A173634 -id:A173634 - cp's OEIS frontend

A204814 Number of decompositions of 2n into an unordered sum of two Ramanujan primes.

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 1, 0, 3, 0, 0, 1, 1, 0, 2, 0, 0, 1, 2, 0, 2, 2, 0, 4, 0, 0, 1, 2, 0, 2, 0, 1, 1, 3, 0, 2, 2, 0, 2, 0, 0, 1, 2, 0, 2, 1, 1, 2, 4, 0, 1, 2

Offset: 1

Donovan Johnson, Jan 27 2012

nonn

Suggested by John W. Nicholson.
There are 95 zeros in the first 10000 terms. Are there more? Related to Goldbach's conjecture. - T. D. Noe, Jan 27 2012
There are no other zeros in the first 10^8 terms. a(n) > 0 for n from 1313 to 10^8. - Donovan Johnson, Jan 27 2012

			a(29) = 3. 2*29 = 58 = 11+47 = 17+41 = 29+29 (11, 17, 29, 41 and 47 are all Ramanujan primes). 58 is the unordered sum of two Ramanujan primes in three ways.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A045917, A104272, A173634.

A205301 Last occurrence of n partitions in A204814.

Original entry on oeis.org

1312, 1501, 2446, 2776, 4381, 3676, 4951, 6541, 5686, 7771, 8326, 7696, 8011, 10471, 9256, 9871, 11041, 11626, 15256, 12751, 15511, 15151, 14956, 19441, 16801, 20596, 20101, 22291, 17611, 17341, 18451, 21856, 19051, 21226, 23761, 20842, 24796, 22651, 22546

Offset: 0

John W. Nicholson, Jan 27 2012

nonn

Conjectured lower bound to be increasing for increasing n.

Related to Goldbach's conjecture.

			1312 is the last observed 0 so for n=0, a(0)=1312.

Donovan Johnson, Table of n, a(n) for n = 0..1000 (from search done for first 25*10^6 terms in A204814)

Cf. A104272, A173634, A204814.

a(10)-a(38) from Donovan Johnson, Jan 28 2012

A205617 Number of decompositions of 2n into an unordered sum of two non-Ramanujan primes (A174635).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 3, 0, 1, 3, 1, 2, 2, 1, 3, 2, 0, 1, 3, 2, 1, 3, 1, 3, 4, 1, 2, 4, 1, 4, 2, 0, 3, 2, 3, 2, 3, 2, 3, 5, 1, 3, 4, 0, 5, 1, 0, 4, 3, 3, 1, 4, 3, 5, 4, 0, 4, 3, 1, 4, 2, 2, 6, 2, 3, 4, 4, 1, 3

Offset: 1

John W. Nicholson and Donovan Johnson, Jan 30 2012

nonn

There are 15 zeros in the first 10^8 terms. a(n) > 0 for n from 315 to 10^8.

			a(25) = 3. 2*25 = 50 = 7+43 = 13+37 = 19+31 (7, 13, 19, 31, 37, and 43 are all non-Ramanujan primes (A174635)). 50 is the unordered sum of two non-Ramanujan primes in three ways.

Donovan Johnson, Table of n, a(n) for n = 1..10000
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009), 630-635.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Cf. A104272, A174635, A173634, A204814, A205301, A205616, A205618.

A205618 Last occurrence of n partitions in A205617.

Original entry on oeis.org

314, 629, 959, 1154, 1424, 4619, 1922, 4094, 2549, 3884, 3989, 5774, 4724, 5669, 6404, 5879, 7664, 5594, 8609, 9239, 9029, 8714, 10562, 10394, 9869, 11549, 9764, 12239, 11444, 11969, 11654, 14279, 14489, 12209, 13229, 15014, 13859, 14804, 15584, 16979, 19634

Offset: 0

John W. Nicholson and Donovan Johnson, Jan 30 2012

nonn

			a(0) = 314 because the last occurrence of a zero in A205617 is at a(314).

Donovan Johnson, Table of n, a(n) for n = 0..1000
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009), 630-635.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Cf. A104272, A174635, A173634, A204814, A205301, A205616, A205617.

A205616 Even numbers that are not the sum of two non-Ramanujan primes (A174635).

Original entry on oeis.org

2, 4, 52, 70, 100, 124, 130, 148, 208, 232, 238, 292, 352, 418, 628

Offset: 1

John W. Nicholson and Donovan Johnson, Jan 30 2012

nonn

No other terms < 2*10^8. Conjectured to be complete.

			70 is a term because no two non-Ramanujan primes (A174635) sum to 70. 72 is not a term because 19 + 53 = 72. 19 and 53 are both non-Ramanujan primes.

J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009), 630-635.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Cf. A104272, A174635, A173634, A204814, A205301, A205617, A205618.

cp's OEIS Frontend

A204814 Number of decompositions of 2n into an unordered sum of two Ramanujan primes.

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A205301 Last occurrence of n partitions in A204814.

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A205617 Number of decompositions of 2n into an unordered sum of two non-Ramanujan primes (A174635).

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A205618 Last occurrence of n partitions in A205617.

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A205616 Even numbers that are not the sum of two non-Ramanujan primes (A174635).

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