A063533 -id:A063533 - cp's OEIS frontend

A184417 p^2 + (p+2)^2 - 1 where (p,p+2) is the n-th twin prime pair.

33, 73, 289, 649, 1801, 3529, 7201, 10369, 20809, 23329, 38089, 45001, 64801, 73729, 78409, 103969, 115201, 145801, 159049, 194689, 242209, 352801, 373249, 426889, 544969, 649801, 720001, 763849, 824329, 871201, 1312201, 1351369, 1371169, 1472329, 1555849, 2080801, 2130049, 2205001, 2255689, 2384929, 2654209

Offset: 1

Robert Mohr, Feb 13 2011

nonn

This seems to have a disproportionately high probability of generating a prime number.

			a(1) = prime(1)^2 + (prime(1)+2)^2 - 1 =  3^2 +  (3+2)^2 - 1 =  33;
a(2) = prime(2)^2 + (prime(2)+2)^2 - 1 =  5^2 +  (5+2)^2 - 1 =  73;
a(3) = prime(3)^2 + (prime(3)+2)^2 - 1 = 11^2 + (11+2)^2 - 1 = 289.

Total/@(Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&]^2)-1  (* Harvey P. Dale, Feb 24 2011 *)

a(n) = A063533(n) - 1.

A224505 Primes p such that p+1 is the sum of the squares of a pair of twin primes.

Original entry on oeis.org

73, 1801, 3529, 10369, 20809, 103969, 115201, 426889, 649801, 2080801, 2205001, 2654209, 3266569, 3328201, 4428289, 5171329, 10017289, 10672201, 11347849, 14709889, 21780001, 22177801, 28395649, 29675809, 30701449, 32320801, 35583049, 40176649, 41368609

Offset: 1

Bruno Berselli, Apr 08 2013

nonn

Primes in A184417.

Obviously, no prime has the form q^2+(q+2)^2+1, where q and q+2 are twin primes.

			3529 (prime) is in the sequence because 3529+1 = 41^2+43^2, where 41 and 43 are twin primes.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A063533 (sums of the squares of a pair of twin primes), A118072 (primes which are sum of a pair of twin primes minus 1), A184417.

[p: r in PrimesUpTo(5000) | IsPrime(r+2) and IsPrime(p) where p is 2*r^2+4*r+3];

A224505:=proc(q) local a,n;
for n from 1 to q do
  if ithprime(n+1)-ithprime(n)=2 then a:=ithprime(n+1)^2+ithprime(n)^2-1;
  if isprime(a) then print(a); fi; fi;
od; end: A224505(10^6); # Paolo P. Lava, Apr 17 2013

Select[(#[[1]]^2 + #[[2]]^2 - 1) & /@ Select[Partition[Prime[Range[700]], 2, 1], #[[2]] - #[[1]] == 2 &], PrimeQ]

A131348 Sum of squares of prime quadruplets.

Original entry on oeis.org

364, 940, 44140, 152140, 2722540, 8820940, 14062540, 17388940, 42380140, 48024940, 127916140, 356076940, 676520140, 979064140, 990360940, 1032336940, 1302488140, 1431108940, 1509322540, 1766520940, 1984702540, 2561372140

Offset: 1

Jonathan Vos Post, Sep 29 2007

This is to prime quadruplets A007530 as sums of squares of twin primes A063533 are to twin primes. This is to prime quadruplets A007530 as A133524 is to four consecutive primes. Note that prime quadruplets are not the same as four consecutive primes. After a(1) these are always multiples of 20, because after A007530(1) = 5, all A007530(n) == 1 mod 10. a(n) is a prime times 20 for an = 1, 2, 3, 12, 16, 21.

			a(1) = 364 = 5^2 + 7^2 + 11^2 + 13^2.
a(2) = 940 = 11^2 + 13^2 + 17^2 + 19^2.
a(3) = 44140 = 101^2 + (103)^2 + (107)^2 + (109)^2 because 101, 103, 107, 109 are a prime quadruplet.

Cf. A000040, A007530, A063533, A133523, A133524.

Total[#^2]&/@Select[Partition[Prime[Range[3000]],4,1],MatchQ[#,{#[[1]],#[[1]]+2,#[[1]]+6,#[[1]]+8}]&]  (* Harvey P. Dale, Feb 03 2011 *)

a(n) = p^2 + (p+2)^2 + (p+6)^2 + (p+8)^2 for p in A007530.

Corrected and extended by Harvey P. Dale, Feb 03 2011

cp's OEIS Frontend

A184417 p^2 + (p+2)^2 - 1 where (p,p+2) is the n-th twin prime pair.

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A224505 Primes p such that p+1 is the sum of the squares of a pair of twin primes.

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A131348 Sum of squares of prime quadruplets.

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