A113832 -id:A113832 - cp's OEIS frontend

A115760 Slowest growing sequence of numbers having the prime-pairwise-average property: if i

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

Offset: 1

Zak Seidov, Jan 30 2006

Inspired by A113875 (case of prime numbers). See A113832 minimal sets of primes having the P-P-A property, A115782 primes in A115760.
Equals 2*A103828(n) + 1. - N. J. A. Sloane, Apr 28 2007. This sequence is surely infinite - see comments in A103828.
After a(4), terms are == 19 mod 60. The sequence may also be defined by "a(1)=3 and for n>1, a(n) is the smallest number of the form 4k+3, a(n)>a(n-1) such that the pairwise sums of all elements are semiprimes." - Don Reble, Aug 17 2021

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

Cf. A113832, A113875, A115782, A175532.

a(n) == 19 (mod 60) for n>4 [consequence of mod 30 congruence of A103828(n).] - Don Reble, Aug 17 2021

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

More terms 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

A113833 Triangle read by rows: row n (n>=2) gives a set of n primes such that the averages of all subsets are distinct primes, having the smallest largest element.

Original entry on oeis.org

3, 7, 7, 19, 67, 5, 17, 89, 1277, 209173, 322573, 536773, 1217893, 2484733

Offset: 2

N. J. A. Sloane, Jan 25 2006

If there is more than one set with the same smallest last element, choose the lexicographically earliest solution.

Note that, in each row, the n primes are equal modulo 4, 12, 12 and 120, respectively. - Row 5 from T. D. Noe, Aug 08 2006

			Triangle begins:
3, 7
7, 19, 67
5, 17, 89, 1277

Antal Balog, The prime k-tuplets conjecture on average, in "Analytic Number Theory" (eds. B. C. Berndt et al.) Birkhäuser, Boston, 1990, pp. 165-204. [Background]

Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
Andrew Granville, Prime number patterns

Cf. A113827-A113831, A113832, A113834, A088430.

Row 5 from T. D. Noe, Aug 08 2006

A115631 Triangle read by rows: row n (n>=2) gives a set of n primes such that the pairwise averages are all distinct primes, having the smallest largest element.

Original entry on oeis.org

3, 7, 3, 7, 19, 3, 11, 23, 71, 5, 29, 53, 89, 173, 3, 11, 83, 131, 251, 383, 5, 17, 41, 101, 257, 521, 881, 11, 83, 251, 263, 443, 1103, 1511, 2111, 257, 269, 509, 857, 1697, 2309, 2477, 2609, 5417, 11, 83, 251, 263, 1511, 2351, 2963, 7583, 8663, 10691

Offset: 2

T. D. Noe, Jan 27 2006

This table is the same as A113832 for rows 2,3,4,6,12 and 13. Note that row 7, {5,17,41,101,257,521,881}, is the same as row 8 of A113832 with 761 deleted.

The table on page 6 of Granville is wrong. - Arkadiusz Wesolowski, Mar 11 2013

			The set of primes generated by {5, 29, 53, 89, 173} is {17, 29, 41, 47, 59, 71, 89, 101, 113, 131}.
Triangle begins:
3, 7;
3, 7, 19;
3, 11, 23, 71;
5, 29, 53, 89, 173;
3, 11, 83, 131, 251, 383;
5, 17, 41, 101, 257, 521, 881;
...

Toshitaka Suzuki, Table of n, a(n) for n = 2..91
Andrew Granville, Prime number patterns
Michel Marcus, Triangle read by rows

A115782 Primes generated by pairwise averages of terms in A115760.

Original entry on oeis.org

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.

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

A155463 Largest element of a set of n primes with the property that the pairwise averages are all distinct primes, having the smallest largest element (A115631).

Original entry on oeis.org

7, 19, 71, 173, 383, 881, 2111, 5417, 10691, 21757, 27611

Offset: 2

Dmitry Kamenetsky, Jan 22 2009

nonn

The solution for n=5 is {5, 29, 53, 89, 173}, so a(5)=173. This sequence is not proved to be optimal, so smaller terms may exist.

Andrew Granville, Prime number patterns

Cf. A115631, A113832.

cp's OEIS Frontend

A115760 Slowest growing sequence of numbers having the prime-pairwise-average property: if i

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

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A113833 Triangle read by rows: row n (n>=2) gives a set of n primes such that the averages of all subsets are distinct primes, having the smallest largest element.

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A115631 Triangle read by rows: row n (n>=2) gives a set of n primes such that the pairwise averages are all distinct primes, having the smallest largest element.

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A115782 Primes generated by pairwise averages of terms in A115760.

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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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A155463 Largest element of a set of n primes with the property that the pairwise averages are all distinct primes, having the smallest largest element (A115631).

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