A136244 -id:A136244 - cp's OEIS frontend

A023036 Smallest positive even integer that is an unordered sum of two primes in exactly n ways.

2, 4, 10, 22, 34, 48, 60, 78, 84, 90, 114, 144, 120, 168, 180, 234, 246, 288, 240, 210, 324, 300, 360, 474, 330, 528, 576, 390, 462, 480, 420, 570, 510, 672, 792, 756, 876, 714, 798, 690, 1038, 630, 1008, 930, 780, 960, 870, 924, 900, 1134, 1434, 840, 990, 1302, 1080

Offset: 0

David W. Wilson, Jun 14 1998

Except for first two terms, same as A001172.
The first occurrence of k in A045917.
The graph looks like a comet. - Daniel Forgues, Jun 12 2014

			a(3) = 22 as 22 = (19+3) = (17+5) = (11+11). There are exactly 3 ways 22 can be expressed as the sum of two primes and no even number less than 22 can be so expressed.
From _Daniel Forgues_, Jun 13 2014: (Start)
Terms for n = 1..6 and corresponding sums:
  a(1) =  4 =  2 + 2;
  a(2) = 10 =  7 + 3 =  5 +  5;
  a(3) = 22 = 19 + 3 = 17 +  5 = 11 + 11;
  a(4) = 34 = 31 + 3 = 29 +  5 = 23 + 11 = 17 + 17;
  a(5) = 48 = 43 + 5 = 41 +  7 = 37 + 11 = 31 + 17 = 29 + 19;
  a(6) = 60 = 53 + 7 = 47 + 13 = 43 + 17 = 41 + 19 = 37 + 23 = 31 + 29.
(End)

Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Index entries for sequences related to Goldbach conjecture

Cf. A045917, A000954, A136244, A258713.

f[n_] := Length@ Select[2n - Prime@ Range@ PrimePi@ n, PrimeQ]; nn = 100; t = Table[0, {nn}]; k = 1; cnt = 0; While[cnt < nn, a = f@k; If[a <= nn && t[[a]] == 0, t[[a]] = 2 k; cnt++]; k++]; t (* Robert G. Wilson v, Mar 15 2011 *)

A258713 A001172(n)/2: Least k such that 2k is a sum of two odd primes in exactly n ways.

Original entry on oeis.org

0, 3, 5, 11, 17, 24, 30, 39, 42, 45, 57, 72, 60, 84, 90, 117, 123, 144, 120, 105, 162, 150, 180, 237, 165, 264, 288, 195, 231, 240, 210, 285, 255, 336, 396, 378, 438, 357, 399, 345, 519, 315, 504, 465, 390, 480, 435, 462, 450, 567, 717, 420, 495, 651

Offset: 0

N. J. A. Sloane, Jun 14 2015

nonn

Up to a(14) also indices of records in A002375, number of ways to write 2n as sum of two odd primes. - M. F. Hasler, Aug 21 2017

Robert Israel, Table of n, a(n) for n = 0..845

Cf. A136244, A001172.

Cf. A002375, A002372, A045917.

g:= add(x^ithprime(i),i=2..1000):
G:= series((g^2+add(x^(2*ithprime(i)),i=2..1000))/2,x,ithprime(1001)+3):
A[0]:= 0:
for k from 1 to (ithprime(1001)+1)/2 do
  m:= coeff(G,x,2*k);
  if not assigned(A[m]) then A[m]:= k fi;
od:
for m from 1 while assigned(A[m]) do od:
seq(A[i],i=0..m-1); # Robert Israel, Aug 21 2017

With[{s = Array[Count[Select[IntegerPartitions[2 #, 2], Length@ # == 2 &], p_ /; AllTrue[p, And[PrimeQ@ #, OddQ@ #] &]] &, 10^3]}, Table[FirstPosition[s, n][[1]] /. 1 -> 0, {n, 0, 53}]] (* Michael De Vlieger, Aug 21 2017 *)

Edited by M. F. Hasler, Aug 21 2017

Edited by Robert Israel, Aug 21 2017

A332981 Smallest semiprime m = p*q such that the sum s = p + q can be expressed as an unordered sum of two primes in exactly n ways.

Original entry on oeis.org

4, 21, 57, 93, 183, 291, 327, 395, 501, 545, 695, 791, 815, 831, 1145, 1205, 1415, 1631, 1461, 1745, 1941, 1865, 2661, 2315, 2615, 2855, 2495, 2285, 3665, 2705, 2721, 3521, 3561, 3351, 3755, 4341, 3545, 4701, 4265, 4881, 3981, 4821, 5601, 5255, 6671, 6041, 4595

Offset: 1

Michel Lagneau, Mar 05 2020

nonn

The unique square and even term of the sequence is a(1) = 4.
For n = 1, the sequence of semiprimes having a unique decomposition as the sum of two primes begins with 4, 6, 9, 10, 14, 15, 22, 26, 34, 35, 38, 46, 58, 62, ... containing the even semiprimes (A100484).
We observe a majority of terms where a(n) == 5 (mod 10).

			a(11) = 695 because 695 = 5*139 and the sum 5 + 139 = 144 = 5+139 = 7+137 = 13+131 = 17+127 = 31+113 = 37+107 = 41+103 = 43+101 = 47+97 = 61+83 = 71+73. There are exactly 11 decompositions of 144 into an unordered sum of two primes.

Michel Marcus, Table of n, a(n) for n = 1..501

Cf. A001358, A002375, A006881, A023036, A100484, A136244.

with(numtheory):
for n from 1 to 50 do:
ii:=0:
for k from 2 to 10^8 while(ii=0) do:
x:=factorset(k):it:=0:
if bigomega(k) = 2
  then
   s:=x[1]+k/x[1]:
    for m from 1 to s/2 do:
     if isprime(m) and isprime(s-m)
      then
       it:=it+1:
       else fi:
     od:
     if it = n
     then
      ii:=1: printf(`%d, `,k):
     else fi:
     fi:
    od:
    od:

nbp(k) = {my(nb = 0); forprime(p=2, k\2, if (isprime(k-p), nb++););nb;}
a(n) = {forcomposite(k=1, oo, if (bigomega(k)==2, my(x=factor(k)[1,1]); if (nbp(x+k/x)==n, return(k));););} \\ Michel Marcus, Apr 26 2020

A354462 a(n) is the least number k such that there are exactly n pairs (p,q) of primes with p

Original entry on oeis.org

1, 4, 15, 315, 420, 825, 2310, 3150, 1785, 8925, 6090, 6405, 8610, 24990, 19305, 12705, 14175, 15015, 18165, 19635, 24255, 48510, 63525, 33915, 48195, 54285, 35490, 50505, 55650, 69615, 71610, 80850, 78540, 103740, 39270, 157920, 60060, 65835, 90090, 147840, 120120, 183645

Offset: 0

J. M. Bergot and Robert Israel, May 31 2022

nonn

a(n) is the least solution to A354449(k) = n.

			a(2) = 15 because for k = 15 there are two such pairs, (7,23) and (13,17): 2*15+7 = 37, 2*15+23 = 53, 7*23-2*15 = 131, 7*23+2*15 = 191, 2*15+13 = 43, 2*15+17 = 47, 13*17-2*15 = 191 and 13*17+2*15 = 251 are all prime; and 15 is the least k that works.

Cf. A045917, A136244, A354449.

f:= proc(n) local count,p,q;
  p:= 2*n-1 ; count:= 0;
  do
    p:= prevprime(p);
    if p < n then return count fi;
    q:= 2*n-p;
    if isprime(q) and isprime(2*n+q) and isprime(2*n+p) and isprime(p*q-2*n) and isprime(p*q+2*n) then count:=count+1 fi;
  od
end proc:
f(1):= 0: f(2):= 0:
V:= Array(0..12): count:= 0:
for n from 1 while count < 13 do
  v:= f(n);
  if v <= 12 and V[v] = 0 then
  count:= count+1; V[v]:= n
fi
od:
convert(V,list);

f[n_] := Sum[If[AllTrue[{k, 2*n - k, 2*n + k, 4*n - k, k*(2 n - k) - 2*n, k*(2 n - k) + 2*n}, PrimeQ], 1, 0], {k, 1, n}]; seq[len_, max_] := Module[{s = Table[0, {len}], n = 1, c = 0, i}, While[c < len && n <= max, i = f[n] + 1; If[i<= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[13, 10^4] (* Amiram Eldar, May 31 2022 *)

a354449(n) = my(x=2*n, i=0); forprime(q=1, x, forprime(p=1, q-1, if(p+q==x && ispseudoprime(x+p) && ispseudoprime(x+q) && ispseudoprime(p*q-x) && ispseudoprime(p*q+x), i++))); i
a(n) = for(k=1, oo, if(a354449(k)==n, return(k))) \\ Felix Fröhlich, May 31 2022

upto(n) = {n*=2; v = vector(n\2); forprime(p = 3, n, forprime(q = 3, min(p, n-p), k2 = p+q; if(ispseudoprime(k2+p) && ispseudoprime(k2+q) && ispseudoprime(p*q-k2) && ispseudoprime(p*q+k2), v[k2\2]++ ) ) ); res = [0]; for(i = 1, #v, if(v[i]+1 > #res, res = concat(res, vector(v[i]+1-#res)) ); if(res[v[i]+1] == 0, res[v[i]+1] = i ) ); res } \\ David A. Corneth, Jun 01 2022

a(13)-a(32) from Amiram Eldar, May 31 2022

More terms from David A. Corneth, Jun 01 2022

cp's OEIS Frontend

A023036 Smallest positive even integer that is an unordered sum of two primes in exactly n ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A258713 A001172(n)/2: Least k such that 2k is a sum of two odd primes in exactly n ways.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A332981 Smallest semiprime m = p*q such that the sum s = p + q can be expressed as an unordered sum of two primes in exactly n ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A354462 a(n) is the least number k such that there are exactly n pairs (p,q) of primes with p

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Extensions