A118072 -id:A118072 - cp's OEIS frontend

A072669 Primes of the form prime(x) + prime(x+1) - 1.

7, 11, 17, 23, 29, 41, 59, 67, 83, 89, 127, 137, 151, 197, 239, 257, 307, 359, 383, 389, 409, 433, 449, 461, 479, 491, 547, 557, 563, 599, 617, 647, 683, 701, 739, 751, 761, 797, 809, 827, 839, 863, 881, 929, 977, 1063, 1087, 1103, 1163, 1229, 1249, 1283, 1289, 1319, 1373

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net), Aug 12 2002

Consider m such that prime(m) + prime(m+1) = prime(k) + 1 for some k; sequence gives prime(k).

A118072 is a subsequence, hence this sequence is infinite on Dickson's conjecture. - Charles R Greathouse IV, Apr 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001043, A092738, A072666, A072667.

f[n_] := Prime[n] + Prime[n + 1] - 1; f[ # ] & /@ Select[ Range[120], PrimeQ[ f[ # ]] &] (* Robert G. Wilson v, Apr 14 2004 *)
Select[Total[#]-1&/@Partition[Prime[Range[200]],2,1],PrimeQ] (* Harvey P. Dale, Aug 06 2012 *)

p=2;forprime(q=3,1e6,if(isprime(p+q-1),print1(p+q-1", "));p=q) \\ Charles R Greathouse IV, Apr 18 2013

Definition reworded by Jorge Coveiro, Apr 12 2004

Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar

A256386 Numbers m such that m-2, m-1, m+1, m+2 cannot all be represented in the form x*y + x + y for values x, y with x >= y > 1.

Original entry on oeis.org

2, 3, 4, 5, 8, 11, 59, 1319, 1619, 4259, 5099, 6659, 6779, 11699, 12539, 21059, 66359, 83219, 88259, 107099, 110879, 114659, 127679, 130199, 140759, 141959, 144539, 148199, 149519, 157559, 161339, 163859, 175079, 186479, 204599, 230939, 249539, 267959, 273899, 312839

Offset: 1

Alex Ratushnyak, Mar 31 2015

nonn

Indices of terms surrounded by pairs of zeros in A255361.
Conjectures:
1. A255361(a(n)) > 0 for n > 4.
2. All terms > 8 are primes.
3. All terms > 8 are terms of these supersequences: A118072, A171667, A176821, A181602, A181669.
From Lamine Ngom, Feb 12 2022: (Start)
For n > 4, a(n) is not a term of A254636. This means that a(n)-2, a(n)-1, a(n)+1 and a(n)+2 are adjacent terms in A254636.
Number of terms < 10^k: 5, 7, 7, 13, 19, 96, 441, 2552, ...
Conjecture 2 would follow if we establish the equivalence "t is in sequence" <=> "t is a term of b(n): lesser of twin primes pair p and q such that (p - 1)/2 and (q + 1)/2 are also a pair of twin primes (A077800)".
It appears that b(n) = a(n) for n > 5. Verified for all terms < 10^9. (End)

			9, 10, 12, 13 cannot be represented as x*y + x + y, where x >= y > 1. Therefore 11 is in the sequence.

Cf. A255361, A254636.
Cf. A118072, A171667, A176821, A181602, A181669.
Cf. A001097, A001359, A006512, A077800, A158870.

a(n) = A158870(n-5) - 2, n > 5 (conjectured). - Lamine Ngom, Feb 12 2022

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]

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A072669 Primes of the form prime(x) + prime(x+1) - 1.

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A256386 Numbers m such that m-2, m-1, m+1, m+2 cannot all be represented in the form x*y + x + y for values x, y with x >= y > 1.

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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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