A109611 -id:A109611 - cp's OEIS frontend

A117708 Numbers that are both lucky numbers and Chen primes.

3, 7, 13, 31, 37, 67, 127, 211, 307, 409, 487, 541, 577, 631, 769, 787, 937, 991, 1009, 1039, 1117, 1201, 1291, 1459, 1471, 1567, 1777, 1801, 2251, 2281, 2467, 2557, 2647, 2971, 3037, 3187, 3259, 3307, 3559, 3709, 3847, 3889, 4441, 4567, 4801, 4969, 4987

Offset: 1

Jani Melik, Apr 27 2006

nonn

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

Intersection of A000959 and A109611.

Subsequence of A031157.

lst = Range[1, 5000, 2]; i = 2; While[ i <= (len = Length@lst) && (k = lst[[i]]) <= len, lst = Drop[lst, {k, len, k}]; i++ ]; chenQ[n_] := PrimeQ[n] && Plus @@ Last /@ FactorInteger[n + 2] < 3; Select[lst, chenQ@# &] (* Robert G. Wilson v, May 12 2006 *)

Corrected and extended by Robert G. Wilson v, May 12 2006

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

A118721 Chen primes for which the sum of the digits is also a Chen prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 47, 67, 83, 89, 101, 113, 131, 137, 139, 157, 179, 191, 197, 199, 227, 263, 269, 281, 311, 317, 337, 353, 359, 379, 401, 409, 443, 449, 461, 467, 487, 557, 571, 577, 599, 641, 647, 683, 719, 751, 797, 809, 821, 827, 829, 863, 881

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 21 2006

			29 is in the sequence because (1) it is a Chen prime and (2) the sum of its digits 2+9=11 is also a Chen prime.

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

Cf. A109611.

With[{ch=Select[Prime[Range[200]],PrimeOmega[#+2]<3&]},Select[ch, MemberQ[ ch, Total[IntegerDigits[#]]]&]] (* Harvey P. Dale, May 10 2018 *)

A118776 Differences between consecutive Chen primes.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 6, 6, 6, 8, 4, 12, 6, 12, 6, 2, 4, 14, 4, 6, 2, 10, 8, 10, 12, 2, 10, 6, 2, 12, 16, 6, 6, 12, 6, 6, 6, 12, 12, 14, 4, 6, 20, 10, 6, 6, 20, 10, 12, 8, 10, 12, 12, 6, 12, 6, 12, 8, 4, 8, 4, 6, 12, 20, 16, 6, 6, 2, 6, 10, 12, 18, 14, 10, 6, 6, 6, 18, 6, 18, 18, 24

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 22 2006

nonn

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

Cf. A001223, A109611.

chenQ[n_] := PrimeQ[n] && PrimeOmega[n + 2] < 3; Differences@ Select[Range[750], chenQ] (* Amiram Eldar, Oct 19 2021 *)

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

A185347 Semiprimes that are the sum of 10 consecutive primes.

Original entry on oeis.org

129, 158, 382, 1114, 1546, 2374, 2582, 3446, 3578, 6218, 6826, 7978, 8266, 9298, 9382, 10202, 12946, 14002, 15178, 15406, 15766, 16382, 16466, 17282, 17362, 18374, 18838, 19226, 19606, 23878, 24074, 25154, 25642, 26206, 29782, 30034, 30638, 32902, 33526, 34862, 34934, 35678, 35978, 36602

Offset: 1

Zak Seidov, Feb 15 2011

nonn

Or, semiprimes in A127337 (Numbers that are the sum of 10 consecutive primes).

a(1) = 3*43, all other terms are of the form 2*prime.

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

Cf. A127337.

(* First run the program for A109611 to define semiPrimeQ *) Select[Table[Plus@@Prime[Range[n, n + 9]], {n, 500}], semiPrimeQ] (* Alonso del Arte, Feb 15 2011 *)
Select[Total/@Partition[Prime[Range[600]],10,1],PrimeOmega[#]==2&] (* Harvey P. Dale, Sep 06 2014 *)

{s=129;for(n=1,2000,if(2==bigomega(s), print1(s", ")); s=s-prime(n)+prime(n+10))}

A211410 Chen triprimes, triprimes (A014612) m such that m+2 is either prime or semiprime.

Original entry on oeis.org

8, 12, 20, 27, 44, 45, 63, 75, 92, 99, 105, 116, 117, 125, 147, 153, 164, 165, 171, 175, 195, 207, 212, 231, 245, 255, 261, 275, 279, 285, 325, 332, 333, 345, 356, 357, 363, 369, 387, 399, 425, 429, 435, 452, 455, 465, 477, 483, 507, 524

Offset: 1

Jonathan Vos Post, Feb 09 2013

			27=3^3 and 45=3^2*9 are in the sequence because 27+2 = 29 and 45+2 = 47 are primes.
8=2^3, 12=2^2*3, and 20=2^2*5 are in the sequence because 8+2=10=2*5, 12+2=14=2*7, and 20+2=22=2*11 are semiprimes (A001358).

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

Cf. A001358, A014612, A109611, A175634.

A211410 := proc(n)
    option remember;
    local a;
    if n = 1 then
        8;
    else
        for a from procname(n-1)+1 do
            if numtheory[bigomega](a) = 3 then
                if isprime(a+2) or numtheory[bigomega](a+2) = 2 then
                    return a;
                end if;
            end if;
        end do:
    end if;
end proc:
seq(A211410(n),n=1..80) ; # R. J. Mathar, Feb 10 2013

Select[Range[600],PrimeOmega[#]==3&&PrimeOmega[#+2]<3&] (* Harvey P. Dale, Jul 15 2019 *)

issemi(n)=bigomega(n)==2
list(lim)=my(v=List(),pq); forprime(p=2,lim\4, forprime(q=2,min(lim\2\p,p), pq=p*q; forprime(r=2,min(lim\pq,q), if(isprime(pq*r+2) || issemi(pq*r+2), listput(v,pq*r))))); Set(v) \\ Charles R Greathouse IV, Aug 23 2017

A221865 The nonprimes k such that k + 2 is either a prime or a semiprime.

Original entry on oeis.org

0, 1, 4, 8, 9, 12, 15, 20, 21, 24, 27, 32, 33, 35, 36, 39, 44, 45, 49, 51, 55, 56, 57, 60, 63, 65, 69, 72, 75, 77, 80, 81, 84, 85, 87, 91, 92, 93, 95, 99, 104, 105, 111, 116, 117, 119, 120, 121, 125, 129, 132, 135, 140, 141, 143, 144, 147, 153, 155, 156, 159, 161, 164, 165, 171, 175, 176, 177, 183

Offset: 1

Gerasimov Sergey, Apr 18 2013

nonn

Chen primes A109611(n) such that A109611(n)-/+ a(n) are both prime: 2, 29, 53, 113, 139,...
Unrelated: Numbers n such that n + 2^bigomega(n) is either a prime or a semiprime: 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 18, 19, 21, 22, 23, 25, 27,...
A179384 is a subsequence. - R. J. Mathar, Apr 26 2013

Cf. A001358, A109611, A175634.

A221865 := proc(n)
    option remember;
    if n =1 then
        0;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) then
            if isprime(a+2) or numtheory[bigomega](a+2) = 2 then
                return a;
            end if;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 26 2013

Select[Range[0,200],!PrimeQ[#]&&PrimeOmega[#+2]<3&] (* Harvey P. Dale, May 05 2013 *)

Corrected by R. J. Mathar, Apr 26 2013

A291525 a(n) is the largest number in an n-term AP of Chen primes.

Original entry on oeis.org

2, 3, 7, 23, 29, 257, 1439, 2351, 26561, 146639, 1891949, 2062889, 341708489, 2062232987

Offset: 1

Charles R Greathouse IV, Aug 25 2017

Zhou proves that a(n) exists for each n, generalizing Green & Tao (2008) from primes to Chen primes and generalizing Green & Tao (2006) from 3-AP to n-AP. Sequence is increasing by definition.

			3, 5, 7 = a(3)
5, 11, 17, 23 = a(4)
5, 11, 17, 23, 29 = a(5)
107, 137, 167, 197, 227, 257 = a(6)
179, 389, 599, 809, 1019, 1229, 1439 = a(7)
881, 1091, 1301, 1511, 1721, 1931, 2141, 2351 = a(8)
4721, 7451, 10181, 12911, 15641, 18371, 21101, 23831, 26561 = a(9)
122069, 124799, 127529, 130259, 132989, 135719, 138449, 141179, 143909, 146639 = a(10)
182549, 353489, 524429, 695369, 866309, 1037249, 1208189, 1379129, 1550069, 1721009, 1891949 = a(11)
182549, 353489, 524429, 695369, 866309, 1037249, 1208189, 1379129, 1550069, 1721009, 1891949, 2062889 = a(12)
205492409, 216843749, 228195089, 239546429, 250897769, 262249109, 273600449, 284951789, 296303129, 307654469, 319005809, 330357149, 341708489 = a(13)
19712507, 176829467, 333946427, 491063387, 648180347, 805297307, 962414267, 1119531227, 1276648187, 1433765147, 1590882107, 1747999067, 1905116027, 2062232987 = a(14)

B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications, J. Théor. Nombres Bordeaux 18:1 (2006), pp. 147-182. arXiv:math/0405581 [math.NT]
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188 [math.NT], 2004-2007.
Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica 138 (2009), pp. 301-315.

Cf. A109611, A005115.

primorial(n)=vecprod(primes(primepi(n)));
listChen(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)
chen=listChen(1e6); \\ Increase as needed to find more terms
a(n,startAt=n)=n--; my(div=lcm(primorial(n+1),n)); for(i=startAt,#chen, for(j=1,i-n, my(d=chen[i]-chen[j],g); if(d%div,next); g=d/n; forstep(k=chen[j]+g, chen[i]-g, g, if(!setsearch(chen,k), next(2))); return(chen[i])))

a(14) from Charles R Greathouse IV, Sep 06 2017

A321855 Number of permutations f of {1,...,n} such that prime(k)*prime(f(k)) - 2 is prime for every k = 1,...,n.

Original entry on oeis.org

1, 1, 2, 3, 5, 12, 2, 3, 65, 248, 448, 1792, 4288, 6468, 27068, 29752, 106066, 447982, 1250762, 6304196, 46613084, 126391780, 504582496, 2270372946, 3028652541, 8941959118, 36442298864, 175008626450, 318369805106, 1974700703920, 6654020288821, 48819526290634, 150577775767875, 574885284627624, 3058310882340228, 15949743649457780

Offset: 1

Zhi-Wei Sun, Nov 19 2018

nonn

Conjecture: a(n) > 0 for all n > 0. Moreover, for each n > 0, there is an even permutation f of {1,...,n} with prime(k)*prime(f(k)) - 2 prime for all k = 1,...,n. Also, for any integer n > 2, there is an odd permutation f of {1,...,n} with prime(k)*prime(f(k)) - 2 prime for all k = 1,...,n.
If we let b(n) denote the number of even permutations f of {1,...,n} with prime(k)*prime(f(k)) - 2 prime for all k = 1,...,n, then (b(1),...,b(11)) = (1,1,1,1,3,6,1,1,33,125,226).
In 1973 J.-R. Chen proved that there are infinitely many primes p with p + 2 a product of at most two primes, such primes p are now called Chen primes.

			a(7) = 2. The only even permutation of {1,...,7} meeting the requirement is (1,5,7,4,2,6,3) with prime(1)*prime(1) - 2 = 2, prime(2)*prime(5) - 2 = 31, prime(3)*prime(7) - 2 = 83, prime(4)*prime(4) - 2 = 47, prime(5)*prime(2) - 2 = 31, prime(6)*prime(6) - 2 = 167 and prime(7)*prime(3) - 2 = 83 all prime. Also, the only odd permutation of {1,...,7} meeting the requirement is (1,5,7,6,2,4,3) with prime(1)*prime(1) - 2 = 2, prime(2)*prime(5) - 2 = 31, prime(3)*prime(7) - 2 = 83, prime(4)*prime(6) - 2 = 89, prime(5)*prime(2) - 2 = 31, prime(6)*prime(4) - 2 = 89 and prime(7)*prime(3) - 2 = 83 all prime.

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.
Zhi-Wei Sun, Chen primes and permutations, Question 315679 on Mathoverflow, Nov. 19, 2018.
Zhi-Wei Sun, On permutations of {1, ..., n} and related topics, arXiv:1811.10503 [math.CO], 2018.

Cf. A000040, A109611, A257926, A321597, A321610, A321611, A321727, A321805.

Permanent[m_List]:=With[{v = Array[x, Length[m]]},Coefficient[Times @@ (m.v), Times @@ v]];
a[n_]:=a[n]=Permanent[Table[Boole[PrimeQ[Prime[i]*Prime[j]-2]],{i,1,n},{j,1,n}]];
Do[Print[n," ",a[n]],{n,1,27}]

a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(prime(i)*prime(j) - 2))); \\ Jinyuan Wang, Jun 13 2020

a(28)-a(29) from Jinyuan Wang, Jun 13 2020

a(30)-a(36) from Vaclav Kotesovec, Aug 20 2021

cp's OEIS Frontend

A117708 Numbers that are both lucky numbers and Chen primes.

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

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A118721 Chen primes for which the sum of the digits is also a Chen prime.

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A118776 Differences between consecutive Chen primes.

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

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A185347 Semiprimes that are the sum of 10 consecutive primes.

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A211410 Chen triprimes, triprimes (A014612) m such that m+2 is either prime or semiprime.

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A221865 The nonprimes k such that k + 2 is either a prime or a semiprime.

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A291525 a(n) is the largest number in an n-term AP of Chen primes.

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