A064397 -id:A064397 - cp's OEIS frontend

A074924 Numbers whose square is the sum of two successive primes.

6, 10, 12, 24, 42, 48, 62, 72, 84, 90, 110, 120, 122, 174, 204, 208, 220, 232, 240, 264, 306, 326, 336, 372, 386, 408, 410, 444, 454, 456, 468, 470, 474, 522, 546, 550, 594, 600, 630, 640, 642, 686, 740, 750, 762, 766, 788, 802, 852, 876, 882, 920, 936, 970

Offset: 1

Zak Seidov, Oct 02 2002

nonn

			6^2 = 17 + 19, 1610^2 = 1296041 + 1296059.

Zak Seidov, Table of n, a(n) for n = 1..22054 (all terms up to 10^6).

Square roots of squares in A001043.
Cf. A062703 (the squares), A061275 (lesser of the primes), A064397 (index of that prime).
Cf. A064397 (numbers n such that prime(n) + prime(n+1) is a square), A071220 (prime(n) + prime(n+1) is a cube), A074925 (n^3 is sum of 2 consecutive primes).

filter:= proc(n) local t; t:= n^2/2; prevprime(ceil(t)) + nextprime(floor(t)) = n^2 end proc:
select(filter, [$3..1000]); # Robert Israel, Nov 19 2024

Select[Sqrt[#]&/@(Total/@Partition[Prime[Range[50000]],2,1]),IntegerQ] (* Harvey P. Dale, Oct 04 2014 *)
f@n_ := Sqrt@Select[(2*Range@n)^2, # == Plus @@ NextPrime[#/2, {-1, 1}] &]; f@485 (* Hans Rudolf Widmer, Nov 19 2024 *)

is(n)=if(n%2, return(0)); nextprime(n^2/2+1)+precprime(n^2/2)==n^2 \\ Charles R Greathouse IV, Apr 29 2015

select( {is_A074924(n)=!bittest(n=n^2,0) && precprime(n\2)+nextprime(n\/2)==n}, [1..999]) \\ M. F. Hasler, Jan 03 2020

A74924=[6]; apply( A074924(n)={while(n>#A74924, my(N=A74924[#A74924]); until( is_A074924(N+=2),);A74924=concat(A74924,N));A74924[n]}, [1..99]) \\ 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 k
print(list(islice(agen(), 54))) # Michael S. Branicky, May 24 2022

a(n) = sqrt(A062703(n)). - Zak Seidov, May 26 2013

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

Crossrefs section corrected and extended by M. F. Hasler, Jan 03 2020

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

Original entry on oeis.org

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

A062703 Squares that are the sum of two consecutive primes.

Original entry on oeis.org

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

A179975 Smallest k such that k*10^n is a sum of two successive primes.

Original entry on oeis.org

5, 3, 1, 6, 6, 6, 14, 6, 9, 19, 21, 21, 42, 93, 21, 6, 11, 2, 12, 111, 37, 39, 63, 38, 42, 24, 15, 15, 60, 6, 39, 82, 47, 58, 337, 49, 72, 25, 34, 21, 6, 107, 128, 96, 20, 2, 63, 231, 70, 7, 62, 144, 28, 151, 157, 33, 98, 55, 134, 162, 87, 201, 124, 303, 64, 106, 130, 13, 43

Offset: 0

Zak Seidov, Aug 04 2010

nonn

From Robert G. Wilson v, Aug 11 2010: (Start)
A179975 n's such that a(n)=1: 3, 335, ..., .
A179975 First occurrence of k: 3, 18, 2, ???, 1, 4, 50, 162, 9, 335, 17, 19, 68, 7, 27, ..., .
Records: 5, 6, 14, 19, 21, 42, 93, 111, 337, 449, 862, 1049, 1062, 1122, 1280, 2278, 3168, 4290, ..., . (End)

			a(0)=5 because 5=2+3
a(1)=3 because 30=13+17
a(2)=1 because 100=47+53
a(3)=6 because 6000=2999+3001.

Robert G. Wilson v, Table of n, a(n) for n = 0..400.
Dario Alejandro Alpern, Brilliant numbers

Cf. A064397, A071220, A074924, A074925.

Cf. A033873, A033874, A005235, A055211, A038804, A084475.

Join[{5,3},Reap[Do[Do[n=10^m k; If[n==PreviousPrime[n/2]+NextPrime[n/2],Sow[k];Break[]],{k,2000}],{m,2,50}]][[2,1]]]
f[n_] := Block[{k = 1, tn = 10^n}, While[h = k*tn/2; NextPrime[h, -1] + NextPrime@h != k*tn, k++ ]; k]; f[1] = 3; Array[f, 70, 0] (* Robert G. Wilson v, Aug 11 2010 *)

More terms from Robert G. Wilson v, Aug 11 2010

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 *)

A074925 Numbers n such that n^3 is a sum of two successive primes.

Original entry on oeis.org

2, 6, 28, 42, 72, 162, 174, 220, 222, 234, 246, 252, 296, 344, 380, 390, 486, 510, 584, 594, 638, 646, 674, 702, 720, 816, 828, 882, 942, 948, 990, 1044, 1056, 1146, 1200, 1314, 1422, 1436, 1554, 1566, 1596, 1602, 1632, 1740, 1770, 1778, 1806, 1818, 1824

Offset: 1

Zak Seidov, Oct 02 2002

nonn

Prime(n)+ prime(n+1) as a square in A064397; n^2 as a sum of two successive primes in A074924; prime(n)+ prime(n+1) as a cube in A071220.

			6^3 = 216 = 107 + 109.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms for n = 1..429 from Zak Seidov)

Cf. A064397, A071220, A074924.

Surd[#,3]&/@Select[Total/@Partition[Prime[Range[150*10^6]],2,1], IntegerQ[ Surd[#,3]]&] (* Harvey P. Dale, Jun 05 2018 *)

from sympy import nextprime, prevprime
A074925_list = [i for i in range(2,10**4,2) if prevprime(i**3//2) + nextprime(i**3//2) == i**3] # Chai Wah Wu, Feb 22 2017

More terms from Zak Seidov, Jul 22 2009

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

A071220 Numbers n such that prime(n) + prime(n+1) is a cube.

Original entry on oeis.org

2, 28, 1332, 3928, 16886, 157576, 192181, 369440, 378904, 438814, 504718, 539873, 847252, 1291597, 1708511, 1837979, 3416685, 3914319, 5739049, 6021420, 7370101, 7634355, 8608315, 9660008, 10378270, 14797144, 15423070, 18450693

Offset: 1

Labos Elemer, May 17 2002

nonn

The corresponding primes are in A061308; n^3 is a sum of two successive primes in A074925.

Prime(n)+ Prime(n+1) is a square in A064397; n^2 is a sum of two successive primes in A074924;

			28 is in the list because prime(28)+prime(29) = 107+109 =216 = 6^3.
n=1291597: prime(1291597)+prime(1291598) = 344*344*344.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A064397, A074925, A074924, A001043.

PrevPrim[n_] := Block[{k = n - 1}, While[ !PrimeQ[k], k-- ]; k]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; Do[ If[ n^3 == PrevPrim[Floor[(n^3)/2]] + NextPrim[Floor[(n^3)/2]], Print[ PrimePi[ Floor[(n^3)/2]]]], {n, 2, 10^4}]
Flatten[Position[Total/@Partition[Prime[Range[20000000]],2,1],?(IntegerQ[ Surd[ #,3]]&)]] (* _Harvey P. Dale, May 28 2014 *)

from _future_ import division
from sympy import isprime, prevprime, nextprime, primepi
A071220_list, i = [], 2
while i < 10**6:
    n = i**3
    m = n//2
    if not isprime(m) and prevprime(m) + nextprime(m) == n:
        A071220_list.append(primepi(m))
    i += 1 # Chai Wah Wu, May 31 2017

A001043(x)=m^3 for some m; if p(x+1)+p(x) is a cube, then x is here.

a(n) = primepi(A061308(n)). - Michel Marcus, Oct 24 2014

Edited and extended by Robert G. Wilson v, Oct 07 2002

cp's OEIS Frontend

A074924 Numbers whose square is the sum of two successive primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A062703 Squares that are the sum of two consecutive primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A179975 Smallest k such that k*10^n is a sum of two successive primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula