A023036 -id:A023036 - cp's OEIS frontend

A352231 Numbers that can be expressed as the sum of two primes in exactly 9 ways.

90, 132, 170, 196, 202, 220, 230, 236, 238, 244, 250, 254, 262, 268, 302, 314, 338, 346, 356, 388, 428, 458, 488

Offset: 1

Wesley Ivan Hurt, Mar 08 2022

			90 = 7+83 = 11+79 = 17+73 = 19+71 = 23+67 = 29+61 = 31+59 = 37+53 = 43+47.

Numbers that can be expressed as the sum of two primes in k ways for k=0..10: A014092 (k=0), A067187 (k=1), A067188 (k=2), A067189 (k=3), A067190 (k=4), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), this sequence (k=9), A352233 (k=10).

Cf. A000954, A023036, A061358.

c[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; Select[Range[1000], c[#] == 9 &] (* _Amiram Eldar, Mar 08 2022 *)

A061358(a(n)) = 9. - Alois P. Heinz, Mar 08 2022

A136244 Least positive integer k such that 2k can be expressed as the sum of two primes in exactly n ways.

Original entry on oeis.org

1, 2, 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, 540, 615, 759, 525, 570, 693, 645

Offset: 0

K. B. Subramaniam (shunya_1950(AT)yahoo.co.in), Dec 24 2007

nonn

It appears that 2, 3, 4, 6 are the only numbers k such that 2k can be expressed as the sum of two primes in only one way.

Except when n = 1, a(n) = A258713(n). The first 11 terms of this sequence are the same as the initial terms of A053033. If a(n) exists for all n then A053033 is a subsequence. - Andrew Howroyd, Jan 28 2020

			a(3) = 11: 22 = 3 + 19 = 5 + 17 = 11 + 11. Also 22 is the least number which could be expressed as the sum of two prime numbers in exactly three ways.

David A. Corneth, Table of n, a(n) for n = 0..16805 (first 1001 terms from Andrew Howroyd)
Index entries for sequences related to Goldbach conjecture

Cf. A023036, A045917, A053033, A126204, A258713.

a(n, lim=oo)={for(i=1, lim, my(s=0); forprime(p=2, i, s+=isprime(2*i-p)); if(s==n, return(i))); -1} \\ Andrew Howroyd, Jan 28 2020

From Andrew Howroyd, Jan 28 2020: (Start)
a(n) = A023036(n) / 2.
A045917(a(n)) = n. (End)

a(0)=1 prepended, a(5) corrected and a(7) and beyond from Andrew Howroyd, Jan 28 2020

A056636 Conjecturally largest even integer which is the sum of two primes in at most n ways.

Original entry on oeis.org

2, 12, 68, 128, 152, 188, 332, 398, 398, 488, 632, 692, 692, 992, 992, 992, 1112, 1112, 1412, 1412, 1448, 1718, 1718, 1718, 1718, 2048, 2252, 2252, 2672, 2672, 2672, 2936, 2936, 2936, 2978, 3092, 3092, 3218, 3272, 3296, 3632, 3632, 3754, 4022, 4058, 4412

Offset: 0

Brian Galebach, Aug 17 2000

The Goldbach conjecture is that every even number >= 4 is the sum of two primes.

			a(1) is 12 because it is the largest even integer having only 1 distinct way to express it as the sum of two primes (7+5) and a(0) < 12.
a(8) = 398 because it is the largest number in A000954 for n <= 8.

Index entries for sequences related to Goldbach conjecture

Cf. A045917, A023036, A000954.

a(n) = max({A000954(i),i=0..n}). - Robert Israel, Mar 21 2016

A109679 Smallest even number which is the unordered sum of two primes in more ways than any previous even number.

Original entry on oeis.org

2, 4, 10, 22, 34, 48, 60, 78, 84, 90, 114, 120, 168, 180, 210, 300, 330, 390, 420, 510, 630, 780, 840, 990, 1050, 1140, 1260, 1470, 1650, 1680, 1890, 2100, 2310, 2730, 3150, 3570, 3990, 4200, 4410, 4620, 5250, 5460, 6090, 6510, 6930, 7980, 8190, 9030, 9240

Offset: 1

Gilmar Rodriguez Pierluissi (gilmarlily(AT)yahoo.com), Aug 30 2005

nonn

Record value of A023036 or A045917.

a(n)== 0 (mod 30) for n > 13.

Vincenzo Librandi, Table of n, a(n) for n = 1..150
Eric Weisstein's World of Mathematics, Goldbach Partition
Index entries for sequences related to Goldbach conjecture

Essentially the same as A082917. Cf. A082918, A002375, A023036, A045917, A000954.

f[n_] := Length[ Select[n - Prime@ Range@ PrimePi[n/2], PrimeQ]]; t = {}; mxm = -1; Do[ If[ f[n] > mxm, AppendTo[t, n]; mxm = f[n]], {n, 2, 9000, 2}]; t

Edited and extended by Robert G. Wilson v, Sep 08 2005

Changed offset from 0 to 1 by Vincenzo Librandi, Apr 18 2013

A281875 Least k such that phi(k) is the sum of two primes in exactly n ways, or 0 if no such k exists.

Original entry on oeis.org

1, 5, 11, 23, 37, 65, 61, 79, 103, 161, 127, 157, 143, 199, 181, 307, 277, 313, 241, 211, 409, 341, 379, 487, 331, 623, 577, 551, 463, 527, 421, 571, 601, 673, 829, 757, 877, 997, 1571, 691, 1039, 631, 1009, 961, 869, 967, 1543, 989, 1057, 1247, 2411, 899, 991, 1303, 1147, 1231, 1999

Offset: 0

Altug Alkan, Feb 01 2017

Note that this sequence is not the subsequence of A037143; i.e., a(318) = 7^3 * 47.

			a(3) = 23 because phi(23) = 22 = 3 + 19 = 5 + 17 = 11 + 11 and 23 is the least number with this property.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A000010, A001172, A023036, A281816.

a(0) = 1 prepended by Chai Wah Wu, Feb 03 2017

A352296 Smallest number that can be expressed as the sum of two primes in exactly n ways or -1 if no such number exists.

Original entry on oeis.org

1, 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

Offset: 0

Chai Wah Wu, Mar 11 2022

nonn

Conjecture: a(n) != -1 for all n.
If n > 0 and a(n) != -1, then a(n) is even.
a(0) = A014092(1)
a(1) = A067187(1)
a(2) = A067188(1)
a(3) = A067189(1)
a(4) = A067190(1)
a(5) = A067191(1)
a(6) = A066722(1)
a(7) = A352229(1)
a(8) = A352230(1)
a(9) = A352231(1)
a(10) = A352233(1)

Cf. A014092, A067187, A067188, A067189, A067190, A067191, A066722, A352229, A352230, A352231, A352233.

Essentially the same as A023036 and A001172.

f[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; seq[max] :=  Module[{s = Table[0, {max}], n = 1, c = 0, k}, While[c < max, k = f[n]; If[k < max && s[[k + 1]] == 0, c++; s[[k + 1]] = n]; n++]; s]; seq[50] (* Amiram Eldar, Mar 11 2022 *)

from itertools import count
from sympy import nextprime
def A352296(n):
    if n == 0:
        return 1
    pset, plist, pmax = {2}, [2], 4
    for m in count(2):
        if m > pmax:
            plist.append(nextprime(plist[-1]))
            pset.add(plist[-1])
            pmax = plist[-1]+2
        c = 0
        for p in plist:
            if 2*p > m:
                break
            if m - p in pset:
                c += 1
        if c == n:
            return m

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

A356442 a(n) is the least positive even number that is the unordered sum of two primes congruent mod 10 in exactly n ways.

Original entry on oeis.org

2, 4, 26, 86, 126, 174, 264, 324, 396, 456, 546, 594, 624, 876, 966, 984, 924, 954, 1326, 1344, 1386, 1512, 1596, 1638, 1848, 1764, 2046, 2226, 2838, 2574, 2706, 2604, 2772, 2436, 3366, 3066, 2964, 3432, 3894, 3738, 3234, 3696, 3654, 4074, 4446, 4158, 4368, 4494, 4788, 5016, 4746, 5754, 4914

Offset: 0

J. M. Bergot and Robert Israel, Aug 07 2022

a(n) is the least even number k such that there are exactly n unordered pairs of primes (p,q) with p + q = k and p and q have the same last decimal digit.

			a(3) = 86 because 86 = 3 + 83 = 13 + 73 = 43 + 43, all summands being prime with last digit 3, and 86 is the least even number that works.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A023036.

f:= proc(m) local d, p;
  if m mod 10 = 0 then return 0 fi;
  d:= chrem([m/2 mod 5, 1],[5,2]);
  nops(select(p -> isprime(p) and isprime(m-p), [seq(p,p=d..m/2,10)]))
end proc:
f(4):= 1:
M:= 100: # to get a(0)..a(M)
V:= Array(0..M): count:= 0:
for m from 2 by 2 while count < M+1 do
  v:= f(m);
  if v <= M and V[v] = 0 then V[v]:= m; count:= count+1 fi
od:
convert(V,list);

A357816 a(n) is the first even number k such that there are exactly n pairs (p,q) where p and q are prime, p<=q, p+q = k, and p+A001414(k) and q+A001414(k) are also prime.

Original entry on oeis.org

2, 16, 60, 72, 220, 132, 374, 276, 492, 638, 636, 852, 620, 854, 996, 1056, 1026, 1212, 2070, 1530, 2610, 3976, 3844, 1488, 1572, 4812, 4770, 3942, 2484, 5028, 3234, 4668, 6036, 3276, 5172, 5532, 6756, 2730, 6084, 4230, 6390, 9132, 14134, 4620, 9674, 10692, 6600, 8910, 10836, 12204, 18852, 9660

Offset: 0

J. M. Bergot and Robert Israel, Oct 13 2022

nonn

			a(3) = 72 because A001414(72) = 12 and there are 3 pairs: (5,67), (11,61) and (31,41) where 5+67 = 11+61 = 31+41 = 72 and 5, 5+12 = 17, 67, 67+12 = 79, 11, 11+12 = 23, 61, 61+12 = 73, 31, 31+12 = 43, 41, and 41+12 = 53 are all prime; and this is the first even number with 3 such pairs.

Robert Israel, Table of n, a(n) for n = 0..500

Cf. A001414, A023036.

sp:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
f:= proc(n) local s,p,q,count;
    s:= sp(n);
    if s::odd then return 0 fi;
    p:= 2; count:= 0;
    do
      p:= nextprime(p);
      q:= n-p;
      if p > q then return count fi;
      if isprime(p+s) and isprime(q) and isprime(q+s) then count:= count+1 fi;
    od;
end proc:
V:= Array(0..60): count:= 0:
for n from 2 by 2 while count < 61 do
  v:= f(n);
  if v <= 60 and V[v] = 0 then V[v]:= n; count:= count+1;  fi
od:
convert(V,list);

a[n_] := Block[{k=2, s}, While[True, s = Plus @@ Times @@@ FactorInteger@ k; If[n == Length@ Select[ Prime@ Range@ PrimePi[k/2], And @@ PrimeQ@ {k-#, #+s, k-#+s} &], Break[]]; k += 2]; k]; a /@ Range[0, 20] (* Giovanni Resta, Oct 24 2022 *)

A376287 Index of first occurrence of n in A129363, or 0 if no such number exists.

Original entry on oeis.org

2, 6, 10, 22, 48, 120, 114, 298, 240, 540, 288, 1620, 210, 300, 702, 840, 660, 2312, 1290, 4284, 1332, 2580, 2070, 2100, 1890, 5100, 2340, 5580, 3720, 6660, 3612, 6240, 2310, 10288, 3540, 4680, 4788, 5460, 4410, 5940, 6120, 10200, 4200, 4620, 3570, 10560, 5700, 16588, 5250

Offset: 0

Robert G. Wilson v, Sep 19 2024

nonn

Twin prime analogous to A023036.
Conjecture: a(n) > 0. Checked to 1010.
Conjectured last occurrence: 4208, 24536, 28916, 21278, 51806, 68078, 73538, 89216, 83978, ..., .
Conjecture number of terms for A129363(k) = n: 35, 115, 285, 327, 557, 537, 723, 652, 882, ..., .
A129363(n) = 0: A007534.

Cf. A001097, A023036, A129363.

tp = Select[Prime@Range@ 16340, PrimeQ[# -2] || PrimeQ[# +2] &]; f[n_] := Length@ IntegerPartitions[n, {2, 2}, tp]; t[_] := 0; k = 2; While[k < 10201, a = f@k; If[ t[a] == 0, t[a] = k]; k += 2]; t /@ Range[0, 75]

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A352231 Numbers that can be expressed as the sum of two primes in exactly 9 ways.

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A136244 Least positive integer k such that 2k can be expressed as the sum of two primes in exactly n ways.

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A056636 Conjecturally largest even integer which is the sum of two primes in at most n ways.

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A109679 Smallest even number which is the unordered sum of two primes in more ways than any previous even number.

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A281875 Least k such that phi(k) is the sum of two primes in exactly n ways, or 0 if no such k exists.

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A352296 Smallest number that can be expressed as the sum of two primes in exactly n ways or -1 if no such number exists.

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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.

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A356442 a(n) is the least positive even number that is the unordered sum of two primes congruent mod 10 in exactly n ways.

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