A130588 -id:A130588 - cp's OEIS frontend

A146295 Integers which are not the sum of a 4-almost prime and a prime.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, 25, 28, 30, 32, 34, 36, 40, 44, 46, 48, 50, 52, 54, 60, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 96, 108, 114, 116, 120, 126, 130, 132, 136, 144, 150, 156, 162, 168, 174, 180, 204, 210, 216, 240, 246, 258

Offset: 2

Donovan Johnson, Nov 05 2008

Largest term is 60060 (see b-file). No more terms < 10^8. Conjectured to be complete.

			20 is in this sequence because no 4-almost prime and a prime sum to 20. 21 is not in this sequence because the sum of 16 (4-almost prime) and 5 (prime) is 21.

Donovan Johnson, Table of n, a(n) for n=2..196

Cf. A000040, A014613, A130588, A146296, A146297.

Complement[Range[1000], Union@Flatten@Outer[Plus, Select[Range[1000], PrimeOmega[#] == 4 &], Prime[Range[PrimePi[1000]]]]] (* Robert Price, Jun 16 2019 *)

A146296 Integers which are not the sum of a 5-almost prime and a prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 38, 40, 41, 42, 44, 46, 47, 48, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 70, 72, 76, 78, 80, 81, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108

Offset: 2

Donovan Johnson, Nov 05 2008

Largest term is 3573570 (see b-file). No more terms < 10^9. Conjectured to be complete.

			36 is in this sequence because no 5-almost prime and a prime sum to 36. 37 is not in this sequence because the sum of 32 (5-almost prime) and 5 (prime) is 37.

Donovan Johnson, Table of n, a(n) for n=2..2008

Cf. A000040, A014614, A130588, A146295, A146297.

Complement[Range[1000], Union@Flatten@Outer[Plus, Select[Range[1000], PrimeOmega[#] == 5 &], Prime[Range[PrimePi[1000]]]]] (* Robert Price, Jun 16 2019 *)

A146297 Integers which are not the sum of a 6-almost prime and a prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 79

Offset: 2

Donovan Johnson, Nov 05 2008

nonn

Largest known term is 446185740 (see b-file). No more terms < 10^9.

			68 is in this sequence because no 6-almost prime and a prime sum to 68. 69 is not in this sequence because the sum of 64 (6-almost prime) and 5 (prime) is 69.

Donovan Johnson, Table of n, a(n) for n=2..20150

Cf. A000040, A046306, A130588, A146295, A146296.

Complement[Range[1000], Union@Flatten@Outer[Plus, Select[Range[1000], PrimeOmega[#] == 6 &], Prime[Range[PrimePi[1000]]]]] (* Robert Price, Jun 16 2019 *)

A322006 a(n) = number of primes of the form p = n - q, where q is a prime or semiprime.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 4, 4, 4, 6, 5, 5, 4, 6, 5, 7, 4, 8, 5, 8, 5, 9, 4, 7, 4, 8, 7, 9, 4, 11, 5, 9, 6, 11, 6, 11, 6, 11, 8, 12, 4, 13, 6, 12, 8, 13, 6, 14, 5, 13, 8, 13, 4, 16, 5, 15, 9, 16, 7, 16, 6, 14, 9, 16, 5, 18, 6, 16, 10, 19, 7, 19, 6, 17, 10, 18, 4, 21, 9, 17, 9, 19, 8

Offset: 0

M. F. Hasler, Jan 06 2019

Related to Chen's theorem (Chen 1966, 1973) which states that every sufficiently large even number is the sum of a prime and another prime or semiprime. Yamada (2015) has proved that this holds for all even numbers larger than exp(exp(36)).
In terms of this sequence, Chen's theorem with Yamada's bound is equivalent to say that a(2*n) > 0 for all n > 1.7 * 10^1872344071119348 (exponent ~ 1.8*10^15).
Sequence A322007(n) = a(2n) lists the bisection corresponding to even numbers only.
A235645 lists the number of decompositions of 2n into a prime p and a prime or semiprime q; this is less than a(2n) because p + q and q + p is the same decomposition (if q is a prime), but this sequence will count the two distinct primes 2n - q and 2n - p (if q <> p).

			a(4) = 1 is the first nonzero term corresponding to 4 = 2 + 2 or, rather, to the prime 2 = 4 - 2.
a(5) = 2 because the primes 2 = 5 - 3 and 3 = 5 - 2 are of the required form n - q where q = 3 resp. q = 2 are primes.
a(6) = 2 because the primes 2 = 6 - 4 and 3 = 6 - 3 are of the required form n - q, since q = 4 is a semiprime and q = 3 is a prime.

Chen, J. R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao. 11 (9): 385-386.
Chen, J. R. (1973). "On the representation of a larger even integer as the sum of a prime and the product of at most two primes". Sci. Sinica. 16: 157-176.

David A. Corneth, Table of n, a(n) for n = 0..10000
Y. C. Cai, Chen's Theorem with Small Primes, Acta Mathematica Sinica 18, no. 3 (2002), pp. 597-604. doi:10.1007/s101140200168.
P. M. Ross, On Chen's theorem that each large even number has the form (p1+p2) or (p1+p2p3), J. London Math. Soc. Series 2 vol. 10, no. 4 (1975), pp. 500-506. doi:10.1112/jlms/s2-10.4.500.
Tomohiro Yamada, Explicit Chen's theorem, preprint arXiv:1511.03409 [math.NT] (2015).
Veritasium, The Simplest Unsolved Problem in Math, YouTube video (2025).

Cf. A322007, A235645, A045917, A130588, A241539.

A322006(n,s=0)={forprime(p=2,n-2,bigomega(n-p)<3&&s++);s}

A322007 a(n) = number of primes of the form p = 2n - q, where q is a prime or semiprime.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 4, 4, 6, 5, 6, 7, 8, 8, 9, 7, 8, 9, 11, 9, 11, 11, 11, 12, 13, 12, 13, 14, 13, 13, 16, 15, 16, 16, 14, 16, 18, 16, 19, 19, 17, 18, 21, 17, 19, 22, 19, 19, 24, 19, 21, 23, 20, 21, 26, 22, 23, 28, 23, 24, 29, 23, 24, 29, 21, 25, 29, 24, 25, 29, 27, 25, 33, 26, 27, 32, 27

Offset: 0

M. F. Hasler, Jan 06 2019

nonn

Related to Chen's theorem (Chen 1966, 1973) which states that every sufficiently large even number is the sum of a prime and another prime or semiprime. Yamada (2015) has proved that this holds for all even numbers larger than exp(exp(36)).
In terms of this sequence, Chen's theorem with Yamada's bound is equivalent to say that a(n) > 0 for all n > 1.7 * 10^1872344071119348 (exponent ~ 1.8*10^15).
A235645 lists the number of decompositions of 2n into a prime p and a prime or semiprime q; this is less than a(n) because p + q and q + p is the same decomposition (if q is a prime), but this sequence will count two distinct primes 2n - q and 2n - p (if q <> p).
Sequence A322006 lists the same for even and odd numbers n, not only for even numbers 2n.

			a(4) = 2 since for n = 4, 2n = 8 = 2 + 6 = 3 + 5 = 5 + 3, i.e., primes 2, 3 and 5 are of the form specified in the definition (since 6 = 2*3 is a semiprime and 5 and 3 are primes).

Chen, J. R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao. 11 (9): 385-386.
Chen, J. R. (1973). "On the representation of a larger even integer as the sum of a prime and the product of at most two primes". Sci. Sinica. 16: 157-176.

Y. C. Cai, Chen's Theorem with Small Primes. Acta Mathematica Sinica 18, no. 3 (2002), pp. 597-604. doi:10.1007/s101140200168.
P. M. Ross, On Chen's theorem that each large even number has the form (p1+p2) or (p1+p2p3), J. London Math. Soc. Series 2 vol. 10, no. 4 (1975), pp. 500-506. doi:10.1112/jlms/s2-10.4.500.
Tomohiro Yamada, Explicit Chen's theorem, preprint arXiv:1511.03409 [math.NT] (2015).

Cf. A322006, A235645, A045917, A130588, A241539.

A322007(n,s=0)={forprime(p=2,-2+n*=2,bigomega(n-p)<3&&s++);s}

a(n) = A322006(2n).

A152165 Largest number which is not the sum of an n-almost prime and a prime.

Original entry on oeis.org

10, 300, 60060, 3573570, 446185740

Offset: 2

Jonathan Vos Post, Mar 30 2009

All the values are conjectural and untrustworthy. - N. J. A. Sloane, Oct 05 2009

For n=1 see A061358. - N. J. A. Sloane, Oct 05 2009

Cf. A130588, A146295, A146296, A146297, A100949.

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A146295 Integers which are not the sum of a 4-almost prime and a prime.

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A146296 Integers which are not the sum of a 5-almost prime and a prime.

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A146297 Integers which are not the sum of a 6-almost prime and a prime.

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A322006 a(n) = number of primes of the form p = n - q, where q is a prime or semiprime.

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A322007 a(n) = number of primes of the form p = 2n - q, where q is a prime or semiprime.

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A152165 Largest number which is not the sum of an n-almost prime and a prime.

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