A194830 -id:A194830 - cp's OEIS frontend

A046927 Number of ways to express 2n+1 as p+2q where p and q are primes.

0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 3, 3, 3, 4, 4, 2, 5, 3, 4, 4, 5, 4, 6, 4, 4, 7, 5, 3, 7, 3, 3, 7, 7, 5, 7, 4, 4, 8, 7, 5, 8, 4, 7, 8, 7, 4, 11, 5, 6, 9, 6, 5, 12, 6, 6, 10, 8, 6, 11, 7, 5, 11, 8, 6, 10, 6, 6, 13, 8, 5, 13, 6, 9, 12, 8, 6, 14, 8, 6, 11, 10, 9, 16, 5, 8, 13, 9, 9, 14, 7, 6, 14

Offset: 0

David W. Wilson

nonn

This is related to a conjecture of Lemoine (also sometimes called Levy's conjecture, although Levy was anticipated by Lemoine 69 years earlier). - Zhi-Wei Sun, Jun 10 2008
The conjecture states that any odd number greater than 5 can be written as p+2q where p and q are primes.
It can be conjectured that 1, 3, 5, 59 and 151 are the only odd integers n such that n + 2p and n + 2q both are composite for all primes p,q with n = p + 2q. (Following an observation from V. Shevelev, cf. link to SeqFan list.) - M. F. Hasler, Apr 10 2017

L. E. Dickson, "History of the Theory of Numbers", Vol. I (Amer. Math. Soc., Chelsea Publ., 1999); see p. 424.

T. D. Noe, Table of n, a(n) for n = 0..10000
L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
E. Lemoine, L'intermédiaire des math., 1 (1894), p. 179; 3 (1896), p. 151.
H. Levy, On Goldbach's Conjecture, Math. Gaz. 47 (1963), 274.
Vladimir Shevelev, Binary additive problems: recursions for numbers of representations, arXiv:0901.3102 [math.NT], 2009-2013.
V. Shevelev, Re: New sequence, SeqFan list, April 2017.
Eric Weisstein's World of Mathematics, Levy's Conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A194831 (records), A194830 (positions of records).

a[n_] := (ways = 0; Do[p = 2k + 1; q = n-k; If[PrimeQ[p] && PrimeQ[q], ways++], {k, 1, n}]; ways); Table[a[n], {n, 0, 91}] (* Jean-François Alcover, Dec 05 2012 *)
Table[Count[FrobeniusSolve[{1, 2}, 2 n + 1], {?PrimeQ}], {n, 0, 91}] (* Jan Mangaldan, Apr 08 2013 *)

a(n)=my(s);n=2*n+1;forprime(p=2,n\2,s+=isprime(n-2*p));s \\ Charles R Greathouse IV, Jul 17 2013

For n >= 1, a(n) = Sum_{3<=p<=n+1, p prime} A((2*n + 1 - p)/2) + Sum_{2<=q<=(n+1)/2, q prime} B(2*n + 1 - 2*q) - A((n+1)/2)*B(n+1) - a(n-1) - ... - a(0), where A(n) = A000720(n), B(n) = A033270(n). - Vladimir Shevelev, Jul 12 2013

Additional references from Zhi-Wei Sun, Jun 10 2008

A194831 Records in the number of ways to express an odd number as a sum 2*p+q, with p, q primes.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 16, 21, 22, 26, 27, 31, 32, 35, 43, 48, 52, 65, 77, 87, 101, 104, 115, 128, 133, 146, 155, 169, 180, 188, 194, 196, 201, 209, 225, 228, 248, 250, 282, 286, 325, 332, 359, 391, 400, 443, 449, 470, 555, 579, 582, 679, 741

Offset: 1

Hugo Pfoertner, Sep 12 2011

nonn

Records in A046927. The growth rate of this sequence makes the slow growth of A194829 plausible, i.e. 2*n+1 can be represented by 2*p+q with q<

			a(1)=1: A194830(1)=7 has 1 representation 7=2*2+3; a(2)=2 representations of A194830(2)=9=2*2+5=2*3+3; a(3)=4 representations of A194830(3)=17=2*2+13=2*3+11=2*5+7=2*7+3.

See A046927.

Hugo Pfoertner, Table of n, a(n) for n = 1..322

Cf. A194830 [record-setting numbers], A046927, A194828, A194829.

ways[n_] := ways[n] = (w = 0; Do[ p = 2k + 1; q = n - k; If[PrimeQ[p] && PrimeQ[q], w++], {k, 1, n}]; w); record = 0; A194831 = Reap[Do[If[ways[n] > record, record = ways[n]; Print["2n+1 = ", 2n + 1, " record = ", record]; Sow[{ways[n], n}]], {n, 0, 12000}]][[2, 1]][[All, 1]] (* Jean-François Alcover, Dec 05 2012 *)

A269329 Number of partitions of a positive integer n into two distinct primes such that for even n, it is of the form n = p + q and for odd n, it is of the form n = 2p' + q'.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 4, 2, 2, 2, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 2, 2, 4, 3, 3, 4, 4, 1, 3, 3, 5, 4, 4, 3, 6, 3, 4, 5, 4, 4, 6, 3, 5, 5, 3, 3, 6, 3, 3, 6, 3, 2, 7, 5, 7, 6, 5, 2, 6, 5, 4, 6, 4, 4, 8, 5, 7, 7, 5, 4, 8, 4, 4, 8, 7, 4, 7, 4, 7, 9, 4, 4, 10, 4, 5, 7, 6, 3, 9

Offset: 1

Frank M Jackson and M. B. Rees, Feb 23 2016

nonn

This sequence combines Levy's conjecture for odd positive numbers with the Goldbach conjecture for even positive numbers and strenghtens both by restricting the prime pairs to be distinct. I.e., every positive integer n > 6 is the sum of two distinct primes p and q such that for n even, it is of the form n = p + q and for n odd, it is of the form n = 2p' + q'.

			a(23)=3. Hence there are 3 partitions (as defined above) of the odd integer 23, namely 19+2+2, 17+3+3 and 13+5+5. a(24)=3. Hence there are 3 partitions of the even integer 24, namely 19+5, 17+7 and 13+11.

Cf. A002375, A061357, A194830, A223893, A225522.

parts[n_, a_, b_] := Select[IntegerPartitions[n, {a+b}, Prime@Range[PrimePi[n]]], Length[Union@#]==2&&MemberQ[Values@Counts@#, a] &]; lst1=Table[Length@parts[2n-1, 1, 2], {n, 1, 200}]; lst2=Table[Length@parts[2n, 1, 1], {n, 1, 200}]; Riffle[lst1, lst2]

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A046927 Number of ways to express 2n+1 as p+2q where p and q are primes.

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A194831 Records in the number of ways to express an odd number as a sum 2*p+q, with p, q primes.

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A269329 Number of partitions of a positive integer n into two distinct primes such that for even n, it is of the form n = p + q and for odd n, it is of the form n = 2p' + q'.

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