A115760 -id:A115760 - cp's OEIS frontend

A115782 Primes generated by pairwise averages of terms in A115760.

5, 11, 13, 29, 31, 37, 71, 73, 79, 97, 431, 433, 439, 457, 499, 1061, 1063, 1069, 1087, 1129, 1489, 56501, 56503, 56509, 56527, 56569, 56929, 57559, 166841, 166843, 166849, 166867, 166909, 167269, 167899, 223339, 5020061, 5020063, 5020069

Offset: 1

Zak Seidov, Jan 30 2006

nonn

Also primes arising from A103828. - N. J. A. Sloane, Apr 28 2007

			The pairwise averages of first three terms in A115760, {3,7,19} produce the set of primes {5,11,13}.

Cf. A113832, A113875, A115760.

A175533 Duplicate of A115760.

Original entry on oeis.org

3, 7, 19, 55, 139, 859, 2119, 112999, 333679, 10040119, 15363619, 548687179, 16632374359, 5733638351299, 14360489685499, 433098704482699, 44258681327079259, 5009018648920510999

Offset: 1

Zak Seidov, Jun 12 2010

dead

Or, for i <> j, (a(i) + a(j))/2 are primes.
From a(5)=139 onward, all terms are 19 (mod 20).
Is sequence finite?
Is this a duplicate of A115760? - R. J. Mathar, Jul 22 2010
Yes, the terms are all 3 (mod 4), just as A175533. - Don Reble, Aug 17 2021

Cf. A115760, A175532.

More terms from Don Reble, Aug 17 2021

A103828 Sequence of odd numbers defined recursively by: a(1)=1 and a(n) is the first odd number greater than a(n-1) such that a(n) + a(i) + 1 is prime for 1<=i<=n-1.

Original entry on oeis.org

1, 3, 9, 27, 69, 429, 1059, 56499, 166839, 5020059, 7681809, 274343589, 8316187179, 2866819175649, 7180244842749, 216549352241349, 22129340663539629, 2504509324460255499

Offset: 1

Walter Kehowski, May 29 2006

Is the sequence infinite? Is each prime a(i)+a(j)+1, i<>j, always distinct?
Except for a(1), a(n) == 3 (mod 6). - Robert G. Wilson v, Jun 02 2006.
The Hardy-Littlewood k-tuple conjecture would imply that this sequence is infinite. Note that, for n>2, a(n)+2 and a(n)+4 are both primes, so a proof that this sequence is infinite would also show that there are infinitely many twin primes. - N. J. A. Sloane, Apr 21 2007
From the mod 30 property of A115760 we conclude that a(n) == 9 (mod 15) for n>4. This implies that either a(n) == 9 (mod 30) or == 24 (mod 30), but == 24 (mod 30) is impossible because then == 0 (mod 6). Therefore a(n) == 9 (mod 30) for n>4. - Don Reble, Aug 17 2021

			a(1)=1, a(2)=3, but 5+1+1=7, 5+3+1=9; 7+1+1=9, 7+3+1=11; 9+1+1=11, 9+3+1=13 so a(3)=9.

G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.

Cf. A093483, A115760, A115782 (primes arising from this sequence), A118818, A128933 (a(n)+1), A291163.

EP:=[]: for w to 1 do for n from 1 to 8*10^6 do s:=2*n-1; Q:=map(z->z+s+1, EP); if andmap(isprime,Q) then EP:=[op(EP),s]; print(nops(EP),s); fi od od; EP;

a[1] = 1; a[2] = 3; a[n_] := a[n] = Block[{k = a[n - 1] + 6, t = Table[ a[i], {i, n - 1}] + 1}, While[ First@ Union@ PrimeQ[k + t] == False, k += 6]; k]; Do[ Print[ a[n]], {n, 15}] (* Robert G. Wilson v, Jun 03 2006 *)

a(n) = (A115760(n) - 1)/2.

a(12) from Robert G. Wilson v, Jun 03 2006
a(13) from Walter Kehowski, Jun 03 2006
Definition corrected by Walter Kehowski, Nov 03 2008
a(14)-a(16) from Don Reble added by N. J. A. Sloane, Sep 18 2012
a(17)-a(18) from Don Reble, Aug 17 2021

A113875 Slowest growing sequence of primes having the prime-pairwise-average property: if i

Original entry on oeis.org

3, 7, 19, 139, 859, 8179, 173059, 1026199, 1827139, 15828679, 13187242759, 18732483199, 912492556939, 9130567625119

Offset: 1

T. D. Noe, Jan 26 2006

Assuming the prime k-tuples conjecture, Granville shows (in section 2.4) that this sequence is infinite.

			The pairwise averages of {3,7,19} are the primes {5,11,13}.

Andrew Granville, Prime number patterns, The American Mathematical Monthly, Vol. 115, No. 4 (2008), pp. 279-296; alternative link.

Cf. A113832, A115760, A119751.

s={3, 7}; i=5; Do[While[ !And@@PrimeQ[(s+Prime[i])/2], i++ ]; AppendTo[s, Prime[i]]; i++, {n, 3, 10}]; s

a(n) = 2*A119751(n)+1. - Don Reble, Aug 17 2021

More terms from Don Reble and Giovanni Resta, Feb 15 2006

a(14) from Amiram Eldar, Jun 27 2024

A180565 Numbers starting with 5 such that the sum of any two distinct entries is two times some prime.

Original entry on oeis.org

5, 9, 17, 29, 77, 197, 689, 44537, 159617, 374249, 695957, 4343237, 8439595349, 196119836669, 45036059849537, 108841069412237, 505069584287297

Offset: 1

Michel Lagneau, Jan 21 2011

The numbers starting with 1 are in the set {1, 5, 9} because if another number q = 2k + 1 exists, then k+1, k+3 and k+5 are primes only if k = 2, but q=5 is already in the set.
The numbers starting with 3 are given by A115760.
The numbers starting with 7 are 7, 15, 19, 67, 127, 187, 547, 607, ...

			For the set of the first three entries, 5+9 = 2*7, 5+17 = 2*11, 9+17 = 2*13.

Cf. A115760.

A180565 := proc(n) option remember; if n = 1 then 5; else for a from procname(n-1)+1 do wrks := true ; for prev from 1 to n-1 do if not type((procname(prev)+a)/2,prime) then wrks := false; break; end if; end do: if wrks then return a; end if; end do: end if; end proc: # R. J. Mathar, Jan 24 2011

a(n) = 2*A093483(n)+1 (follows from the definition). - Chris Boyd, Mar 16 2014

a(14)-a(17) from Chris Boyd, Mar 16 2014

A175532 a(1) = 1; a(2) = 9; for n>2, a(n) = smallest number of the form 4k+1 and a(n) > a(n-1) such that the pairwise sums of elements are all semiprimes.

Original entry on oeis.org

1, 9, 13, 25, 133, 193, 18673, 57313, 1032313, 4387273, 3450430573, 11717813053, 21691948933

Offset: 1

Zak Seidov, Jun 12 2010

Or, for i<>j, (a(i)+a(j))/2 are primes.
Starting with a(5)=133 on all terms === 13 mod 20.
Is this sequence finite?

Cf. A115760.

a(11)-a(13) from Jason Yuen, Oct 13 2024

A114845 Slowest growing sequence of semiprimes having the semiprime-pairwise-average property: for any i,j, (a(i)+a(j))/2 is semiprime.

Original entry on oeis.org

4, 14, 38, 134, 254, 13238, 252254, 691958, 952814, 3316238, 30364918838, 210339665174, 575167942574

Offset: 1

Jonathan Vos Post, Feb 20 2006

Semiprime analog of A113875.

			The pairwise average of the semiprimes {4 = 2^2, 14 = 2*7} is {9 = 3^2}.
The pairwise averages of the semiprimes {4, 14, 38} are {9, 21, 26}.
The pairwise averages of the semiprimes {4, 14, 38, 134} are {9, 21, 26, 69, 74, 86}.
The pairwise averages of the semiprimes {4, 14, 38, 134, 254} are {9, 21, 26, 69, 74, 86, 129, 134, 146, 194}.

Cf. A001358, A113832, A113875, A115760, A164979.

a(n) = 2*A164979(n).

More terms from Zak Seidov, Feb 21 2006
Corrected and extended by Zak Seidov, Sep 03 2009
a(11)-a(12) from Amiram Eldar, Jun 27 2024
a(13) from Jinyuan Wang, May 29 2025

A162662 Sequence of alternating increasing odd and increasing even numbers such that the sum of any two terms of opposite parity is a prime number.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 27, 70, 57, 100, 267, 1060, 1227, 27790, 1479, 146380, 3459, 2508040, 49527, 35506900, 470079

Offset: 1

Michel Lagneau, Jan 27 2011

a(n+1) is taken to be the smallest number, greater than a(n-2), of opposite parity to a(n) that satisfies the condition.

A000034: Period 2: repeat [1, 2] is another sequence satisfying the definition without the increasing constraint. - Michel Marcus, Dec 22 2014

			1060 + 267 = 1327 is prime;
1060 + 27 = 1087 is prime;
1060 + 9 = 1069 is prime;
1060 + 3 = 1063 is prime;
1060 + 1 = 1061 is prime.

Cf. A180743, A115760.

with(numtheory):nn:=30:T:=array(1..nn): T[1]:=1:a:=1:for k from 2 to nn do:id:=0:for
  n from k to 1000000 while(id=0) do:n1:=irem(n,2):i:=0:for p from 1 to a do:
  if n=T[p] then i:=0:else fi: x:=n+T[p]:if type(x, prime)=true then i:=i+1:else
  fi:od: if i=ceil(a/2) then T[k]:=n:print(n):a:=a+1:id:=1:else fi:od:od:

ok(k, m, v) = {if (k % 2, js = 2, js = 1); forstep(j=js, m, 2, if (! isprime(k + v[j]), return (0));); return (1);}
findval(n, v) = {if (n <=2, k = n, k = v[n-2]+2); while (!ok (k, n-1, v), k+= 2); k;}
lista(nn) = {a = vector(nn); a[1] = 1; print1(a[1], ", "); for (n=2, nn, a[n] = findval(n, a); print1(a[n], ", "););} \\ Michel Marcus, Dec 22 2014

a(18)-a(21) from Michel Marcus, Dec 22 2014

Name clarified by Michel Marcus, Dec 22 2014

cp's OEIS Frontend

A115782 Primes generated by pairwise averages of terms in A115760.

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A175533 Duplicate of A115760.

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A103828 Sequence of odd numbers defined recursively by: a(1)=1 and a(n) is the first odd number greater than a(n-1) such that a(n) + a(i) + 1 is prime for 1<=i<=n-1.

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A113875 Slowest growing sequence of primes having the prime-pairwise-average property: if i

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A180565 Numbers starting with 5 such that the sum of any two distinct entries is two times some prime.

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A175532 a(1) = 1; a(2) = 9; for n>2, a(n) = smallest number of the form 4k+1 and a(n) > a(n-1) such that the pairwise sums of elements are all semiprimes.

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A114845 Slowest growing sequence of semiprimes having the semiprime-pairwise-average property: for any i,j, (a(i)+a(j))/2 is semiprime.

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A162662 Sequence of alternating increasing odd and increasing even numbers such that the sum of any two terms of opposite parity is a prime number.

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