A133559 -id:A133559 - cp's OEIS frontend

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

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

A133560 Primes which have a partition as the sum of squares of seven consecutive primes.

Original entry on oeis.org

1543, 3271, 4519, 7591, 9439, 11719, 23599, 39631, 45319, 51031, 56599, 90199, 151471, 173359, 210319, 222919, 235159, 261463, 313879, 367711, 402511, 459223, 478831, 499711, 610567, 634327, 732967, 760519, 819319, 883087, 939439, 968959

Offset: 1

Artur Jasinski, Sep 16 2007

nonn

For sums of squares of two consecutive primes, only 2^2 + 3^2 = 13 is prime.
For sums of squares of three consecutive primes (see A133529), it seems that only 3^2 + 5^2 + 7^2 = 83 is prime.
Sums of squares of four (and all even numbers of) consecutive primes are even numbers except for 2^2 + 3^2 + 5^2 + 7^2 = 87 = 3*29, which is not prime.
For sums of squares of five of consecutive primes see A133559.
For every prime p > 3, p^2 mod 3 = 1, so the sum of the squares of any 3 such primes will be divisible by 3. - Jon E. Schoenfield, Sep 04 2023

			a(3)=4519 because 13^2 + 17^2 + 19^2 + 23^2 + 29^2 + 31^2 + 37^2 = 4519 is prime.

Muniru A Asiru, Table of n, a(n) for n = 1..3000

Cf. A133529, A133538, A133559, A133561.

select(isprime,[seq(add(ithprime(n+k)^2,k=0..6),n=1..80)]); # Muniru A Asiru, Jul 19 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; If[PrimeQ[k], AppendTo[b, k]], {n, 1, 100}]; b
(* Second program: *)
Select[Map[Total, Partition[Prime@ Range@ 80, 7, 1]^2], PrimeQ] (* Michael De Vlieger, Jul 20 2018 *)

A133561 Numbers n for which sum of squares of seven consecutive primes(n,n+1,n+2,n+3,n+4,n+5,n+6) is prime.

Original entry on oeis.org

3, 5, 6, 8, 9, 10, 14, 18, 19, 20, 21, 26, 32, 34, 37, 38, 39, 41, 44, 47, 49, 52, 53, 54, 59, 60, 63, 64, 66, 68, 70, 71, 75, 83, 88, 89, 91, 92, 97, 100, 107, 108, 110, 112, 113, 117, 122, 125, 128, 129, 131, 135, 141, 142, 150, 151, 157, 158, 165, 168, 169, 178, 183

Offset: 1

Artur Jasinski, Sep 16 2007

nonn

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 seems that only 83 belonging (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.
Sums of squares of five of consecutive primes A133559.
Sums of squares of seven of consecutive primes A133562.

			a(3)=6 because prime(6)^2+prime(7)^2+prime(8)^2+prime(9)^2+prime(10)^2+prime(11)^2+prime(12)^2 = 13^2+17^2+19^2+23^2+29^2+31^2+37^2=4519 is prime.

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

Cf. A133538, A133559, A133562.

select(n->isprime(add(ithprime(n+k)^2,k=0..6)),[$1..200]); # Muniru A Asiru, Jul 28 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; If[PrimeQ[k], AppendTo[b, n]], {n, 1, 100}]; b

is(n) = ispseudoprime(sum(i=0, 6, prime(n+i)^2)) \\ Felix Fröhlich, Jul 28 2018

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

A263724 Least prime p = prime(n)^2 + prime(n+1)^2 + prime(n+2)^2 + prime(n+3)^2 + q^2, where q > prime(n+3) is also prime.

Original entry on oeis.org

373, 653, 1997, 1901, 2309, 3389, 4373, 5381, 6701, 8069, 10589, 12269, 18269, 18461, 19541, 22973, 24821, 29021, 32909, 38261, 46589, 45869, 50021, 53549, 56909, 66029, 69389, 77261, 87629, 93581, 102101, 107741, 118901, 128981, 131837, 145517, 152909, 159869, 170021, 188261, 184901, 196661, 214469, 229781, 237821, 252509, 277157, 281429, 291101, 305933, 317693, 333029, 344021, 359981, 370661, 387341, 395069, 418349, 460949

Offset: 2

Jonathan Sondow, Oct 24 2015

nonn

The corresponding prime q is in A263725.
The prime p exists for all n > 1 under Schinzel's Hypothesis H; see Sierpinski (1988), p. 221.
If q = prime(n+4), then p is in A133559 (prime sums of squares of 5 consecutive primes). The converse holds if a(n) != a(m) when n != m (which holds if a(n) < a(n+1), as appears to be true).

			The primes 373 = 3^2 + 5^2 + 7^2 + 11^2 + 13^2, 653 = 5^2 + 7^2 + 11^2 + 13^2 + 17^2, and 1997 = 7^2 + 11^2 + 13^2 + 17^2 + 37^2 lead to a(1) = 373, a(2) = 653, and a(3) = 1997.

W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988; see p. 221.

Wikipedia, Schinzel's Hypothesis H

Cf. A133559, A263725.

Table[k = 4;
While[p = Sum[Prime[n + j]^2, {j, 0, 3}] + Prime[n + k]^2; ! PrimeQ[p],
  k++]; p, {n, 2, 60}]

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

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A133560 Primes which have a partition as the sum of squares of seven consecutive primes.

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A133561 Numbers n for which sum of squares of seven consecutive primes(n,n+1,n+2,n+3,n+4,n+5,n+6) is prime.

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

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A263724 Least prime p = prime(n)^2 + prime(n+1)^2 + prime(n+2)^2 + prime(n+3)^2 + q^2, where q > prime(n+3) is also prime.

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