A067187 -id:A067187 - 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

A352233 Numbers that can be expressed as the sum of two primes in exactly 10 ways.

Original entry on oeis.org

114, 126, 162, 260, 290, 304, 316, 328, 344, 352, 358, 374, 382, 416, 542, 632

Offset: 1

Wesley Ivan Hurt, Mar 08 2022

All terms are even. Conjecture: 632 is the last term. Hardy and Littlewood conjectured a growth rate of the number of decompositions for large even numbers (see Conjecture A in page 32 of Hardy and Littlewood reference), implying this sequence is finite. - Chai Wah Wu, Mar 10 2022

			114 = 5+109 = 7+107 = 11+103 = 13+101 = 17+97 = 31+83 = 41+73 = 43+71 = 47+67 = 53+61.

G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Mathematica, volume 44, pages 1-70 (1923).

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), A352231 (k=9), this sequence (k=10).

c[n_] := Count[IntegerPartitions[n, {2}], ?(And @@ PrimeQ[#] &)]; Select[Range[1000], c[#] == 10 &] (* _Amiram Eldar, Mar 08 2022 *)

A048974 Odd numbers that are the sum of 2 primes.

Original entry on oeis.org

5, 7, 9, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259

Offset: 1

N. J. A. Sloane

A048974, A052147, A067187 and A088685 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003

Robert Price, Table of n, a(n) for n = 1..10000

Cf. A014092, A065091.

Select[Flatten@Table[Prime[i] + Prime[j], {i, 100}, {j, 1, i}], # < Prime[100] && OddQ[#] &] (* Robert Price, Apr 21 2025 *)

One of the primes must be 2, so this is simply the odd primes + 2.

a(n) = A065091(n) + 2. - Sean A. Irvine, Jul 15 2021

A088685 Records for the sum-of-primes function sopfr(n) if sopfr(prime) is taken to be 0.

Original entry on oeis.org

0, 4, 5, 6, 7, 9, 10, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259, 265, 271

Offset: 1

Eric W. Weisstein, Oct 05 2003

nonn

A048974, A052147 and A067187 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003

Robert Price, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sum of Prime Factors

Cf. A001414, A048974, A052147, A067187, A088686.

Union@ FoldList[Max, Table[Total@ Flatten@ Map[ConstantArray[#1, #2] /. 1 -> 0 & @@ # &, FactorInteger@ n] - n Boole[PrimeQ@ n], {n, 540}]] (* Michael De Vlieger, Jun 29 2017 *)

sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
lista(nn) = {my(record = -1); for (n=1, nn, if (! isprime(n), if ((x=sopfr(n)) > record, record = x; print1(record, ", "));););} \\ Michel Marcus, Jun 29 2017

from sympy import factorint, isprime
def sopfr(n):
    f=factorint(n)
    return sum([i*f[i] for i in f])
l=[]
record=-1
for n in range(1, 501):
    if not isprime(n):
        x=sopfr(n)
        if x>record:
            record=x
            l.append(record)
print(l) # Indranil Ghosh, Jun 29 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

A370090 Numbers that can be expressed in exactly one way as the unordered sum of two distinct primes.

Original entry on oeis.org

5, 7, 8, 9, 10, 12, 13, 14, 15, 19, 21, 25, 31, 33, 38, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259, 265

Offset: 1

Wesley Ivan Hurt, Feb 11 2024

nonn

Apparently, a number that is the predecessor or successor of a prime number does not have a sum as defined here, except for a finite number of primes, which may be {7, 11, 13, 37}. - Peter Luschny, Feb 16 2024

			5 = 2+3; 7 = 2+5; 8 = 3+5; 9 = 2+7; 10 = 3+7 (10 = 5+5 is not considered).

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A117929, A048974, A065091, A067187 (not necessarily distinct).

If we change 1 way (this sequence) we get A077914 (2 ways), A077969 (3 ways), A078299 (4 ways), A080854 (5 ways), and A080862 (6 ways).

select(n -> A117929(n) = 1, [seq(1..265)]);  # Peter Luschny, Feb 16 2024

tdpQ[{a_,b_}]:=AllTrue[{a,b},PrimeQ]&&a!=b; Select[Range[300],Count[IntegerPartitions[#,{2}],?tdpQ]==1&] (* _Harvey P. Dale, Dec 30 2024 *)

from sympy import sieve
from collections import Counter
from itertools import combinations
def aupton(max):
    sieve.extend(max)
    a = Counter(c[0]+c[1] for c in combinations(sieve._list, 2))
    return [n for n in range(1, max+1) if a[n] == 1]
print(aupton(265)) # Michael S. Branicky, Feb 16 2024

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

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A352233 Numbers that can be expressed as the sum of two primes in exactly 10 ways.

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A048974 Odd numbers that are the sum of 2 primes.

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A088685 Records for the sum-of-primes function sopfr(n) if sopfr(prime) is taken to be 0.

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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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A370090 Numbers that can be expressed in exactly one way as the unordered sum of two distinct primes.

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