A240708 -id:A240708 - cp's OEIS frontend

A237628 a(n) is the smallest product of prime numbers such that all numbers from 6 and 2n can be written as the sum of two prime factors (duplication allowed) of a(n).

3, 15, 15, 105, 105, 1155, 1155, 1365, 15015, 15015, 15015, 255255, 255255, 596505, 4849845, 4849845, 4849845, 10140585, 10140585, 179444265, 229474245, 229474245, 242777535, 640049865, 5898837945, 7357214865, 7357214865, 7357214865, 13350001665, 196656364905

Offset: 3

Lei Zhou, May 02 2014

The prime factors of a(n) make a subset of prime numbers that satisfies the Goldbach Conjecture for even numbers from 6 to 2n.

			n=4: 2*4=8. 8=3+5.  This is the only possible two-prime decomposition which contains prime numbers 3 and 5, while 6=3+3, 3 is an element of set {3,5}.  So a(4)=3*5=15.
n=5: 2*5=10. 6=3+3, 8=3+5, 10=5+5.  So two selections of prime numbers in set {3,5} (reuse allowed) can be summed into all three numbers 6, 8, and 10.  So a(5)=3*5=15.
...
n=8: 2n=16. We will be able to find two sets, {3,5,7,11} and {3,5,7,13}, that have such feature:
  for set {3,5,7,11}, 6=3+3, 8=3+5, 10=5+5, 12=5+7, 14=7+7, and 16=5+11;
  for set {3,5,7,13}, 6=3+3, 8=3+5, 10=5+5, 12=5+7, 14=7+7, and 16=3+13.
  3*5*7*11=1155, and 3*5*7*13=1365.  1155<1365, so a(8)=1155.  Here we did not count set {3,5,7,11,13} which also has the desired feature since the two shorter sets are its subsets such that the products of the elements in the subsets are obviously smaller than the product of elements in this larger set.

Lei Zhou, Table of n, a(n) for n = 3..79

Cf. A000040, A002375, A240708.

a = {{{3}}}; Table[n2 = 2*n; na = {}; la = Last[a]; lo = Length[la]; Do[ok = 0; Do[p1 = la[[i, j]]; p2 = n2 - p1; If[MemberQ[la[[i]], p2], ok = 1], {j, 1, Length[la[[i]]]}];
  If[ok == 1, na = Sort[Append[na, la[[i]]]], Do[p1 = la[[i, j]]; p2 = n2 - p1; If[PrimeQ[p2], ng = Sort[Append[la[[i]], p2]]; big = 0; If[Length[na] > 0, Do[If[Intersection[na[[k]], ng] == na[[k]], big = 1], {k, 1, Length[na]}]]; If[big == 0, na = Sort[Append[na, ng]]]], {j, 1, Length[la[[i]]]}]], {i, 1, lo}]; AppendTo[a, na]; b = {};
lna = Length[na]; Do[prd = Times @@ na[[k]]; AppendTo[b, prd], {k, 1, lna}]; Min[b], {n, 4, 32}](*Program lists the 4th item and beyond*)

A276034 a(n) is the number of decompositions of 2n into an unordered sum of two primes in A274987.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 3, 2, 1, 2, 2, 2, 1, 2, 1, 0, 2, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 1, 2, 4, 3, 1, 5, 3, 2, 5, 1, 2, 2, 2, 5, 2, 3, 4, 5, 3, 2, 5, 2, 1, 4, 0, 1, 5, 3, 1, 3, 5, 4, 4, 3, 2, 4, 3, 3, 4, 2, 3, 7, 2, 2, 3, 2, 2, 2

Offset: 1

Lei Zhou, Nov 15 2016

The two primes are allowed to be the same.
It is conjectured that the primes in A274987 (a subset of all primes) are sufficient to decomposite even numbers into two primes in A274987 when n > 958.
This sequence provides a very tight alternative of the Goldbach conjecture for all positive integers, in which indices of zero terms form a complete sequence {1, 2, 16, 26, 64, 97, 107, 122, 146, 167, 194, 391, 451, 496, 707, 856, 958}.
There is no more zero terms of a(n) tested up to n = 100000.

			A274987 = {3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 59, 61, 73, 79, 83, 89, 101, 103, 109, ...}.
For n=3, 2n=6 = 3+3, one case of decomposition, so a(3)=1;
for n=4, 2n=8 = 3+5, one case of decomposition, so a(4)=1;
...
for n=17, 2n=34 = 3+31 = 11+23 = 17+17, three cases of decompositions, so a(17)=3.

Lei Zhou, Table of n, a(n) for n = 1..10000
Lei Zhou, List plot of the first 10000 terms of a(n).

Cf. A002375, A045917, A001031, A274987, A171611, A240708, A240712, A230443.

p = 3; sp = {p}; a = Table[m = 2*n; l = Length[sp]; While[sp[[l]] < m, While[p = NextPrime[p]; cp = 2*3^(Floor[Log[3, 2*p - 1]]) - p; ! PrimeQ[cp]]; AppendTo[sp, p]; l++]; ct = 0; Do[If[(2*sp[[i]] <= m) && (MemberQ[sp, m - sp[[i]]]), ct++], {i, 1, l}]; ct, {n, 1, 87}]

A276520 a(n) is the number of decompositions of n into unordered form p + c*q, where p, q are terms of A274987, c=1 for even n-s and c=2 for odd n-s.

Original entry on oeis.org

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

Offset: 1

Lei Zhou, Nov 11 2016

p=q is allowed.
It is conjectured that the primes p, q in A274987 (a subset of all primes) are sufficient to decomposite all numbers into p and c*q (c=1 when n is even, 2 when c is odd) when n > 2551.
This sequence provides a very tight alternative of the Goldbach conjecture for all positive integers, in which indices of zero terms form a complete sequence {1, 2, 3, 4, 5, 7, 32, 52, 55, 61, 128, 194, 214, 244, 292, 334, 388, 782, 902, 992, 1414, 1571, 1712, 1916, 2551}.
There are no more zero terms of a(n) up to n = 100000.

			A274987 = {3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 59, 61, 73, 79, 83, 89, 101, 103, 109, ...}
For n=6, 6 = 3+3, one case of decomposition, so a(6)=1;
For n=7, 7 < 3+2*3=9, no eligible case could be found, so a(7)=0;
...
For n=17, 17 = 3+2*7 = 7+2*5 = 11+2*3, three cases of decompositions, so a(17)=3.

Lei Zhou, Table of n, a(n) for n = 1..10000
Lei Zhou, List plot of the first 10000 terms of a(n).

Cf. A002375, A045917, A001031, A274987, A171611, A240708, A240712, A230443, A276034, A103151, A001031.

p = 3; sp = {p}; Table[l = Length[sp]; While[sp[[l]] < n, While[p = NextPrime[p]; cp = 2*3^(Floor[Log[3, 2*p - 1]]) - p; ! PrimeQ[cp]]; AppendTo[sp, p]; l++]; c = 2 - Mod[n + 1, 2]; ct = 0; Do[If[MemberQ[sp, n - c*sp[[i]]], If[c == 1, If[(2*sp[[i]]) <= n, ct++], ct++]], {i, 1, l}]; ct, {n, 1, 87}]

A237638 a(n) is the number of prime sets such that each set contains enough prime numbers to decompose every even number from 6 to 2n into the sum of two of its elements (reuse allowed), while none of the sets is a subset of another such set.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 9, 11, 11, 11, 13, 16, 23, 25, 31, 47, 57, 63, 70, 74, 79, 82, 122, 131, 129, 180, 215, 219, 323, 367, 446, 501, 531, 661, 867, 897, 1311, 1471, 1691, 1695, 2130, 2288, 2833, 3363, 3891, 5435, 8068, 8867, 13476, 15451, 15897

Offset: 3

Lei Zhou, May 02 2014

			n=4, 2n=8. There is only one set of primes {3,5} such that 6=3+3, 8=3+5. So a(4)=1.
...
n=8, 2n=16. We can find two sets, {3,5,7,11} and {3,5,7,13} that have such features. So a(8)=2. Here any set with more primes either contains an unused prime number or one of these two sets is a subset of them, like {3,5,7,11,13}, and thus is not considered. So a(8)=2.
...
n=13, 2n=26. Five such sets are found: {3,5,7,11,13}, {3,5,7,13,17},{3,5,7,13,19}, {3,5,7,11,17,19}, {3,5,7,11,17,23}. So a(13)=5.

Lei Zhou, Table of n, a(n) for n = 3..79

Cf. A000040, A002375, A240708, A237628.

a = {{{3}}}; Table[n2 = 2*n; na = {}; la = Last[a]; lo = Length[la]; Do[ok = 0; Do[p1 = la[[i, j]]; p2 = n2 - p1; If[MemberQ[la[[i]], p2], ok = 1], {j, 1, Length[la[[i]]]}];
  If[ok == 1, na = Sort[Append[na, la[[i]]]], Do[p1 = la[[i, j]]; p2 = n2 - p1; If[PrimeQ[p2], ng = Sort[Append[la[[i]], p2]]; big = 0; If[Length[na] > 0, Do[If[Intersection[na[[k]], ng] == na[[k]], big = 1], {k, 1, Length[na]}]]; If[big == 0, na = Sort[Append[na, ng]]]], {j, 1, Length[la[[i]]]}]], {i, 1, lo}]; AppendTo[a, na]; Length[na], {n, 4, 60}](* Program lists the 4th item and beyond *)

A242189 a(n) is the smallest prime number such that every even number from 6 to 2n can be written as the sum of two primes less than or equal to a(n).

Original entry on oeis.org

3, 5, 5, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, 19, 19, 19, 23, 23, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 47, 47, 47, 47, 47, 47, 47, 47, 61, 61, 61, 61, 61, 61, 61, 61, 61, 67, 67, 67, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 83

Offset: 3

Lei Zhou, May 06 2014

nonn

The two primes stated in the name can be equal.

			n=3, 2*3=6=3+3. Since 3 is the smallest prime needed, a(3)=3.
n=4, 2*3=6=3+3, 2*4=8=5+3, Since 5 is the smallest prime needed, a(4)=5.
...
n=14, we need to consider the even numbers from 6 to 2*14=28, while trying to minimize the larger prime number used to decompose such even numbers. 6=3+3; 8=5+3; 10=5+5; 12=7+5; 14=7+7; 16=11+5; 18=11+7; 20=13+7; 22=11+11; 24=13+11; 26=13+13; 28=17+11. The maximum prime number used is 17. So a(14)=17.

Lei Zhou, Table of n, a(n) for n = 3..5881

Cf. A000040, A002375, A234345, A237628, A237638, A240708.

f:= proc(m) local p,p0;
   p0:= m/2; if p0::even then p0:= p0+1 fi;
   for p from p0 by 2 do if isprime(p) and isprime(m-p) then return p fi od
end proc:
R:= 3: m:= 3:
for i from 8 to 200 by 2 do
  v:= f(i);
  if v > m then R:= R,v; m:= v
  else R:= R,m
  fi
od:
R; # Robert Israel, Oct 10 2024

a = {2}; Table[found = 0; While[la = Length[a]; xx = 1; Do[yy = 0; Do[If[MemberQ[a, i*2 - a[[j]]], yy = 1], {j, 1, la}]; If[yy == 0, xx = 0], {i, 3, n}]; If[xx == 1, found = 1]; found == 0, AppendTo[a, NextPrime[Last[a]]]]; Last[a], {n, 3, 68}]

a(n) = max_{3 <= i <= n} A234345(i). - Robert Israel, Oct 10 2024

Name corrected by Robert Israel, Oct 10 2024

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A237628 a(n) is the smallest product of prime numbers such that all numbers from 6 and 2n can be written as the sum of two prime factors (duplication allowed) of a(n).

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A276034 a(n) is the number of decompositions of 2n into an unordered sum of two primes in A274987.

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A276520 a(n) is the number of decompositions of n into unordered form p + c*q, where p, q are terms of A274987, c=1 for even n-s and c=2 for odd n-s.

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A237638 a(n) is the number of prime sets such that each set contains enough prime numbers to decompose every even number from 6 to 2n into the sum of two of its elements (reuse allowed), while none of the sets is a subset of another such set.

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A242189 a(n) is the smallest prime number such that every even number from 6 to 2n can be written as the sum of two primes less than or equal to a(n).

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