A063637 -id:A063637 - cp's OEIS frontend

A109611 Chen primes: primes p such that p + 2 is either a prime or a semiprime.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409

Offset: 1

Paul Muljadi, Jul 31 2005

nonn

43 is the first prime which is not a member (see A102540).
Contains A001359 = lesser of twin primes.
A063637 is a subsequence. - Reinhard Zumkeller, Mar 22 2010
In 1966 Chen proved that this sequence is infinite; his proof did not appear until 1973 due to the Cultural Revolution. - Charles R Greathouse IV, Jul 12 2016
Primes p such that p + 2 is a term of A037143. - Flávio V. Fernandes, May 08 2021
Named after the Chinese mathematician Chen Jingrun (1933-1996). - Amiram Eldar, Jun 10 2021

			a(4) = 7 because 7 + 2 = 9 and 9 is a semiprime.
a(5) = 11 because 11 + 2 = 13, a prime.

R. J. Mathar, Table of n, a(n) for n = 1..34076
Jing Run Chen, 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 (1973), pp. 157-176.
Ben Green and Terence Tao, Restriction theory of the Selberg sieve, with applications, arXiv:math/0405581 [math.NT], 2004-2005, pp. 5, 14, 18-19, 21.
Ben Green and Terence Tao, Restriction theory of the Selberg sieve, with applications, J. Théor. Nombres Bordeaux, Vol. 18, No. 1 (2006), pp. 147-182.
Eric Weisstein's World of Mathematics, Chen's Theorem.
Eric Weisstein's World of Mathematics, Chen Prime.
Wikipedia, Chen prime.
Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica, Vol. 138 (2009), pp. 301-315.

Union of A001359 and A063637.
Cf. A001358, A112021, A112022, A139689, A269256.
Cf. A037143.

A109611 := proc(n)
    option remember;
    if n =1 then
        2;
    else
        a := nextprime(procname(n-1)) ;
        while true do
            if isprime(a+2) or numtheory[bigomega](a+2) = 2 then
                return a;
            end if;
            a := nextprime(a) ;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 26 2013

semiPrimeQ[x_] := TrueQ[Plus @@ Last /@ FactorInteger[ x ] == 2]; Select[Prime[Range[100]], PrimeQ[ # + 2] || semiPrimeQ[ # + 2] &] (* Alonso del Arte, Aug 08 2005 *)
SequencePosition[PrimeOmega[Range[500]], {1, , 1|2}][[All, 1]] (* _Jean-François Alcover, Feb 10 2018 *)

isA001358(n)= if( bigomega(n)==2, return(1), return(0) );
isA109611(n)={ if( ! isprime(n), return(0), if( isprime(n+2), return(1), return( isA001358(n+2)) ); ); }
{ n=1; for(i=1,90000, p=prime(i); if( isA109611(p), print(n," ",p); n++; ); ); } \\ R. J. Mathar, Aug 20 2006

list(lim)=my(v=List([2]),semi=List(),L=lim+2,p=3); forprime(q=3,L\3, forprime(r=3,min(L\q,q), listput(semi,q*r))); semi=Set(semi); forprime(q=7,lim, if(setsearch(semi,q+2), listput(v,q))); forprime(q=5,L, if(q-p==2, listput(v,p)); p=q); Set(v) \\ Charles R Greathouse IV, Aug 25 2017

from sympy import isprime, primeomega
def ok(n): return isprime(n) and (primeomega(n+2) < 3)
print(list(filter(ok, range(1, 410)))) # Michael S. Branicky, May 08 2021

a(n)+2 = A139690(n).

Sum_{n>=1} 1/a(n) converges (Zhou, 2009). - Amiram Eldar, Jun 10 2021

Corrected by Alonso del Arte, Aug 08 2005

A063638 Primes p such that p-2 is a semiprime.

Original entry on oeis.org

11, 17, 23, 37, 41, 53, 59, 67, 71, 79, 89, 97, 113, 131, 157, 163, 179, 211, 223, 239, 251, 269, 293, 307, 311, 331, 337, 367, 373, 379, 383, 397, 409, 419, 439, 449, 487, 491, 499, 503, 521, 547, 593, 599, 613, 631, 673, 683, 691, 701, 709, 719, 733, 739

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Primes of form p*q + 2, where p and q are primes.
11 is the only prime of this form where p=q. For prime p>3, 3 divides p^2+2. - T. D. Noe, Mar 01 2006
The asymptotic growth of this sequence is relevant for A204142. We have a(10^k) = (11, 79, 1571, 27961, 407741, 5647823, ...). - M. F. Hasler, Feb 13 2012

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A005385, A001358, A063637, A109611 (Chen primes), A204142, A064911, A040976, A241809.

a063638 n = a063638_list !! (n-1)
a063638_list = map (+ 2) $ filter ((== 1) . a064911) a040976_list
-- Reinhard Zumkeller, Feb 22 2012

Take[Select[ # + 2 & /@ Union[Flatten[Outer[Times, Prime[Range[100]], Prime[Range[100]]]]], PrimeQ], 60]
Select[Prime[Range[200]],PrimeOmega[#-2]==2&] (* Paolo Xausa, Oct 30 2023 *)

n=0; for (m=2, 10^9, p=prime(m); if (bigomega(p - 2) == 2, write("b063638.txt", n++, " ", p); if (n==1000, break))) \\ Harry J. Smith, Aug 26 2009

forprime(p=3,9999, bigomega(p-2)==2 & print1(p","))

p=2; for(n=1,1e4, until(bigomega(-2+p=nextprime(p+1))==2,); write("b063638.txt", n" "p)) \\ M. F. Hasler, Feb 13 2012

list(lim)=my(v=List(), t); forprime(p=3, (lim-2)\3, forprime(q=3, min((lim-2)\p, p), t=p*q+2; if(isprime(t), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Aug 05 2016

a(n) = A241809(n) + 2. - Hugo Pfoertner, Oct 30 2023

A063643 Primes with 2 representations: pq - 2 = uv + 2 where p, q, u and v are primes.

Original entry on oeis.org

23, 37, 53, 67, 89, 113, 131, 157, 211, 251, 293, 307, 337, 379, 409, 449, 487, 491, 499, 503, 631, 683, 701, 719, 751, 769, 787, 919, 941, 953, 991, 1009, 1039, 1117, 1193, 1201, 1259, 1381, 1399, 1439, 1459, 1471, 1499, 1511, 1567, 1709, 1733, 1759, 1801

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Or, primes p such that p+/-2 are semiprimes. - Zak Seidov, Mar 08 2006

			A063643(25) = 751: 751 = A063637(60)= 753 - 2 = 3*251 - 2, 751 = A063638(55)= 749 + 2 = 7*107 + 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 term from T. D. Noe)

Cf. A063637, A063638, A001358.

Cf. A109611 (Chen primes).

q:= p-> isprime(p) and map(numtheory[bigomega], {p-2, p+2})={2}:
select(q, [$2..2000])[];  # Alois P. Heinz, Apr 01 2024

Select[Prime[Range[300]], PrimeOmega[#+2] == PrimeOmega[#-2] == 2&] (* Jean-François Alcover, Mar 02 2019 *)

{ n=0; for (m=2, 10^9, p=prime(m); if (bigomega(p + 2) == 2 && bigomega(p - 2) == 2, write("b063643.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

Intersection of A063637 and A063638. - Zak Seidov, Mar 14 2011

A062721 Numbers k such that k is a product of two primes and k-2 is prime.

Original entry on oeis.org

4, 9, 15, 21, 25, 33, 39, 49, 55, 69, 85, 91, 111, 115, 129, 133, 141, 159, 169, 183, 201, 213, 235, 253, 259, 265, 295, 309, 319, 339, 355, 361, 381, 391, 403, 411, 445, 451, 469, 481, 489, 493, 501, 505, 511, 543, 559, 565, 573, 579, 589, 633, 649, 655, 679

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 14 2001

This sequence is a subsequence of A107986, which only requires k to be composite. The first term in that sequence which is not in this sequence is 45, a number with three prime factors. - Alonso del Arte, May 03 2014

Zak Seidov, Table of n, a(n) for n = 1..10000.

Cf. A006512, A001358, A043326.

Cf. A063637, A046315, A089268.

Select[ Range[ 2, 1500 ], Plus @@ Last@Transpose@FactorInteger[ # ] == 2 && PrimeQ[ # - 2 ] & ]
Select[Range[700], PrimeOmega[#] == 2 && PrimeQ[# - 2]&] (* Harvey P. Dale, Mar 25 2013 *)

{ n=0; for (m=1, 10^9, a=prime(m) + 2; f=factor(a)~; if ((length(f)==1 && f[2, 1]==2) || (length(f)==2 && f[2, 1]==1 && f[2, 2]==1), write("b062721.txt", n++, " ", a); if (n==10000, break)) ) } \\ Harry J. Smith, Aug 09 2009

A189827 a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.

Original entry on oeis.org

3, 5, 4, 7, 4, 8, 5, 8, 5, 10, 4, 10, 6, 9, 6, 11, 4, 12, 6, 10, 6, 12, 5, 12, 7, 10, 6, 14, 4, 14, 6, 10, 8, 13, 6, 13, 6, 12, 6, 16, 4, 14, 8, 10, 8, 14, 5, 16, 7, 12, 6, 14, 6, 16, 8, 12, 6, 16, 4, 16, 8, 11, 10, 15, 6, 14, 6, 14, 6, 20, 4, 16, 8, 10, 10

Offset: 2

T. D. Noe, Apr 28 2011

nonn

d(n-1) + d(n+1) is a measure of the compositeness of the numbers next to n. Sequence A189825 lists the first occurrence of each number.

It is conjectured that every number greater than 3 occurs an infinite number of times. Note that an infinite number of 4's is equivalent to there being an infinite number of twin primes (A001097). An infinite number of 5's is equivalent to there being an infinite number of primes of the form p^2-2 (A028871) or p^2+2 (A056899) for prime p. An infinite number of 6's is equivalent to there being an infinite number of primes of the form p^3-2 (A066878), p^3+2 (A048636), p*q-2 (A063637), or p*q+2 (A063638), where p and q are distinct primes.

			a(5) = d(4) + d(6) = 3 + 4 = 7.

T. D. Noe, Table of n, a(n) for n = 2..10000

Cf. A175144, A189825.

Table[DivisorSigma[0,n-1] + DivisorSigma[0,n+1], {n, 2, 100}]
First[#]+Last[#]&/@Partition[DivisorSigma[0,Range[80]],3,1] (* Harvey P. Dale, May 27 2013 *)

A115093 Primes of the form p*q-2, where p and q are distinct primes.

Original entry on oeis.org

13, 19, 31, 37, 53, 67, 83, 89, 109, 113, 127, 131, 139, 157, 181, 199, 211, 233, 251, 257, 263, 293, 307, 317, 337, 353, 379, 389, 401, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 541, 557, 563, 571, 577, 587, 631, 647, 653, 677, 683, 701, 719, 743

Offset: 1

T. D. Noe, Mar 01 2006

nonn

This sequence is a subset of A063637, which is the union of this sequence and A049002.

Cf. A049002 (primes of the form p^2-2, where p is prime), A063637 (primes of the form p*q-2 where p and q are primes).

SemiPrimeQ[n_] := (n>1) && (2==Plus@@(Transpose[FactorInteger[n]][[2]])); Select[Prime[Range[150]], SemiPrimeQ[ #+2] && !IntegerQ[Sqrt[ #+2]]&]

A176229 The smaller members p of cousin prime pairs (p,p+4) with a semiprime arithmetic mean p+2.

Original entry on oeis.org

7, 13, 19, 37, 67, 109, 127, 307, 379, 487, 499, 769, 877, 937, 1009, 1297, 1567, 2269, 2389, 2659, 2857, 3037, 3187, 3457, 3847, 3907, 3919, 4447, 4789, 4969, 4999, 5077, 5167, 5347, 5737, 6007, 6997, 7039, 7669, 8689, 8779, 9199, 10597, 11467, 11827

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 12 2010

nonn

By definition a subsequence of A063637 and of A023200.
The associated p+4 are members of A063638.
Because all members of A023200 are == 1 (mod 3), the semiprimes p+2 are all == 0 (mod 3), so one of their two factors is 3.
The least-significant digit (LSD) of p > 13 in A023200 is always 3, 7 or 9, but those with LSD equal to 3 demand p+2 to have LSD 5 and therefore divisor 5 which contradicts the semiprime property above, so 13 is the only member of the sequence with LSD equal to 3.

			7 = prime(4), 11 = prime(5), (7+11)/2 = 3^2 = semiprime(3), so 7 is in the sequence.
13 = prime(6), 17 = prime(7), (13+17)/3 = 3 * 5 = semiprime(6), so 13 is in the sequence.
19 = prime(8), 23 = prime(9), (19+23)/3 = 3 * 7 = semiprime(7), so 19 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001358, A023200, A046132, A063637, A063638.

aQ[n_] := PrimeQ[n] && PrimeOmega[n + 2] == 2 && PrimeQ[n + 4]; Select[Range[12000], aQ] (* Amiram Eldar, Sep 08 2019 *)
Select[Prime[Range[1500]],PrimeQ[#+4]&&PrimeOmega[#+2]==2&] (* Harvey P. Dale, May 15 2023 *)

A241659 Primes p such that p^3 + 2 is semiprime.

Original entry on oeis.org

2, 11, 13, 17, 19, 23, 31, 41, 53, 59, 89, 101, 131, 137, 149, 193, 211, 223, 227, 229, 233, 239, 251, 271, 293, 317, 331, 359, 401, 449, 461, 557, 563, 571, 593, 599, 619, 641, 659, 677, 691, 719, 739, 751, 809, 821, 853, 929, 971, 991, 1009, 1013, 1039, 1051

Offset: 1

K. D. Bajpai, Apr 26 2014

nonn

			11 is prime and appears in the sequence because 11^3 + 2 = 1333 = 31 * 43, which is a semiprime.
17 is prime and appears in the sequence because 17^3 + 2 = 4915 =  5 * 983, which is a semiprime.
37 is prime but does not appear in the sequence because 37^3 + 2 = 50655 =  3 * 5 * 11 * 983, which is not a semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001358, A063637, A063638, A228183, A237627, A241483, A241493.

with(numtheory): KD:= proc() local a, b, k; k:=ithprime(n); a:=bigomega(k^3+2); if a=2 then RETURN (k); fi; end: seq(KD(), n=1..500);

A241659 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2, AppendTo[A241659, t]], {n, 500}]; A241659
(*For the b-file*) c = 0; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2, c++; Print[c, "  ", t]], {n, 1,6*10^4}];
Select[Prime[Range[200]],PrimeOmega[#^3+2]==2&] (* Harvey P. Dale, Feb 05 2025 *)

s=[]; forprime(p=2, 1200, if(bigomega(p^3+2)==2, s=concat(s, p))); s \\ Colin Barker, Apr 27 2014

A207526 Primes p such that p-2 or p+2 are semiprimes.

Original entry on oeis.org

2, 7, 11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 109, 113, 127, 131, 139, 157, 163, 167, 179, 181, 199, 211, 223, 233, 239, 251, 257, 263, 269, 293, 307, 311, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 439

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 18 2012

nonn

			2+2=2^2(semiprime), 7+2=3^2(semiprime), 11-2=3^2(semiprime), 13+2=3*5(semiprime), 17-2=3*5(semiprime)

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A063637, A063638, A063643.

Select[Table[Prime[n], {n,200}], (Last /@ FactorInteger[#-2] == {1,1} || Last /@ FactorInteger[#-2] == {2}) || (Last /@ FactorInteger[#+2] == {1,1} || Last /@ FactorInteger[#+2] == {2}) &]

A241716 Primes p such that p^3 - 2 is semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 41, 43, 47, 61, 79, 89, 101, 107, 139, 157, 181, 199, 239, 271, 307, 311, 331, 337, 347, 349, 379, 397, 409, 421, 431, 479, 487, 499, 521, 523, 541, 571, 607, 613, 641, 643, 661, 673, 701, 719, 761, 769, 811, 823, 829, 839, 877, 881, 883

Offset: 1

K. D. Bajpai, Apr 27 2014

nonn

			11 is prime and appears in the sequence because 11^3 - 2 = 1329 = 3 * 443, which is a semiprime.
17 is prime and appears in the sequence because 17^3 - 2 = 4911 = 3 * 1637, which is a semiprime.
23 is prime but does not appear in the sequence because 23^3 - 2 = 12165 =  3 * 5 * 811, which is not a semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..9380

Cf. A001358, A063637, A063638, A072381, A082919, A145292, A228183, A237627, A241483, A241493, A241659.

with(numtheory):A241716:= proc() local k; k:=ithprime(x); if bigomega(k^3-2)=2 then RETURN (k); fi; end: seq(A241716(), x=1..500);

A241716 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 - 2] == 2, AppendTo[A241716, t]], {n, 500}]; A241716
Select[Prime[Range[200]],PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Dec 09 2018 *)

cp's OEIS Frontend

A109611 Chen primes: primes p such that p + 2 is either a prime or a semiprime.

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A063638 Primes p such that p-2 is a semiprime.

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A063643 Primes with 2 representations: p*q - 2 = u*v + 2 where p, q, u and v are primes.

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A062721 Numbers k such that k is a product of two primes and k-2 is prime.

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A189827 a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.

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A115093 Primes of the form p*q-2, where p and q are distinct primes.

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A176229 The smaller members p of cousin prime pairs (p,p+4) with a semiprime arithmetic mean p+2.

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A063643 Primes with 2 representations: pq - 2 = uv + 2 where p, q, u and v are primes.