A102540 -id:A102540 - cp's OEIS frontend

A115606 Partial sums of A102540 (primes that are not Chen primes).

0, 43, 104, 177, 256, 353, 456, 607, 770, 943, 1136, 1359, 1588, 1829, 2100, 2377, 2660, 2973, 3304, 3653, 4020, 4393, 4776, 5173, 5594, 6027, 6466, 6923, 7386, 7909, 8456, 9049, 9650, 10257, 10870, 11489, 12132, 12793, 13466, 14157, 14866, 15593

Offset: 0

Jonathan Vos Post, Mar 09 2006

See also A109611 (Chen primes: primes p such that p + 2 is either a prime or a semiprime), A102540 (primes that are not Chen primes). a(n) is prime for a(1) = 43, a(5) = 353 [Chen prime], a(7) = 607, a(15) = 2377, a(31) = 9049, a(35) = 11489 [Chen prime], a(49) = 22013. a(n) is semiprime for a(3) = 177, a(9) = 943, a(13) = 1829, a(17) = 2973, a(19) = 3653, a(21) = 4393, a(23) = 5173, a(24) = 5594, a(29) = 7909, a(37) = 12793, a(38) = 13466, a(40) = 14866, a(41) = 15593, a(43) = 17065, a(45) = 18595.

			a(5) = 43 + 61 + 73 + 79 + 97 = 353, which happens to be the Chen prime A109611(52).

Cf. A000040, A001358, A007504, A102540, A109611.

a(n) = SUM[k=1..n] A102540(k).

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

Original entry on oeis.org

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

A118499 Numbers k such that the k-th prime number is not a Chen prime.

Original entry on oeis.org

14, 18, 21, 22, 25, 27, 36, 38, 40, 44, 48, 50, 53, 58, 59, 61, 65, 67, 70, 73, 74, 76, 78, 82, 84, 85, 88, 90, 99, 101, 108, 110, 111, 112, 114, 117, 121, 122, 125, 127, 129, 130, 131, 134, 137, 143, 147, 149, 153, 155, 158, 163, 168, 170

Offset: 1

Jani Melik, May 05 2006

nonn

			97 is the 25th prime number but not a Chen prime since 99 = 3*3*11, therefore 25 is in the sequence.

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

Cf. A102540, A109611.

ts_inde_nonchen:= proc(n) local i, ans, inde; ans:=[ ]: inde := 0; for i from 1 to n do if ( isprime(i) = 'true') then inde:=inde+1: if (isprime(i+2) = 'false' and numtheory[bigomega](i+2) <> 2) then ans:=[ op(ans), inde ] fi fi od: return ans end: ts_inde_nonchen(2000);

Select[Range[180],Sum[FactorInteger[Prime[ # ]+2][[i,2]],{i,1,Length[ FactorInteger[Prime[ # ] + 2]]}] > 2 &]

isok(k) = (bigomega(prime(k)+2) > 2); \\ Michel Marcus, Oct 19 2021

Edited by Stefan Steinerberger, Jul 19 2007

A153167 Numbers n such that n+2 is not a Chen prime.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 88, 89, 90

Offset: 1

Vincenzo Librandi, Dec 20 2008

nonn

Contains all strictly positive even numbers A005843.
For each odd k>1 we can accumulate the numbers == k^2-2 (mod 2k) in a row, the last entry equal to A073577(k):
7; (k=3)
13, 23; (k=5)
19, 33, 47; (k=7)
25, 43, 61, 79; (k=9)
31, 53, 75, 97, 119; (k=11)
7, 63, 89, 115, 141, 167; (k=13)
43, 73, 103, 133, 163, 193,223; (k=17)
49, 83, 17, 151,185, 219, 253, 287; (k=19)
Each element T of this table has the format T= k^2-2-j*2*k, so T+2 is of the form k*(k-2*j), therefore not prime, and consequently all elements T are in the sequence.

Cf. A109611, A040976, A073577, A067076.

Edited, 41, 59 (see A102540) etc. inserted by R. J. Mathar, Oct 16 2009

A280010 Orders of consecutive clusters of non-Chen primes.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2

Offset: 1

Enrique Navarrete, Feb 21 2017

nonn

			a(1)=1 since the first cluster consists of the single value p=43.
a(3)=2 since the third cluster consists of the consecutive non-Chen primes {73,79}.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A102540, A280009.

Length /@ DeleteCases[Split@ Table[Boole@ Nand[# != 1, PrimeOmega@ # <= 2] &[# + 2] &@ Prime@ n, {n, 300}], k_ /; Total@ k == 0] (* Michael De Vlieger, Feb 27 2017 *)

do(lim)=my(v=List(), u=v, r, s); forprime(p=2, lim\2, forprime(q=2, min(lim\p, p), listput(u, p*q))); u=Set(u); r=3; forprime(p=5, lim, if(p-r==2 || setsearch(u, r+2), if(s, listput(v, s); s=0), s++); r=p); u=0; Vec(v) \\ Charles R Greathouse IV, Feb 27 2017

A115719 Products of two primes that are not Chen primes.

Original entry on oeis.org

1849, 2623, 3139, 3397, 3721, 4171, 4429, 4453, 4819, 5329, 5767, 5917, 6241, 6283, 6493, 7009, 7081, 7439, 7519, 7663, 8137, 8299, 9211, 9409, 9589, 9847, 9943, 9991, 10363, 10553, 10609, 11023, 11653, 11773, 11899, 11911, 11929, 12169, 12629, 12877

Offset: 1

Jonathan Vos Post, Mar 09 2006

Subset of semiprimes (A001358) such that neither prime factor is a Chen prime (A109611).

			a(1) = 1849 = A102540(1)*A102540(1) = 43*43.
a(2) = 2623 = A102540(1)*A102540(2) = 43*61.
a(36) = 14701 = A102540(2)*A102540(13) = 61 * 241.

Cf. A001358, A109611, A102540.

A118483 Partial sums of primes that are not Chen primes (starting with 1).

Original entry on oeis.org

1, 44, 105, 178, 257, 354, 457, 608, 771, 944, 1137, 1360, 1589, 1830, 2101, 2378, 2661, 2974, 3305, 3654, 4021, 4394, 4777, 5174, 5595, 6028, 6467, 6924, 7387, 7910, 8457, 9050, 9651, 10258, 10871, 11490, 12133, 12794, 13467, 14158, 14867, 15594

Offset: 0

Jani Melik, May 05 2006

nonn

Cf. A014284, A102540, A109611.

ischenprime:=proc(n); if (isprime(n) = 'true') then if (isprime(n+2) = 'true' or numtheory[bigomega](n+2) = 2) then RETURN('true') else RETURN('false') fi fi end: ts_partsum_notchenprime:=proc(n) local i,ans,tren; ans:=1: tren:=1: for i from 1 to n do if (ischenprime(i)='false') then tren := tren+i: ans:=[op(ans), tren]: fi od; RETURN(ans) end: ts_partsum_notchenprime(1000);

Accumulate[Join[{1},Select[Prime[Range[200]],PrimeOmega[#+2]>2&]]] (* Harvey P. Dale, Dec 14 2012 *)

A118491 Product of first n Chen primes.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 14299762385778870, 757887406446280110, 44715356980330526490, 2995928917682145274830

Offset: 0

Jani Melik, May 05 2006

nonn

This first differs from primorials A002110 at a(14) = 14299762385778870 = 47*a(13) rather than 43*a(13) because 43 is the smallest prime that is not a Chen prime (A102540). - Jonathan Vos Post, Dec 25 2008

			a(0) = 1 by definition. a(1) = 2, 2 is first Chen prime, a(2) = 6 since it is the product of the first two Chen primes 2 and 3, ...

Cf. A002110, A109611.

Cf. A102540. - Jonathan Vos Post, Dec 25 2008

ischenprime:=proc(n); if (isprime(n) = 'true') then if (isprime(n+2) = 'true' or numtheory[bigomega](n+2) = 2) then RETURN('true') else RETURN('false') fi fi end: ts_chen_prim_numbers:=proc(n) local i,ans,tren; ans:=[1]: tren:=1: for i from 1 to n do if (ischenprime(i) = 'true') then tren := i*tren: ans:=[op(ans), tren]: fi od; RETURN(ans) end: ts_chen_prim_numbers(140);

FoldList[Times,Join[{1},Select[Prime[Range[50]],PrimeOmega[#+2]<3&]]] (* Harvey P. Dale, Jun 06 2022 *)

A118497 Primes that are not Chen primes written backwards.

Original entry on oeis.org

34, 16, 37, 97, 79, 301, 151, 361, 371, 391, 322, 922, 142, 172, 772, 382, 313, 133, 943, 763, 373, 383, 793, 124, 334, 934, 754, 364, 325, 745, 395, 106, 706, 316, 916, 346, 166, 376, 196, 907, 727, 337, 937, 757, 377, 328, 358, 958, 388, 709, 929, 769, 799

Offset: 1

Jani Melik, May 05 2006

Cf. A004087, A102540, A109611.

# Check if number is Chen prime ischenprime:=proc(n); if (isprime(n) = 'true') then if (isprime(n+2) = 'true' or numtheory[bigomega](n+2) = 2) then return 'true' else return 'false' fi fi end: #Reverse digits obrni_stev:=proc(n) local i, tren, tren1, st, ans; ans:=[ ]: tren:=n: tren1:=0: for i while (tren>0) do st:=round(10*frac(tren/10)): ans:=[op(ans), st]: tren:=trunc(tren/10): od: for i from 0 to nops(ans)-1 do tren1:= tren1 + op(nops(ans)-i, ans)*10^(i): od: return tren1 end: ts_inv_nonchen_pra:= proc(n) local i, trens, ans; trens:= [ ]; ans:=[ ]; for i from 1 to n do if (ischenprime( i ) = 'false') then ans:=[op(ans),obrni_stev(i)] fi: od: return ans end: ts_inv_nonchen_pra(2000);

A321420 Primes p whose reversal is a Chen prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 71, 73, 101, 107, 113, 131, 149, 157, 167, 179, 181, 191, 199, 311, 347, 353, 359, 389, 701, 733, 739, 743, 751, 761, 787, 797, 919, 941, 953, 967, 971, 983, 991, 1009, 1021, 1031, 1061, 1091, 1097, 1103, 1109, 1151, 1153, 1217, 1223

Offset: 1

Paolo Galliani, Nov 09 2018

73 is the smallest non-Chen prime whose reversal is a Chen prime.

			73 is in the sequence because its reversal is 37 which is a Chen prime (because 37 + 2 = 39 has at most two prime factors).

Cf. A118725, A109611, A102540.

cpQ[n_] := Module[{rev = FromDigits[Reverse[IntegerDigits[n]]]}, PrimeQ[rev] && PrimeOmega[rev + 2] < 3]; Select[Prime[Range[400]], cpQ] (* Amiram Eldar, Nov 09 2018 after Harvey P. Dale at A118725 *)

is(n) = if(isprime(n), rn = fromdigits(Vecrev(digits(n))); return(isprime(rn) && bigomega(rn+2) <= 2), 0) \\ David A. Corneth, Nov 09 2018

cp's OEIS Frontend

A115606 Partial sums of A102540 (primes that are not Chen primes).

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

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A118499 Numbers k such that the k-th prime number is not a Chen prime.

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A153167 Numbers n such that n+2 is not a Chen prime.

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A280010 Orders of consecutive clusters of non-Chen primes.

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A115719 Products of two primes that are not Chen primes.

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A118483 Partial sums of primes that are not Chen primes (starting with 1).

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A118491 Product of first n Chen primes.

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A118497 Primes that are not Chen primes written backwards.