A234695 -id:A234695 - cp's OEIS frontend

A236832 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q and r terms of A234695.

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

Offset: 1

Zhi-Wei Sun, Jan 31 2014

nonn

Conjecture: a(n) > 0 for all n > 3.

This is stronger than Goldbach's weak conjecture which was finally proved by H. A. Helfgott in 2013.

			a(4) = 1 since 2*4 - 1 = 2 + 2 + 3 with 2 and 3 terms of A234695.
a(5) = 2 since 2*5 - 1 = 2 + 2 + 5 = 3 + 3 + 3 with 2, 3, 5 terms of A234695.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012-2013.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A068307, A230219, A234695, A235189.

p[n_]:=PrimeQ[Prime[n]-n+1]
q[n_]:=PrimeQ[n]&&p[n]
a[n_]:=Sum[If[p[Prime[i]]&&p[Prime[j]]&&q[2n-1-Prime[i]-Prime[j]],1,0],{i,1,PrimePi[(2n-1)/3]},{j,i,PrimePi[(2n-1-Prime[i])/2]}]
Table[a[n],{n,1,100}]

A235703 Number of ordered ways to write n = p + q with p a term of A234695 and q a term of A235592.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 3, 3, 3, 3, 4, 2, 3, 2, 2, 4, 3, 3, 3, 3, 5, 5, 4, 3, 4, 4, 3, 5, 3, 1, 5, 5, 3, 5, 2, 4, 4, 3, 5, 4, 4, 4, 6, 5, 4, 6, 5, 3, 6, 6, 6, 5, 2, 3, 4, 3, 5, 5, 4, 5, 6, 4, 3, 6, 4, 3, 6, 4, 4, 5, 3, 5, 3, 5, 6, 6, 5, 3, 6, 4, 2, 4, 1, 4, 5, 4, 5, 7, 5, 4, 6, 9, 5, 6, 4, 2, 6, 6, 2, 6

Offset: 1

Zhi-Wei Sun, Jan 14 2014

nonn

Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 4, 30, 83.

			 a(4) = 1 since 4 = 2 + 2 with 2, prime(2) - 2 + 1 = 2 and 2*3 - prime(2) = 3 all prime.
a(30) = 1 since 30 = 3 + 27 with 3, prime(3) - 3 + 1 = 3 and 27*28 - prime(27) = 756 - 103 = 653 all prime.
a(83) = 1 since 83 = 13 + 70 with 13, prime(13) - 13 + 1 = 29 and 70*71 - prime(70) = 4970 - 349 = 4621 all prime.

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

Cf. A000040, A234695, A235330, A235508, A235592.

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

A234694 a(n) = |{0 < k < n: p = k + prime(n-k) and prime(p) - p + 1 are both prime}|.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 29 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 9. Also, for any integer n > 51 there is a positive integer k < n such that p = k + prime(n-k) and prime(p) + p + 1 are both prime.
(ii) If n > 9 (or n > 21), then there is a positive integer k < n such that m - 1 and prime(m) + m (or prime(m) - m, resp.) are both prime, where m = k + prime(n-k).
(iii) If n > 483, then for some 0 < k < n both prime(m) + m and prime(m) - m are prime, where m = k + prime(n-k).
(iv) If n > 3, then there is a positive integer k < n such that prime(k + prime(n-k)) + 2 is prime.
Clearly, part (i) of the conjecture implies that there are infinitely many primes p with prime(p) - p + 1 (or prime(p) + p + 1) also prime.
See A234695 for primes p with prime(p) - p + 1 also prime.

			a(5) = 1 since 2 + prime(3) = 7 and prime(7) - 6 = 11 are both prime.
a(25) = 1 since 20 + prime(5) = 31 and prime(31) - 30 = 97 are both prime.
a(27) = 1 since 18 + prime(9) = 41 and prime(41) - 40 = 139 are both prime.
a(45) = 1 since 6 + prime(39) = 173 and prime(173) - 172 = 859 are both prime.
a(49) = 1 since 26 + prime(23) = 109 and prime(109) - 108 = 491 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A014688, A014689, A014692, A064269, A064270, A232861, A233150, A233183, A233206, A233296, A234695.

f[n_,k_]:=k+Prime[n-k]
q[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[Prime[f[n,k]]-f[n,k]+1]
a[n_]:=Sum[If[q[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236119 Primes p with prime(p) - p - 1 and prime(p) - p + 1 both prime.

Original entry on oeis.org

5, 17, 23, 41, 71, 83, 173, 293, 337, 353, 563, 571, 719, 811, 911, 953, 1201, 1483, 1579, 1877, 2081, 2089, 2309, 2579, 2749, 2803, 3329, 3343, 3511, 3691, 3779, 3851, 3881, 3907, 4021, 4049, 4093, 4657, 4813, 5051, 5179, 5333, 5519, 5591, 6053, 6547, 6841, 7151, 7723, 8209

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

By the conjecture in A236097, this sequence should have infinitely many terms.

			a(1) = 5 since neither prime(2) - 2 - 1 = 0 nor prime(3) - 3 - 1 = 1 is prime, but prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A001359, A006512, A014574, A234695, A235925, A236097.

p[n_]:=PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
n=0;Do[If[p[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1100}]

s=[]; forprime(p=2, 10000, if(isprime(prime(p)-p-1) && isprime(prime(p)-p+1), s=concat(s, p))); s \\ Colin Barker, Jan 19 2014

A235189 Number of ways to write n = (1 + (n mod 2))*p + q with p < n/2 such that p, q and prime(p) - p + 1 are all prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 4, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 4, 2, 2, 4, 4, 1, 3, 2, 3, 2, 3, 3, 4, 3, 4, 3, 3, 3, 5, 2, 4, 4, 2, 2, 6, 2, 2, 4, 1, 1, 5, 4, 5, 4, 4, 2, 4, 3, 3, 3, 4, 4, 5, 4, 5, 4, 3, 2, 4, 2, 3, 6, 5, 3, 6, 3, 5, 5, 2, 3, 9, 3, 3, 5, 3, 1, 6, 3

Offset: 1

Zhi-Wei Sun, Jan 04 2014

nonn

Conjecture: a(n) > 0 for all n > 6.

This implies both Goldbach's conjecture (A045917) and Lemoine's conjecture (A046927). For primes p with prime(p) - p + 1 also prime, see A234695.

			a(10) = 1 since 10 = 3 + 7 with 3, 7 and prime(3) - 3 + 1 = 3 all prime.
a(28) = 1 since 28 = 5 + 23 with 5, 23 and prime(5) - 4 = 7 all prime.
a(61) = 1 since 61 = 2*7 + 47 with 7, 47 and prime(7) - 6 = 11 all prime.
a(98) = 1 since 98 = 31 + 67 with 31, 67 and prime(31) - 30 = 97 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A045917, A046927, A234694, A234695, A235187.

p[n_]:=PrimeQ[Prime[n]-n+1]
a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[n-(1+Mod[n,2])*Prime[k]],1,0],{k,1,PrimePi[(n-1)/2]}]
Table[a[n],{n,1,100}]

A235925 Primes p with q = prime(p) - p + 1 and r = prime(q) - q + 1 both prime.

Original entry on oeis.org

2, 3, 5, 17, 23, 41, 61, 83, 181, 271, 311, 337, 757, 953, 1277, 1451, 1753, 1777, 2027, 2081, 2341, 2707, 2713, 2749, 2819, 2861, 2879, 2909, 2971, 3121, 3163, 3329, 3697, 3779, 3833, 3881, 3907, 4027, 4051, 4129, 4363, 4549, 5333, 5483, 5659, 5743, 5813, 5897, 6029, 6053

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

By the conjecture in A235924, this sequence should have infinitely many terms.

Conjecture: For any integer m > 1, there are infinitely many chains p(1) < p(2) < ... < p(m) of m primes with p(k+1) = prime(p(k)) - p(k) + 1 for all 0 < k < m.

			a(1) = 2 since prime(2) - 2 + 1 = 2 is prime.
a(2) = 3 since prime(3) - 3 + 1 = 3 is prime.
a(3) = 5 since 5, prime(5) - 5 + 1 = 7 and prime(7) - 7 + 1 = 11 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A234695, A235924.

f[n_]:=Prime[n]-n+1
n=0;Do[If[PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]
bpQ[n_]:=Module[{q=Prime[n]-n+1},AllTrue[{q,Prime[q]-q+1},PrimeQ]]; Select[Prime[Range[800]],bpQ](* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 07 2014 *)

A235924 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/3 + 1, q = prime(p) - p + 1 and r = prime(q) - q + 1 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

Conjecture: a(n) > 0 for all n > 37.

This implies that there are infinitely many primes p with q = prime(p) - p + 1 and r = prime(q) - q + 1 both prime.

			a(20) = 1 since phi(6) + phi(14)/3 + 1 = 5, prime(5) - 4 = 11 - 4 = 7 and prime(7) - 6 = 17 - 6 = 11 are all prime.
a(77) = 1 since phi(59) + phi(18)/3 + 1 = 61, prime(61) - 60 = 283 - 60 = 223 and prime(223) - 222 = 1409 - 222 = 1187 are all prime.
a(1471) = 1 since phi(25) + phi(1446)/3 + 1 = 181, prime(181) - 180 = 1087 - 180 = 907 and prime(907) - 906 = 7057 - 906 = 6151 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A234694, A234695.

q[n_]:=Prime[n]-n+1
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/3+1
p[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[q[f[n,k]]]&&PrimeQ[q[q[f[n,k]]]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

Conjecture: a(n) > 0 for all n > 31.

This implies that there are infinitely many primes p with {prime(p) - p - 1, prime(p) - p + 1} a twin prime pair.

			a(20) = 1 since phi(2) + phi(18)/2 + 1 = 5, prime(5) - 5 - 1 = 5 and prime(5) - 5 + 1 = 7 are all prime.
a(36) = 1 since phi(21) + phi(15)/2 + 1 = 17, prime(17) - 17 - 1 = 41 and prime(17) - 17 + 1 = 43 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A001359, A006512, A014574, A234694, A234695, A235924, A236074, A236119.

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n-1]&&PrimeQ[Prime[n]-n+1]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2+1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A235661 Primes p with p*(p+1) - prime(p) prime.

Original entry on oeis.org

2, 3, 5, 11, 19, 29, 37, 41, 53, 61, 71, 89, 131, 137, 149, 157, 233, 263, 271, 281, 293, 331, 337, 359, 389, 431, 433, 439, 457, 487, 499, 571, 617, 631, 659, 701, 739, 751, 761, 809, 859, 877, 907, 911, 1009, 1019, 1031, 1033, 1087, 1093

Offset: 1

Zhi-Wei Sun, Jan 13 2014

nonn

This sequence is a subsequence of A235592.

By the conjecture in A232353, this sequence should have infinitely many terms.

			2 is a term because 2*3 - prime(2) = 3 is prime.
3 is a term because 3*4 - prime(3) = 7 is prime.
5 is a term because 5*6 - prime(5) = 19 is prime.

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

Cf. A000040, A232353, A234695, A235592, A235613, A235614.

PQ[n_]:=PrimeQ[n(n+1)-Prime[n]]
n=0;Do[If[PQ[Prime[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]
Select[Prime[Range[200]],PrimeQ[#(#+1)-Prime[#]]&] (* Harvey P. Dale, Feb 26 2025 *)

A238766 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that prime(prime(k)) - prime(k) + 1, prime(prime(2k+1)) - prime(2k+1) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 05 2014

nonn

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 11, 44, 55, 149, 371.

This suggests that there are infinitely many prime pairs {p, q} with 2*pi(p) + 1 = pi(q) such that prime(p) - p + 1 and prime(q) - q + 1 are both prime.

			a(3) = 1 since 3 = 1 + 2 with prime(prime(1)) - prime(1) + 1 = prime(2) - 2 + 1 = 2, prime(prime(2*1+1)) - prime(2*1+1) + 1 = prime(5) - 5 + 1 = 7 and prime(prime(2)) - prime(2) + 1 = prime(3) - 3 + 1 = 3 all prime.
a(371) = 1 since 371 = 66 + 305 with prime(prime(66)) - prime(66) + 1 = prime(317) - 317 + 1 = 2099 - 316 = 1783, prime(prime(2*66+1)) - prime(2*66+1) + 1 = prime(751) - 751 + 1 = 5701 - 750 = 4951 and prime(prime(305)) - prime(305) + 1 = prime(2011) - 2011 + 1 = 17483 - 2010 = 15473 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A234694, A234695, A235189, A236832, A238134, A238756, A238776.

pq[k_]:=PrimeQ[Prime[Prime[k]]-Prime[k]+1]
a[n_]:=Sum[If[pq[k]&&pq[2k+1]&&pq[n-k],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

cp's OEIS Frontend

A236832 Number of ways to write 2*n - 1 = p + q + r (p <= q <= r) with p, q and r terms of A234695.

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A235703 Number of ordered ways to write n = p + q with p a term of A234695 and q a term of A235592.

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A234694 a(n) = |{0 < k < n: p = k + prime(n-k) and prime(p) - p + 1 are both prime}|.

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A236119 Primes p with prime(p) - p - 1 and prime(p) - p + 1 both prime.

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A235189 Number of ways to write n = (1 + (n mod 2))*p + q with p < n/2 such that p, q and prime(p) - p + 1 are all prime.

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A235925 Primes p with q = prime(p) - p + 1 and r = prime(q) - q + 1 both prime.

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A235924 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/3 + 1, q = prime(p) - p + 1 and r = prime(q) - q + 1 are all prime}|, where phi(.) is Euler's totient function.

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A236097 a(n) = |{0 < k < n-2: p = phi(k) + phi(n-k)/2 + 1, prime(p) - p - 1 and prime(p) - p + 1 are all prime}|, where phi(.) is Euler's totient function.

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A238766 Number of ordered ways to write n = k + m (k > 0 and m > 0) such that prime(prime(k)) - prime(k) + 1, prime(prime(2k+1)) - prime(2k+1) + 1 and prime(prime(m)) - prime(m) + 1 are all prime.