A157931 -id:A157931 - cp's OEIS frontend

A014091 Numbers that are the sum of 2 primes.

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, 98

Offset: 1

N. J. A. Sloane

nonn

Sequence consists of all primes + 2 and, conjecturally (Goldbach), of all even integers larger than 2. The Goldbach conjecture is that every even number is the sum of two primes. - Emeric Deutsch, Jul 14 2004

T. D. Noe, Table of n, a(n) for n = 1..1000
David Eisenbud and Brady Haran, Goldbach Conjecture, Numberphile video (2017)
T. Estermann, Proof that every large integer is the sum of two primes and a square, Proc. Lond. Math. Soc. 42 (1937) 501-516.

Complement = A014092.

Cf. A010051, A000040, A157931 (semiprimes).

a014091 n = a014091_list !! (n-1)
a014091_list = filter (\x -> any ((== 1) . a010051) $
                      map (x -) $ takeWhile (< x) a000040_list) [1..]
-- Reinhard Zumkeller, Oct 15 2014

sort({seq(2+ithprime(j),j=1..21)} union {seq(2*k,k=2..ceil(ithprime(21)/2))}); # Emeric Deutsch, Jul 14 2004

Take[ Union@ Flatten@ Table[ Prime@p + Prime@q, {p, 25}, {q, p}], 71] (* Robert G. Wilson v, Aug 31 2008 *)

isA014091(n)= my(i,p); i=1; p=prime(i); while(pA014091(a), print(n," ",a); n++)) \\ R. J. Mathar, Aug 20 2006

is(n)=if(n%2,isprime(n-2),n>2) \\ on Goldbach's conjecture; Charles R Greathouse IV, Oct 22 2013

More terms from Robert G. Wilson v, Aug 31 2008

A158318 Primes p such that 5p-2 is prime.

Original entry on oeis.org

3, 5, 11, 17, 23, 47, 53, 59, 71, 89, 101, 113, 131, 137, 149, 173, 191, 197, 233, 239, 257, 311, 317, 347, 383, 401, 431, 443, 449, 467, 479, 509, 569, 593, 617, 641, 683, 719, 761, 773, 827, 857, 929, 941, 947, 1031, 1061, 1097, 1163, 1181, 1223, 1229

Offset: 1

Zak Seidov, Mar 16 2009

nonn

Hence 5p are terms in A157931, A062721 and (except of 25) in A043326.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A043326 Numbers n such that n is a product of two different primes and n-2 is prime, A062721 Numbers n such that n is a product of two primes and n-2 is prime, A157931 Numbers that are both the sum and the product of two primes.

Select[Prime[Range[300]],PrimeQ[5#-2]&] (* Harvey P. Dale, Apr 10 2015 *)

isok(p) = isprime(p) && isprime(5*p-2);  \\ Michel Marcus, Oct 16 2013

A100962 Numbers that can neither be written as the sum nor as the product of two primes.

Original entry on oeis.org

1, 2, 3, 11, 17, 23, 27, 29, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 107, 113, 117, 125, 127, 131, 135, 137, 147, 149, 157, 163, 167, 171, 173, 179, 189, 191, 197, 207, 211, 223, 227, 233, 239, 245, 251, 255, 257, 261, 263, 269, 275, 277, 281, 293, 297

Offset: 1

Reinhard Zumkeller, Nov 24 2004

nonn

Intersection of A014092 and A100959.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084997.

Cf. A064911, A014092, A157931, A100959.

a100962 n = a100962_list !! (n-1)
a100962_list = filter ((== 0) . a064911) a014092_list
-- Reinhard Zumkeller, Oct 15 2014

A167629 The odd composites c such that c=qgjy/2 and q+g=jy where q,g,j,y are distinct primes.

Original entry on oeis.org

105, 195, 231, 399, 627, 897, 935, 1023, 1443, 1581, 1729, 2465, 2915, 2967, 4123, 4301, 4623, 4715, 5487, 7055, 7685, 7881, 8099, 9717, 10707, 11339, 12099, 12995, 14993, 16377, 16383, 17353, 17423, 19599, 20213, 20915, 23779, 24963, 25327

Offset: 1

Juri-Stepan Gerasimov, Nov 07 2009

nonn

			a(1) =  3 *  7 *  5 = 105 (q=3, g= 7, j=2, y=5)
a(2) = 13 *  3 *  5 = 195 (q=2, g=13, j=3, y=5)
a(3) =  3 * 11 *  7 = 231 (q=3, g=11, j=2, y=7)
a(4) = 19 *  3 *  7 = 329 (q=2, g=19, j=3, y=7)
a(5) =  3 * 19 * 11 = 627 (q=3, g=19, j=2, y=11)

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A000040, A157931.

from sympy import primerange, primepi
k_upto = 25327
A167629, primeset = set(), set(primelist:= list(primerange(3, int(k_upto**0.5)+1)))
for x in range (primepi(k_upto**(1/3))):
    limit, y = k_upto // (a:=primelist[x]), x
    while (b:= primelist[(y:=y+1)]) * (c1:=(a * b - 2)) <= limit:
        if c1 in primeset : A167629.add(a * b * c1)
        if (c2 := b * 2 - a) in primeset : A167629.add(a * b * c2)
    y -= 1
    while (b:= primelist[(y:=y+1)]) * (c2:=(b * 2 - a)) <= limit:
        if c2 in primeset : A167629.add(a * b * c2)
print(A167629:=sorted(A167629)) # Karl-Heinz Hofmann, Jan 30 2025

Corrected and extended by D. S. McNeil, Dec 10 2009

A167690 The even composites c such that c=qgjy and q+g=jy where q,g,j,y are primes.

Original entry on oeis.org

16, 54, 126, 210, 250, 390, 462, 686, 798, 1150, 1254, 1794, 1870, 2046, 2662, 2886, 3162, 3458, 4394, 4606, 4930, 5830, 5934, 8246, 8602, 9246, 9430, 9826, 10974, 13718, 14110, 15370, 15762, 16198, 19434, 21414, 22678, 24198, 24334, 25990

Offset: 1

Juri-Stepan Gerasimov, Nov 09 2009

nonn

			a(1) = 2 * 2 * 2 * 2 =  16
a(2) = 3 * 3 * 2 * 3 =  54
a(3) = 2 * 7 * 3 * 3 = 126
a(4) = 3 * 7 * 2 * 5 = 210.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A000040, A157931, A167629.

from sympy import isprime, nextprime, integer_nthroot
c_upto = 100000
A167690, q, lim_q, lim_g = set(), 2, integer_nthroot(c_upto//2,3)[0], integer_nthroot(c_upto//3,2)[0]//2
while (g:=q) <= lim_q:
    while g <= lim_g:
        fac = 2 * q * g
        for j in [2 * q - g, 2 * g - q, q * g - 2 ]:
            if isprime(j) and (an:= fac * j) < c_upto : A167690.add(an)
        g = nextprime(g)
    q = nextprime(q)
print((A167690:=sorted(A167690))) # Karl-Heinz Hofmann, Feb 21 2025

Corrected (250, 686, 1794 etc inserted, 9486, 15782 removed) by R. J. Mathar, May 30 2010

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

A227030 Numbers that are neither the difference nor the product of two primes.

Original entry on oeis.org

7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 63, 67, 73, 75, 79, 83, 89, 97, 103, 109, 113, 117, 127, 131, 139, 151, 153, 157, 163, 167, 173, 175, 181, 193, 199, 207, 211, 223, 229, 233, 241, 243, 245, 251, 257, 263, 271, 273, 277, 283, 285, 293, 297, 307, 313, 317

Offset: 1

Gerasimov Sergey, Jun 28 2013

nonn

Cf. A007921, A157931.

A007921 \ A001358.

cp's OEIS Frontend

A014091 Numbers that are the sum of 2 primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Extensions

A158318 Primes p such that 5p-2 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A100962 Numbers that can neither be written as the sum nor as the product of two primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A167629 The odd composites c such that c=q*g*j*y/2 and q+g=j*y where q,g,j,y are distinct primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

A167690 The even composites c such that c=q*g*j*y and q+g=j*y where q,g,j,y are primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A227030 Numbers that are neither the difference nor the product of two primes.

Views

Author

Keywords

Crossrefs

Formula

A167629 The odd composites c such that c=qgjy/2 and q+g=jy where q,g,j,y are distinct primes.

A167690 The even composites c such that c=qgjy and q+g=jy where q,g,j,y are primes.