A133529 -id:A133529 - cp's OEIS frontend

A133940 Numbers n such that (prime(n)^2 + prime(n+1)^2 + prime(n+2)^2)/3 is prime (A084951).

4, 5, 8, 13, 15, 26, 46, 47, 50, 55, 57, 59, 61, 65, 66, 69, 77, 82, 89, 91, 94, 101, 105, 116, 134, 136, 137, 138, 144, 157, 194, 216, 219, 221, 224, 225, 229, 230, 234, 249, 257, 261, 263, 271, 272, 275, 306, 316, 319, 323

Offset: 1

Artur Jasinski, Sep 30 2007

nonn

With the exception of the first two terms, all numbers in A133529 are divisible by 3.

			a(1)=4 because (prime(4)^2 + prime(5)^2 + prime(6)^2)/3 = 113 is prime.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A133529, A084951.

select(n -> isprime((ithprime(n)^2 + ithprime(n+1)^2 + ithprime(n+2)^2)/3), [$3 .. 1000]); # Robert Israel, Apr 21 2015

b = {}; a = 2; Do[k = (Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a)/3; If[PrimeQ[k], AppendTo[b, n]], {n, 1, 200}]; b

is(n)=my(p=prime(n),q=nextprime(p+1),r=nextprime(q+1)); n>3 && isprime((p^2+q^2+r^2)/3) \\ Charles R Greathouse IV, Apr 21 2015

Corrected and edited by Zak Seidov, Apr 21 2015

A259772 Primes p such that p^3 + q^2 + r is also prime, where p,q,r are consecutive primes.

Original entry on oeis.org

3, 17, 19, 43, 53, 89, 107, 149, 293, 401, 439, 449, 659, 809, 821, 937, 1009, 1031, 1091, 1097, 1123, 1163, 1181, 1259, 1277, 1367, 1427, 1657, 1721, 1777, 1789, 1811, 1987, 2027, 2063, 2207, 2333, 2417, 2503, 2657, 2713, 3067, 3079, 3083, 3251, 3389, 3491, 3527

Offset: 1

K. D. Bajpai, Jul 05 2015

nonn

			a(2) = 17 is prime: 17^3 + 19^2 + 23 = 5297 which is also prime.
a(3) = 19 is prime: 19^3 + 23^2 + 29 = 7417 which is also prime.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000040, A034962, A133529, A133530, A258269, A304292.

[p: p in PrimesUpTo (3000) | IsPrime(k) where k is (p^3 + NextPrime(p)^2 + NextPrime(NextPrime(p)))];

select(n -> isprime(n) and isprime((n)^3+nextprime(n)^2+nextprime(nextprime((n)))), [seq(n, n=1..10000)]);

Select[Prime[Range[1000]], PrimeQ[#^3 + NextPrime[#]^2 + NextPrime[NextPrime[#]]]&]
Select[Partition[Prime[Range[500]],3,1],PrimeQ[#[[1]]^3+ #[[2]]^2+ #[[3]]]&][[All,1]] (* Harvey P. Dale, Dec 23 2021 *)

forprime(p=1, 3000, q=nextprime(p+1); r=nextprime(q+1); k=(p^3 + q^2 + r); if(isprime(k), print1(p,", ")))

A131686 Sum of squares of five consecutive primes.

Original entry on oeis.org

208, 373, 653, 989, 1469, 2189, 2981, 4061, 5381, 6701, 8069, 9917, 12029, 14069, 16709, 19541, 22061, 24821, 27989, 31421, 35789, 40661, 45029, 49589, 53549, 56909, 62837, 69389, 76709, 84149, 93581, 100253, 107741, 115541, 124109, 131837

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=208 because 2^2+3^2+5^2+7^2+11^2=208

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133539, A133540, A133541, A133542, A133543.

a = 2; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]

A133562 Numbers which are the sum of the squares of seven consecutive primes.

Original entry on oeis.org

666, 1023, 1543, 2359, 3271, 4519, 6031, 7591, 9439, 11719, 14359, 17119, 20239, 23599, 27079, 31111, 35191, 39631, 45319, 51031, 56599, 62719, 68359, 74239, 82447, 90199, 98767, 107479, 118231, 129151, 141031, 151471, 162199, 173359

Offset: 1

Artur Jasinski, Sep 16 2007

nonn

For primes in this sequence see A133560.
For sum of squares of two consecutive primes only 2^2 + 3^2 = 13 is prime.
For sum of squares of three consecutive primes A133529 it seems that only 83 is a prime (checked for all n < 1000000).
Sums of squares of four (and all even number) of consecutive primes are even numbers with exception n=1 but 2^2 + 3^2 + 5^2 + 7^2 = 87 = 3*29 is not prime.
For primes that are sums of squares of five consecutive primes see A133559.

			a(6) = 13^2 + 17^2 + 19^2 + 23^2 + 29^2 + 31^2 + 37^2 = 4519.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A133529, A133538, A133558, A133559, A133560, A133561.

seq(add(ithprime(n+k)^2,k=0..6),n=1..35); # Muniru A Asiru, Jul 08 2018

b = {}; a = 2; Do[k = Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a + Prime[n + 5]^a + Prime[n + 6]^a; AppendTo[b, k], {n, 1, 100}]; b
Total/@Partition[Prime[Range[40]]^2,7,1] (* Harvey P. Dale, Jan 01 2025 *)

Edited by Michel Marcus, Jul 08 2018

A242218 Semiprimes which are the arithmetic mean of three consecutive primes.

Original entry on oeis.org

511, 537, 1073, 1461, 1501, 1541, 1763, 2071, 2181, 2449, 4101, 4387, 4399, 4467, 4559, 4607, 4681, 4705, 5089, 5257, 5429, 6415, 6621, 6671, 7097, 7111, 7261, 7391, 7447, 7811, 7831, 7897, 7909, 7969, 8079, 8129, 8193, 8333, 8639, 8915, 9101, 9113, 9123, 9211

Offset: 1

K. D. Bajpai, May 07 2014

nonn

			a(1) = 511 = (503 + 509 + 521)/3 = 7 * 73 is semiprime.
a(2) = 537 = (523 + 541 + 547)/3 = 3 * 179 is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001358, A133529, A216432, A234297.

with(numtheory): A242218:= proc()local k ; k:=(ithprime(x)+ithprime(x+1)+ithprime(x+2))/3; if k=floor(k) and bigomega(k)=2 then RETURN (k);  fi; end: seq(A242218 (),x=2..2000);

for(k=1, 2000, t=prime(k)+prime(k+1)+prime(k+2); if(t%3==0 && bigomega(t/3)==2, print1(t/3, ", "))) \\ Colin Barker, May 08 2014

A258269 Primes of the form p^3 + q^2 + r, where p, q, r are consecutive primes.

Original entry on oeis.org

59, 5297, 7417, 81769, 152419, 714479, 1237037, 3330907, 25248317, 64648901, 84801217, 90728159, 286628773, 530133671, 554065817, 823543381, 1028270917, 1096980919, 1299792317, 1321357391, 1417523659, 1574410169, 1648622903, 1997248987, 2084078057, 2556384373

Offset: 1

K. D. Bajpai, May 25 2015

nonn

			a(1) = 59 is prime of the form 3^3 + 5^2 + 7.
a(2) = 5297 is prime of the form 17^3 + 19^2 + 23.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A034962, A133529, A133530.

[k: p in PrimesUpTo (3000) | IsPrime(k) where k is (p^3 + NextPrime(p)^2 + NextPrime(NextPrime(p)))];

A258269:= n-> (ithprime(n)^3+ithprime(n+1)^2+ithprime(n+2)): select(isprime, [seq((A258269(n), n=1..5000))]);

Select[Table[p = Prime[n]; q = NextPrime[p]; r = NextPrime[q]; p^3 + q^2 + r, {n, 5000}], PrimeQ]

forprime(p=1, 5000, q=nextprime(p+1); r=nextprime(q+1);  k=(p^3 + q^2 + r); if(isprime(k), print1(k,", ")))

A164130 Sums s of squares of three consecutive primes, such that s-+2 are primes.

Original entry on oeis.org

195, 5739, 18459, 32259, 33939, 60291, 74019, 169491, 187131, 244899, 276819, 388179, 783531, 902139, 3588339, 5041491, 5145819, 5193051, 8687091, 9637491, 10227291, 10910019, 11341491, 11757339, 14834379, 15354651, 16115091

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 10 2009

nonn

			5^2 + 7^2 + 11^2 = 195 is a sum of the squared consecutive primes 5, 7 and 11, and 193 and 197 are primes, so 195 is a member of the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A075893, A133529, A133530, A034962, A164129

q:= 2: r:= 3: R:= NULL: count:= 0:
while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  s:= p^2+q^2+r^2;
  if isprime(s-2) and isprime(s+2) then
    count:= count+1; R:= R,s;
  fi;
od:
R; # Robert Israel, Apr 21 2023

lst={};Do[p=Prime[n]^2+Prime[n+1]^2+Prime[n+2]^2;If[PrimeQ[p-2]&&PrimeQ[p+2], AppendTo[lst,p]],{n,8!}];lst

A133529 INTERSECT A087679. - R. J. Mathar, Aug 27 2009

Comment turned into example by R. J. Mathar, Aug 27 2009

A242209 Semiprimes sp = p^2 + q^2 + r^2 where p, q and r are consecutive primes.

Original entry on oeis.org

38, 339, 579, 1731, 5739, 8499, 32259, 133851, 145779, 163851, 207579, 222531, 235779, 260187, 308019, 323619, 366819, 469731, 550491, 644979, 684699, 743091, 926427, 1003539, 1242939, 1743531, 1808259, 1852107, 1909059, 2075091, 2585571, 4226979, 5358291

Offset: 1

K. D. Bajpai, May 07 2014

nonn

Subsequence of A133529.

All the terms in the sequence, except a(1), are divisible by 3.

			a(1) = 38 = 2^2 + 3^2 + 5^2 = 2*19 is semiprime.
a(2) = 339 = 7^2 + 11^2 + 13^2 = 3*113 is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..5100

Cf. A001358, A133529, A216432.

with(numtheory): A242209:= proc()local k ; k:=(ithprime(x)^2+ithprime(x+1)^2+ithprime(x+2)^2); if bigomega(k)=2 then RETURN (k); fi;end: seq(A242209 (),x=1..500);

Select[Total[#^2]&/@Partition[Prime[Range[300]],3,1],PrimeOmega[#]==2&] (* Harvey P. Dale, Nov 05 2015 *)

for(k=1, 500, sp=prime(k)^2+prime(k+1)^2+prime(k+2)^2; if(bigomega(sp)==2, print1(sp, ", "))) \\ Colin Barker, May 07 2014

A129592 The smallest in a triple of three consecutive primes such that the ceiling of the square root of their sums-of-squares is prime.

Original entry on oeis.org

2, 7, 13, 43, 53, 59, 127, 241, 271, 317, 331, 349, 367, 439, 487, 491, 607, 659, 719, 733, 757, 773, 821, 857, 881, 929, 971, 1087, 1193, 1259, 1289, 1303, 1409, 1427, 1453, 1607, 1663, 1693, 1723, 1747, 1789, 1949, 2053, 2087, 2089, 2131, 2251, 2333, 2393, 2467, 2549, 2633, 2671, 2719

Offset: 1

J. M. Bergot, May 30 2007

nonn

Can three squares with consecutive prime sides prime(i), i=k,...,k+2, be contained/morphed in a larger square also with prime sides just slightly greater than required?

The areas are the squares of the prime sides; the total area is their sum prime(k)^2 + prime(k+1)^2 + prime(k+2)^2, and pulling the square root is the diagonal of the hosting square. The sequence lists the first, prime(k), if this diagonal (rounded up) is a prime number, indicating that a rather tight enclosing square with (again) a prime side length can be found.

			Take 13,17,19 with summed squares 169 + 289 + 361 = 819 = A133529(6). The square root is approximately 28.6 and rounding up to 29 yields a prime, so 13 is a term.

Select[Partition[Prime[Range[400]],3,1],PrimeQ[Ceiling[ Sqrt[ Total[ #^2]]]]&][[All,1]] (* Harvey P. Dale, Feb 05 2019 *)

{A000040(n): ceiling(sqrt(A133529(n))) in A000040}. - R. J. Mathar, Jul 10 2011

Edited and extended by R. J. Mathar, Jul 10 2011

A133557 Numbers k for which the sum of squares of five consecutive primes starting with prime(k) is prime (A133559).

Original entry on oeis.org

2, 3, 9, 10, 11, 16, 18, 25, 26, 28, 31, 33, 36, 42, 43, 46, 47, 54, 56, 58, 63, 68, 76, 87, 91, 93, 99, 101, 105, 106, 114, 127, 131, 145, 153, 159, 183, 186, 196, 201, 206, 229, 230, 232, 233, 238, 239, 241, 244, 245, 246, 248, 253, 256, 257, 264, 265, 266, 268

Offset: 1

Artur Jasinski, Sep 16 2007

nonn

For sums of squares of two consecutive primes, only k=1 yields a prime.
For sums of squares of three consecutive primes A133529, it seems that only k=2 yields a prime (checked for all k < 1000000).
Sums of squares of four (and all even numbers of) consecutive primes are even numbers except at k=1.

			a(1)=2 because prime(2)^2 + prime(3)^2 + prime(4)^2 + prime(5)^2 + prime(6)^2 = 3^2 + 5^2 + 7^2 + 11^2 + 13^2 = 373 is prime.

Cf. A133538, A133559.

b = {}; a = 2; Do[k = Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a; If[PrimeQ[k], AppendTo[b, n]], {n, 1, 100}]; b (* Artur Jasinski *)

Name and example corrected by Jonathan Sondow, Nov 04 2015

cp's OEIS Frontend

A133940 Numbers n such that (prime(n)^2 + prime(n+1)^2 + prime(n+2)^2)/3 is prime (A084951).

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A259772 Primes p such that p^3 + q^2 + r is also prime, where p,q,r are consecutive primes.

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A131686 Sum of squares of five consecutive primes.

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A133562 Numbers which are the sum of the squares of seven consecutive primes.

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A242218 Semiprimes which are the arithmetic mean of three consecutive primes.

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A258269 Primes of the form p^3 + q^2 + r, where p, q, r are consecutive primes.

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A164130 Sums s of squares of three consecutive primes, such that s-+2 are primes.

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