A014091 -id:A014091 - cp's OEIS frontend

A084997 Numbers which can be written as the sum as well as the product of 2 primes, not necessarily the same.

4, 6, 10, 14, 15, 21, 22, 26, 33, 34, 38, 39, 46, 55, 58, 62, 69, 74, 82, 85, 86, 91, 94, 106, 111, 115, 118, 122, 129, 133, 134, 141, 142, 146, 158, 159, 166, 178, 183, 194, 201, 202, 206, 213, 214, 218, 226, 235, 253, 254, 259, 262, 265, 274, 278

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 30 2003

Intersection of A014091 and A001358; A100484 is a subsequence.

			n=14: 11 + 3 = 14 and 2 * 7 = 14, therefore 14 is a term;
n=15: 13 + 2 = 15 and 3 * 5 = 15, therefore 15 is a term.
E.g. 21 = 19 + 2, 19 and 2 are prime and 21 = 7 * 3, 7 and 3 are primes.
Example: 9 = 3*3 and 2+7

Eric Weisstein's World of Mathematics, Goldbach Conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A014091, A100962.

Corrected and extended by Michael Lahm (mpl148(AT)psu.edu), Apr 24 2006

More terms from Joseph A. Agnew (jaa249(AT)psu.edu), Apr 30 2006

A173664 Sums of 2 primes that are not product of 2 primes.

Original entry on oeis.org

5, 7, 8, 12, 13, 16, 18, 19, 20, 24, 28, 30, 31, 32, 36, 40, 42, 43, 44, 45, 48, 50, 52, 54, 56, 60, 61, 63, 64, 66, 68, 70, 72, 73, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 103, 104, 105, 108, 109, 110, 112, 114, 116, 120, 124, 126

Offset: 1

Juri-Stepan Gerasimov, Nov 24 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A157931, A089268.

a:= proc(n) option remember; local k;
      if n=1 then 5
      else for k from a(n-1)+1 do
             if add (i[2], i=ifactors(k)[2])=2 then next fi;
             if irem (k, 2)=0 or isprime (k-2) then break fi
           od; k
      fi
    end:
seq (a(n), n=1..60);  # Alois P. Heinz, Nov 24 2010

Select[Union[Flatten[Table[Prime[i] + Prime[j], {i, 25}, {j, 25}]]], PrimeOmega[#] != 2 &] (* Alonso del Arte, Feb 08 2013 *)

is(n)=if(n%2,isprime(n-2)&&bigomega(n)!=2,n>2&&!isprime(n/2)) \\ above 4 * 10^18, conditional on the Goldbach conjecture Charles R Greathouse IV, Feb 09 2013

A014091 \ A001358. - R. J. Mathar, Nov 24 2010

More terms from Alois P. Heinz, Nov 24 2010

A243624 Numbers that are the sum of 2 different primes, with repetitions.

Original entry on oeis.org

5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 16, 18, 18, 19, 20, 20, 21, 22, 22, 24, 24, 24, 25, 26, 26, 28, 28, 30, 30, 30, 31, 32, 32, 33, 34, 34, 34, 36, 36, 36, 36, 38, 39, 40, 40, 40, 42, 42, 42, 42, 43, 44, 44, 44, 45, 46, 46, 46, 48, 48, 48, 48, 48, 49, 50, 50, 50, 50, 52, 52, 52, 54, 54, 54, 54, 54, 55, 56, 56, 56, 58, 58, 58

Offset: 1

Zak Seidov, Mar 07 2015

nonn

			16=3+13=5+11 hence 16 occurs twice.
24=5+19=7+13=11+13 hence 24 occurs 3 times.
50=p+q with {p,q}={{3,47},{7,43},{13,37},{19,31}}, 4 representations.
48=p+q with {p,q}={{5,43},{7,41},{11,37},{17,31},{19,29}}, 5 representations.

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

Cf. A014091, A038609.

f:= proc(n) local i;
    if n::odd then
      if isprime(n-2) then n else NULL fi
    else
      n$nops(select(t -> isprime(t) and isprime(n-t), [seq(i,i=3..floor((n-1)/2), 2)]))
    fi
end proc:
map(f, [$5..100]); # Robert Israel, Mar 09 2025

Flatten[Table[Table[n,Count[IntegerPartitions[n,{2}],?(AllTrue[#,PrimeQ]&&#[[1]]!=#[[2]]&)]],{n,60}]] (* _Harvey P. Dale, Jun 20 2025 *)

A287961 Numbers that are the sum of two palindromic primes (A002385).

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 18, 22, 103, 104, 106, 108, 112, 133, 134, 136, 138, 142, 153, 154, 156, 158, 162, 183, 184, 186, 188, 192, 193, 194, 196, 198, 202, 232, 252, 262, 282, 292, 302, 312, 315, 316, 318, 320, 322, 324, 332, 342, 355, 356, 358, 360, 362, 364, 372, 375, 376, 378, 380

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

Eric Weisstein's World of Mathematics, Palindromic Prime
Index entries for sequences related to palindromes

Cf. A002113, A002385, A014091, A035137, A260254, A260255, A261906.

nmax = 380; f[x_] := Sum[Boole[PalindromeQ[k] && PrimeQ[k]] x^k, {k, 1, nmax}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

A290135 Numbers that are the sum of two proper prime powers (A246547).

Original entry on oeis.org

8, 12, 13, 16, 17, 18, 20, 24, 25, 29, 31, 32, 33, 34, 35, 36, 40, 41, 43, 48, 50, 52, 53, 54, 57, 58, 59, 64, 65, 68, 72, 73, 74, 76, 80, 81, 85, 89, 90, 91, 96, 97, 98, 106, 108, 113, 125, 128, 129, 130, 132, 133, 134, 136, 137, 141, 144, 145, 146, 148, 150, 152, 153, 155, 157, 160, 162, 170, 173, 174, 177, 178

Offset: 1

Ilya Gutkovskiy, Jul 20 2017

nonn

Is 2213 the largest prime term that can be expressed as the sum of two proper prime powers in more than one way? - Altug Alkan, Jul 22 2017

			13 is in the sequence because 13 = 2^2 + 3^2.

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

Cf. A014091, A070049, A071330, A071331, A225102, A225103, A246547.

N:= 1000: # to get all terms <= N
P:= select(isprime, [$2..floor(sqrt(N))]):
PP:= {seq(seq(p^j, j=2..floor(log[p](N))),p=P)}:
A:= select(`<=`,{seq(seq(PP[i]+PP[j],j=1..i),i=1..nops(PP))},N):
sort(convert(A,list)); # Robert Israel, Jul 21 2017

nmax = 180; f[x_] := Sum[Boole[PrimePowerQ[k] && PrimeOmega[k] > 1] x^k, {k, 1, nmax}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

Exponents in expansion of (Sum_{k>=1} x^A246547(k))^2.

A084991 Palindromes which can be written as a sum of two prime numbers.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 22, 33, 44, 55, 66, 88, 99, 111, 141, 151, 181, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 313, 333, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 525, 565, 595, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 808, 818, 828, 838, 848, 858, 868, 878, 888, 898, 909, 939, 949, 969, 979, 999

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 27 2003

			E.g. 181 is a palindrome and is 179 + 2. 179 and 2 are both prime.

Cf. A014091.

With[{upto=1000},Select[Select[Union[Total/@Tuples[Prime[ Range[ upto]],2]],# == IntegerReverse[#]&],#<=upto&]] (* Harvey P. Dale, Nov 24 2017 *)

More terms from Harvey P. Dale, Nov 24 2017

A200677 Smallest semiprime such that the sum of the two prime factors equals n, or zero if impossible.

Original entry on oeis.org

0, 0, 0, 4, 6, 9, 10, 15, 14, 21, 0, 35, 22, 33, 26, 39, 0, 65, 34, 51, 38, 57, 0, 95, 46, 69, 0, 115, 0, 161, 58, 87, 62, 93, 0, 155, 0, 217, 74, 111, 0, 185, 82, 123, 86, 129, 0, 215, 94, 141, 0, 235, 0, 329, 106, 159, 0, 265, 0, 371, 118, 177, 122, 183, 0

Offset: 1

Michel Lagneau, Nov 20 2011

nonn

For n > 3, a(n) = 0 if n-2 is an odd composite.
The sequence without zeros is a subsequence of A189553. - Manfred Scheucher, Aug 08 2015
The two prime factors are not necessarily distinct; a(6) = 9, both of whose prime factors are 3s. - Jon E. Schoenfield, Aug 09 2015

			a(10) = 21 because 21 = 3*7 and 3+7 = 10, and there is no semiprime smaller than 21 whose two prime factors sum to 10.

Cf. A001358, A014091, A014092, A135093.

with(numtheory):for n from 1 to 65 do:ii:=0:for k from 1 to 1000 while(ii=0)do:m1:=bigomega(k):x:=factorset(k): m2:=nops(x):if m1=2 and m2=2 and x[1]+x[2]= n or m1=2 and m2=1 and 2*x[1]= n then ii:=1: printf(`%d, `,k):else fi:od:if ii=0 then printf(`%d, `,0):else fi:od:

a(A014091(n)) > 0; a(A014092(n)) = 0. - Michel Marcus, Aug 10 2015

Edited by Jon E. Schoenfield and Manfred Scheucher, Aug 09 2015

A284646 Variation on Leyland numbers: k = x'^y + y'^x, where x' and y' are the arithmetic derivative of x and y.

Original entry on oeis.org

2, 17, 26, 37, 50, 65, 82, 101, 126, 145, 170, 197, 217, 226, 257, 325, 344, 362, 401, 442, 485, 512, 513, 577, 626, 677, 730, 785, 901, 962, 1001, 1025, 1090, 1157, 1297, 1445, 1522, 1601, 1682, 1729, 1765, 1850, 1937, 2026, 2117, 2198, 2305, 2402, 2501, 2602

Offset: 1

Paolo P. Lava, Mar 31 2017

nonn

Another similar variation on Leyland numbers is k = x^y' + y^x' that leads to A014091.

			2' = 1, 4' = 4, 1^4 + 4^2 = 1 + 16 = 17.

Cf. A014091, A076980.

with(numtheory): N:= 10^5: A:={}: for x from 2 to floor(N^(1/2)) do
for y from 2 do yd:=y*add(op(2,p)/op(1,p),p=ifactors(y)[2]); xd:=x*add(op(2,p)/op(1,p),p=ifactors(x)[2]); a:= xd^y + yd^x;
if a>N then break fi; A:=A union {a}; od; od; sort([op(A)]);
# based on Robert Israel code in A076980.

A308040 Numbers k such that k - prevprime(k-1) is prime where prevprime(n) is the largest prime < n.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 58, 60, 61, 62, 63, 64, 66, 68, 69, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 94, 96, 99, 100

Offset: 1

Wesley Ivan Hurt, May 10 2019

nonn

Contains all odd numbers k >= 5 such that k - 2 is prime.

If Goldbach's conjecture is true, the sequence contains an even number k iff there exists a Goldbach partition of k that includes the largest prime < k - 1. This sequence agrees with A014091 (numbers that are the sum of two primes) up to k = 96, but does not include 98 since the largest prime strictly below 97 is 89, which is paired with 9 (and thus, not a Goldbach partition).

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A014091.

Select[Range[4, 100], PrimeQ[# - NextPrime[# - 1, -1]] &]

A330210 Numbers that can be expressed as the sum of 2 prime numbers in a prime number of different ways.

Original entry on oeis.org

10, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 38, 40, 44, 48, 52, 54, 56, 62, 64, 68, 70, 74, 76, 78, 82, 86, 94, 96, 98, 104, 112, 124, 128, 130, 136, 140, 144, 148, 156, 158, 164, 168, 174, 176, 178, 186, 188, 192, 194, 198, 206, 208, 210, 216, 218, 222, 224

Offset: 1

Pietro Saia, Dec 05 2019

nonn

			24 can be expressed as the sum of 2 prime numbers in 3 different ways (5+19, 7+17, and 11+13), and 3 is prime.

Cf. A000040, A014091, A061358.

Select[Range[2, 224, 2], PrimeQ@ Length@ IntegerPartitions[#, {2}, Prime@ Range@ PrimePi@ #] &] (* Giovanni Resta, Dec 06 2019 *)

import math
from sympy import isprime
def main(n):
    x = {}
    a = 1
    b = 1
    for i in range(2, n):
        x[i] = []
        while a < i:
            if a + b == i:
                x[i].append(str(a) + "+" + str(b))
            b += 1
            if b == i:
                a += 1
                b = 1
        a = 1
        b = 1
    for i in x:
        x[i] = x[i][0:math.ceil(len(x[i])/2)]
    x[2] = ["1+1"]
    newdict = {}
    for i in x:
        newdict[i] = []
        for j in x[i]:
            if isprime(int(j.split("+")[0])) and isprime(int(j.split("+")[1])):
                newdict[i].append(j)
    finaloutput = []
    for i in newdict:
        if isprime(len(newdict[i])):
            finaloutput.append(i)
    return finaloutput
def a(n):
    x = 0
    while len(main(x)) != n:
        x += 1
    return main(x)[-1]

cp's OEIS Frontend

A084997 Numbers which can be written as the sum as well as the product of 2 primes, not necessarily the same.

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A173664 Sums of 2 primes that are not product of 2 primes.

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Maple

Mathematica

PARI

Formula

Extensions

A243624 Numbers that are the sum of 2 different primes, with repetitions.

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Maple

Mathematica

A287961 Numbers that are the sum of two palindromic primes (A002385).

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Keywords

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Mathematica

A290135 Numbers that are the sum of two proper prime powers (A246547).

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Maple

Mathematica

Formula

A084991 Palindromes which can be written as a sum of two prime numbers.

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Mathematica

Extensions

A200677 Smallest semiprime such that the sum of the two prime factors equals n, or zero if impossible.

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Comments

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Programs

Maple

Formula

Extensions

A284646 Variation on Leyland numbers: k = x'^y + y'^x, where x' and y' are the arithmetic derivative of x and y.

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