Sqrt[#]&/@Select[Total/@Partition[Prime[Range[250000]],8,1], IntegerQ[ Sqrt[#]]&] (* Harvey P. Dale, Nov 28 2018 *)

A363281 Numbers which are the sum of 4 squares of distinct primes.

Original entry on oeis.org

87, 159, 183, 199, 204, 207, 231, 247, 252, 303, 319, 324, 327, 343, 348, 351, 364, 367, 372, 399, 423, 439, 444, 463, 468, 471, 484, 487, 492, 495, 511, 516, 532, 535, 540, 543, 556, 559, 564, 567, 583, 588, 591, 604, 607, 612, 628, 655, 660, 663, 676, 679, 684, 700, 703, 708

Offset: 1

Zhining Yang, May 25 2023

			87 is a term as 87 = 2^2 + 3^2 + 5^2 + 7^2.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A133524, A051395.

Select[Range@1000,
 Length[PowersRepresentations[#, 4, 2] // Select[AllTrue@PrimeQ] //
     Select[DuplicateFreeQ]] > 0 &]

upto(n) = {if(n <= 86, return([])); my(pr = primes(primepi(sqrtint(n - 38))), res = List()); forvec(v = vector(4, i, [1, #pr]), c = sum(i = 1, #v, pr[v[i]]^2); if(c <= n, listput(res, c)), 2); listsort(res, 1); res} \\ David A. Corneth, Jul 12 2023

from itertools import combinations as comb
ps=[p**2 for p in [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31]]
a=[n for n in range(1001) if n in [sum(n) for n in list(comb(ps,4))]]
print(a)

A217443 Primes p such that p^2 is an arithmetic average of 4 consecutive primes.

Original entry on oeis.org

3, 179, 443, 487, 523, 1237, 1481, 1571, 1933, 2293, 2801, 3307, 3863, 4073, 4493, 4787, 4909, 5399, 5683, 5693, 6427, 7433, 7789, 9067, 9623, 10607, 10883, 11437, 11497, 11579, 12149, 12263, 12553, 13337, 13669, 13841, 14071, 14723, 14741, 15569, 15901, 16381

Offset: 1

Zak Seidov, Oct 03 2012

nonn

2*p is a term of A051395: 2*3 = A051395(1), 2*179 = A051395(10), 2*443 = A051395(22).

			3^2 = (5+7+11+13)/4, 179^2 = (32027+32029+32051+32057)/4.

Cf. A051395.

Select[Sqrt[Mean[#]]&/@Partition[Prime[Range[147*10^5]],4,1],PrimeQ] (* Harvey P. Dale, Jul 27 2019 *)

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

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

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A252018 Numbers n such that n^2 is a sum of 6 consecutive primes.

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A252066 Numbers n such that n^2 is a sum of 2 and also of 4 consecutive primes.

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Mathematica

A252019 Numbers n such that n^2 is a sum of 2, 4 and 6 consecutive primes.

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A251056 Numbers n such that n^2 is a sum of 8 consecutive primes.

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A363281 Numbers which are the sum of 4 squares of distinct primes.

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A217443 Primes p such that p^2 is an arithmetic average of 4 consecutive primes.