A002375 -id:A002375 - cp's OEIS frontend

A284919 Even integers E such that there is no prime p < E with E - p and E + p both prime.

0, 2, 4, 6, 28, 38, 52, 58, 62, 68, 74, 80, 82, 88, 94, 98, 112, 118, 122, 124, 128, 136, 146, 148, 152, 158, 164, 166, 172, 178, 182, 184, 188, 190, 206, 208, 212, 214, 218, 220, 224, 238, 242, 244, 248, 250, 256, 262, 268, 278, 284, 290, 292, 296, 298

Offset: 1

Claudio Meller, Apr 05 2017

nonn

Or, even integers k such that k + p is composite for all primes p, q with p + q = k.
The two initial terms vacuously satisfy the definition, but all even numbers >= 4 are the sum of two primes, according to the Goldbach conjecture.
All odd numbers except for numbers m such that m-2 and m+2 are prime (= A087679) would satisfy the definition. - M. F. Hasler, Apr 05 2017
Conjecture: except for a(4)=6, all terms are coprime to 3. - Bob Selcoe, Apr 06 2017
If E is an even number not divisible by 3, then E is in the sequence unless E-3 and at least one of E+3 and 2E-3 are prime. - Robert Israel, Apr 10 2017
Consider a subsequence with the additional condition: n+odd part of p-1 is composite (for example, for p=19 it is 9). I found that this subsequence begins 0,2,118 and up to 300000 Peter J. C. Moses found only more one term 868. Is this subsequence finite? - Vladimir Shevelev, Apr 12 2017
One can compare the theoretical maxima with the actual sequence numbers of terms.  Doing this at powers of 10, we see at powers {2,3,4,5,6} the ratio progression {2.33, 1.51, 1.25, 1.15, 1.096}.  This implies that the excluded even numbers become increasingly rare (those coprime to 3). - Bill McEachen, Apr 17 2017
From Robert Israel's comment and the distribution of primes, the proportion of even numbers not divisible by 3 that are in the sequence tends to 1. - Peter Munn, Apr 23 2017
Moreover, If n is not divisible by 3 and 2*n - 3 is composite, then 2*n+p is also composite. Indeed, for these 2*n all primes p such that 2*n-p is prime are in the interval (3, 2*n-3). Then either 2*n-p or 2*n+p should be divisible by 3, but 2*n-p is a prime > 3. So 2*n+p is composite and 2*n is in the sequence. - Vladimir Shevelev, Apr 28 2017

			k=28 is in the sequence because 5+23 = 28 and 11+17 = 28, and 28 + {5,11,17,23} are composite; k=26 is not in the sequence because 3+23 = 26, 7+19 = 26 and 13+13 = 26, but 26+3 = 29 (prime).  - _Bob Selcoe_, Apr 06 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms from M. F. Hasler)
Claudio Meller and others, New sequence, SeqFan list, April 5, 2017. (Click "next" for subsequent contributions.)
Robert G. Wilson v, Graph of all terms < 100000

Cf. A284928 (terms/2), A002375 (number of decompositions p + q = 2k), A020481 (least p: p + q = 2k), A277688 (an analog for decompositions odd k as 2p+q).

fQ[n_] :=  Select[Select[Prime@Range@PrimePi@n, PrimeQ[n - #] &],    PrimeQ[n + #] &] == {}; Select[2 Range[0, 150], fQ] (* Robert G. Wilson v, Apr 05 2017 *)

is(n)=!bittest(n,0)&&!forprime(p=2,n\2, isprime(n-p) && (isprime(n+p) || isprime(2*n-p)) && return) \\ Charles R Greathouse IV and M. F. Hasler, Apr 05 2017

a(n) = 2*A284928(n). - M. F. Hasler, Apr 06 2017

A227908 Number of ways to write 2*n = p + q with p, q and (p-1)^2 + q^2 all prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 3, 2, 2, 0, 2, 6, 1, 3, 5, 2, 3, 2, 1, 2, 2, 5, 4, 3, 2, 3, 8, 1, 4, 3, 3, 2, 5, 1, 2, 4, 5, 3, 4, 4, 2, 6, 1, 4, 5, 3, 3, 6, 2, 6, 5, 4, 5, 7, 3, 1, 9, 2, 3, 6, 1, 2, 5, 4, 7, 2, 7, 6, 6, 2, 4, 10, 3, 3, 6, 1, 7, 9, 5, 4, 5, 4, 3, 5, 3, 5, 8, 4, 4, 5, 2, 11, 9, 4

Offset: 1

Zhi-Wei Sun, Oct 12 2013

nonn

Conjecture: a(n) > 0 except for n = 1, 16, 292.
This implies not only Goldbach's conjecture for even numbers, but also Ming-Zhi Zhang's conjecture (cf. A036468) that any odd number greater than one can be written as x + y (x, y > 0) with x^2 + y^2 prime.
We have verified the conjecture for n up to 10^7.
Conjecture verified for n up tp 10^9. - Mauro Fiorentini, Sep 21 2023

			a(7) = 1 since 2*7 = 11 + 3, and (11-1)^2 + 3^2 = 109 is prime.
a(19) = 1 since 2*19 = 7 + 31, and (7-1)^2 + 31^2 = 997 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A002375, A036468, A220554, A230224.

a[n_]:=Sum[If[PrimeQ[2n-Prime[i]]&&PrimeQ[(Prime[i]-1)^2+(2n-Prime[i])^2],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

A228956 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all the 2*n+2 numbers |i_0 +/- i_1|, |i_1 +/- i_2|, ..., |i_{n-1} +/- i_n|, |i_n +/- i_0| have the form (p-1)/2 with p an odd prime.

Original entry on oeis.org

1, 1, 1, 1, 5, 9, 17, 84, 30, 127, 791, 2404, 11454, 27680, 25942, 137272, 515947, 2834056, 26583034, 82099932, 306004652, 4518630225, 11242369312, 8942966426, 95473633156, 533328765065

Offset: 1

Zhi-Wei Sun, Sep 09 2013

Conjecture: a(n) > 0 for all n > 0.
Note that if i-j = (p-1)/2 and i+j = (q-1)/2 for some odd primes p and q then 4*i+2 is the sum of the two primes p and q. So the conjecture is related to Goldbach's conjecture.
Zhi-Wei Sun also made the following similar conjecture:  For any integer n > 5, there exists a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that all the 2*n+2 numbers 2*|i_k-i_{k+1}|+1 and  2*(i_k+i_{k+1})-1 (k = 0,...,n) (with i_{n+1} = i_0) are primes.

			a(n) = 1 for n = 1,2,3 due to the natural circular permutation (0,...,n).
a(4) = 1 due to the circular permutation (0,1,4,2,3).
a(5) = 5 due to the circular permutations (0,1,2,4,5,3), (0,1,4,2,3,5), (0,1,4,5,3,2), (0,2,1,4,5,3), (0,3,2,1,4,5).
a(6) = 9 due to the circular permutations
  (0,1,2,4,5,3,6), (0,1,2,4,5,6,3), (0,1,4,2,3,5,6),
  (0,1,4,2,3,6,5), (0,1,4,5,6,3,2), (0,2,1,4,5,3,6),
  (0,2,1,4,5,6,3), (0,3,2,1,4,5,6), (0,5,4,1,2,3,6).
a(7) = 17 due to the circular permutations
  (0,1,2,7,4,5,3,6), (0,1,2,7,4,5,6,3), (0,1,4,7,2,3,5,6),
  (0,1,4,7,2,3,6,5), (0,1,7,2,4,5,3,6), (0,1,7,2,4,5,6,3),
  (0,1,7,4,2,3,5,6), (0,1,7,4,2,3,6,5), (0,1,7,4,5,6,3,2),
  (0,2,1,7,4,5,3,6), (0,2,1,7,4,5,6,3), (0,2,7,1,4,5,3,6),
  (0,2,7,1,4,5,6,3), (0,3,2,1,7,4,5,6), (0,3,2,7,1,4,5,6),
  (0,5,4,1,7,2,3,6), (0,5,4,7,1,2,3,6).

Z.-W. Sun, Some new problems in additive combinatorics, arXiv preprint arXiv:1309.1679 [math.NT], 2013-2014.

Cf. A000040, A051252, A002375, A228917, A228886.

(* A program to compute required circular permutations for n = 7. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction; for example, (0,6,3,5,4,7,2,1) is identical to (0,1,2,7,4,5,3,6) if we ignore direction. Thus a(7) is half of the number of circular permutations yielded by this program. *)
p[i_,j_]:=PrimeQ[2*Abs[i-j]+1]&&PrimeQ[2(i+j)+1]
V[i_]:=Part[Permutations[{1,2,3,4,5,6,7}],i]
m=0
Do[Do[If[p[If[j==0,0,Part[V[i],j]],If[j<7,Part[V[i],j+1],0]]==False,Goto[aa]],{j,0,7}]; m=m+1;Print[m,":"," ",0," ",Part[V[i],1]," ",Part[V[i],2]," ",Part[V[i],3]," ",Part[V[i],4]," ",Part[V[i],5]," ",Part[V[i],6]," ",Part[V[i],7]];Label[aa];Continue,{i,1,7!}]

a(10)-a(26) from Max Alekseyev, Sep 17 2013

A230252 Number of ways to write n = x + y (x, y > 0) with 2*x + 1, x^2 + x + 1 and y^2 + y + 1 all prime.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 4, 4, 3, 4, 1, 3, 3, 3, 5, 5, 4, 3, 6, 4, 7, 7, 2, 4, 6, 4, 4, 6, 3, 1, 4, 2, 4, 7, 4, 1, 4, 4, 2, 6, 4, 3, 4, 2, 3, 5, 3, 2, 1, 2, 3, 6, 2, 6, 6, 3, 5, 4, 5, 3, 7, 2, 4, 6, 2, 4, 5, 3, 5, 8, 5, 2, 10, 4, 4, 8, 5, 6, 7, 8, 4, 11, 4, 3, 6, 4, 2, 4, 8, 8, 11, 5, 3, 11, 5, 3, 6, 4, 5

Offset: 1

Zhi-Wei Sun, Oct 13 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) Any integer n > 3 can be written as p + q with p, 2*p - 3 and q^2 + q + 1 all prime. Also, each integer n > 3 not equal to 30 can be expressed as p + q with p, q^2 + q - 1 and q^2 + q + 1 all prime.
(iii) Any integer n > 1 can be written as x + y (x, y > 0) with x^2 + 1 (or 4*x^2+1) and y^2 + y + 1 (or 4*y^2 + 1) both prime.
(iv) Each integer n > 3 can be expressed as p + q (q > 0) with p, 2*p - 3 and 4*q^2 + 1 all prime.
(v) Any even number greater than 4 can be written as p + q with p, q and p^2 + 4 (or p^2 - 2) all prime. Also, each even number greater than 2 and not equal to 122 can be expressed as p + q with p, q and (p-1)^2 + 1 all prime.
We have verified the first part for n up to 10^8.

			a(5) = 2 since 5 = 2 + 3 = 3 + 2, and 2*2+1 = 5, 2*3+1 = 7, 2^2+2+1 = 7, 3^2+3+1 = 13 are all prime.
a(31) = 1 since 31 = 14 + 17, and 2*14+1 = 29, 14^2+14+1 = 211 and 17^2+17+1 = 307 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A002383, A002384, A002375, A219864, A227908, A227909, A230230, A230241, A230254.

a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[i^2+i+1]&&PrimeQ[(n-i)^2+n-i+1],1,0],{i,1,n-1}]
Table[a[n],{n,1,100}]

A235645 From Goldbach's conjecture and Chen's theorem: number of decompositions of 2n as the sum of either two primes, or a prime and a semiprime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 5, 4, 3, 3, 4, 5, 5, 5, 5, 5, 5, 4, 6, 6, 5, 6, 6, 4, 6, 7, 8, 8, 8, 7, 9, 8, 8, 7, 9, 8, 9, 8, 6, 9, 9, 10, 10, 9, 9, 12, 12, 11, 13, 12, 10, 11, 11, 9, 12, 11, 12, 11, 11, 12, 15, 14, 12, 12, 12

Offset: 1

Jean-François Alcover, Jan 13 2014

nonn

The first 15 terms from this sequence and from A045917 are identical.

			40 = 23+17 = 29+11 = 37+3, so a(20) = 3.
Compare with 40 = 23+17 = 29+11 = 31+9 = 37+3 and A045917(20) = 4.

Jean-François Alcover, Table of n, a(n) for n = 1..10000
Eric Weisstein's MathWorld, Chen's theorem
Wikipedia, Chen's theorem

Cf. A002375, A045917.

a[n_] := Count[IntegerPartitions[2*n, {2}], {p_, q_} /; PrimeQ[p] && (PrimeQ[q] || Length[FactorInteger[q]] == 2)]; Table[a[n], {n, 1, 100}]

A237168 Number of ways to write 2n - 1 = 2p + q with p, q, phi(p+1) - 1 and phi(p+1) + 1 all prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 2, 2, 2, 2, 3, 1, 3, 3, 2, 2, 4, 1, 1, 3, 2, 2, 3, 1, 1, 3, 2, 2, 2, 1, 2, 3, 2, 2, 4, 1, 4, 5, 2, 1, 6, 3, 3, 2, 3, 2, 5, 1, 2, 5, 3, 3, 4, 3, 2, 6, 4, 4, 5, 2, 3, 7, 2, 2

Offset: 1

Zhi-Wei Sun, Feb 04 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 12.
(ii) Any even number greater than 4 can be written as p + q with p, q, phi(p+2) - 1 and phi(p+2) + 1 all prime.
Part (i) implies both Lemoine's conjecture (cf. A046927) and the twin prime conjecture, while part (ii) unifies Goldbach's conjecture and the twin prime conjecture.

			a(9) = 1 since 2*9 - 1 = 2*7 + 3 with 7, 3, phi(7+1) - 1 = 3 and phi(7+1) + 1 = 5 all prime.
a(934) = 1 since 2*934 - 1 = 2*457 + 953 with 457, 953, phi(457+1) - 1 = 227 and phi(457+1) + 1 = 229 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Unification of Goldbach's conjecture and the twin prime conjecture, a message to Number Theory List, Jan. 29, 2014.

Cf. A000010, A000040, A001359, A002375, A002372, A006512, A046927, A072281, A236566, A237127, A237130.

PQ[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[PQ[Prime[k]+1]&&PrimeQ[2n-1-2*Prime[k]],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,70}]

A240708 Number of decompositions of 2n into an unordered sum of two terms of A240699.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3, 7, 9, 6, 5, 8, 7, 8, 11, 6, 5, 12, 4, 8, 11, 5, 8, 10, 5, 6, 13, 9, 6, 11, 7, 7, 14, 6, 8, 13, 5, 8, 11, 7, 9, 13, 8, 9, 14, 7, 7, 19, 6, 7, 12, 6, 9

Offset: 1

Lei Zhou, Apr 10 2014

The first different term of this sequence to A002375 is a(107).
Conjecture: for n >= 3, this sequence is always positive.
This is a stronger version of the Goldbach Conjecture.

			For n <= 106, refer to examples in A002375.
For n = 107, 2n=214. A240699 up to 214 gives {3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199}.  We have 214 = 17+197 = 23+191 = 41+173 = 47+167 = 83+131 = 101+113 = 107+107. Seven instances found. So a(107)=7.
Where as for A002375, there is one more instance as 3+211, however 211 is not a term in A240699.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A000040, A002375, A045917, A240699.

a240699 = {3}; Table[s = 2*n; While[a240699[[-1]] < s, p = a240699[[-1]]; While[p = NextPrime[p]; ((NextPrime[p] - p) > 6) && (6 < (p - NextPrime[p, -1]))]; AppendTo[a240699, p]]; pos = 0; ct = 0; While[pos++; pos <= Length[a240699], p = a240699[[pos]]; If[p <= n, If[MemberQ[a240699, s - p], ct++]]]; ct, {n, 1, 110}]

A240712 Number of decompositions of 2n into an unordered sum of two terms of A240710.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 1, 3, 4, 2, 2, 4, 2, 3, 5, 3, 3, 5, 2, 4, 6, 2, 4, 6, 2, 4, 6, 4, 4, 7, 4, 4, 8, 4, 4, 9, 3, 5, 7, 3, 5, 8, 4, 5, 8, 5, 6, 10, 5, 6, 12, 4, 5, 10, 3, 6, 9, 5, 5, 8, 6, 7, 11, 6, 5, 12, 3, 7, 11, 5, 7, 10, 5, 5, 13, 8

Offset: 1

Lei Zhou, Apr 10 2014

a(n) differs from A171611 beginning at term a(264).  To show the difference, the first 270 terms are listed.
Conjecture: a(n) > 0 for all n > 4.
This is a much stronger version of the Goldbach Conjecture.

			For n < 264, please refer to examples at A171611.
For n = 264, 2n=528. A240710 has terms {5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521} up to 528, where prime number 523 < 528 is not in the set, such that 528 = 5 + 523 is not counted in this sequence but is counted in A171611. So a(264) = A171611(264)-1 = 25-1 = 24.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A000040, A002375, A171611, A240710.

a240710 = {5}; Table[s = 2*n; While[a240710[[-1]] < s, p = a240710[[-1]]; While[p = NextPrime[p]; ok = 0; a1 = p - 12; a2 = p - 6; a3 = p + 6; a4 = p + 12; If[a1 > 0, If[PrimeQ[a1], ok = 1]]; If[a2 > 0, If[PrimeQ[a2], ok = 1]]; If[PrimeQ[a3], ok = 1]; If[PrimeQ[a4], ok = 1]; ok == 0]; AppendTo[a240710, p]]; pos = 0; ct = 0; While[pos++; pos <= Length[a240710], p = a240710[[pos]]; If[p <= n, If[MemberQ[a240710, s - p], ct++]]]; ct, {n, 1, 270}]

A258140 Number of ways to write 6*n + 2 as p^2 + q with p and q both prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 1, 1, 3, 3, 3, 0, 2, 2, 3, 2, 1, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 0, 4, 4, 5, 1, 4, 4, 2, 2, 2, 3, 3, 3, 5, 1, 3, 3, 4, 4, 1, 2, 3, 4, 3, 1, 5, 4, 5, 1, 1, 3, 4, 6, 4, 2, 3, 2, 6, 7, 3, 2, 2, 3, 5, 3, 4, 4, 4, 5, 2, 5, 2, 4, 6, 1, 5, 2, 5, 5, 2, 3, 3, 4, 4, 2, 4, 5, 6, 3, 2, 4, 5, 6

Offset: 0

Zhi-Wei Sun, May 22 2015

nonn

Conjecture: a(n) > 0 except for n = 0, 1, 12, 28, 102, 117, 168, 4079.

A258141 Number of ways to write n as p^2 + q with p, q and 2*p + 3 all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0

Offset: 1

Zhi-Wei Sun, May 22 2015

nonn

Conjecture: For any integer n > 0, we have a(n+r) > 0 for some r = 0,1,2,3,4,5.
We have verified this for n up to 10^8. See also A258139 for a weaker version of this conjecture.
The conjecture is somewhat similar to Goldbach's Conjecture. It implies that there are infinitely many primes p with 2*p + 3 prime.

			a(11) = 1 since 11 = 2^2 + 7 with 2, 7 and 2*2 + 3 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A002375, A023204, A258139, A258140.

Do[r=0;Do[If[PrimeQ[2Prime[k]+3]&&PrimeQ[n-Prime[k]^2],r=r+1],{k,1,PrimePi[Sqrt[n]]}];Print[n," ",r];Continue,{n,1,100}]

cp's OEIS Frontend

A284919 Even integers E such that there is no prime p < E with E - p and E + p both prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A227908 Number of ways to write 2*n = p + q with p, q and (p-1)^2 + q^2 all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A228956 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all the 2*n+2 numbers |i_0 +/- i_1|, |i_1 +/- i_2|, ..., |i_{n-1} +/- i_n|, |i_n +/- i_0| have the form (p-1)/2 with p an odd prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A230252 Number of ways to write n = x + y (x, y > 0) with 2*x + 1, x^2 + x + 1 and y^2 + y + 1 all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A235645 From Goldbach's conjecture and Chen's theorem: number of decompositions of 2n as the sum of either two primes, or a prime and a semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A237168 Number of ways to write 2*n - 1 = 2*p + q with p, q, phi(p+1) - 1 and phi(p+1) + 1 all prime, where phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A240708 Number of decompositions of 2n into an unordered sum of two terms of A240699.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A240712 Number of decompositions of 2n into an unordered sum of two terms of A240710.

Views

Author

Keywords

Comments

Examples

A237168 Number of ways to write 2n - 1 = 2p + q with p, q, phi(p+1) - 1 and phi(p+1) + 1 all prime, where phi(.) is Euler's totient function.