A235924 -id:A235924 - cp's OEIS frontend

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

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 *)

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}]

A235934 Primes p with f(p), f(f(p)) and f(f(f(p))) all prime, where f(n) = prime(n) - n + 1.

Original entry on oeis.org

2, 3, 23, 311, 1777, 2341, 2861, 3329, 3833, 4051, 8753, 9007, 11587, 13093, 13309, 14551, 16001, 19687, 23143, 26993, 37309, 41981, 44131, 45491, 54623, 56431, 56821, 57991, 60223, 61643, 66413, 66883, 67511, 68767, 69029, 70003, 75743, 76261, 76819, 80021

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

By the general conjecture in A235925, this sequence should have infinitely many terms.

			a(3) = 23 with 23, f(23) = 61, f(61) = 223 and f(223) = 1187 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A234694, A234695, A235924, A235925.

f[n_]:=Prime[n]-n+1
p[k_]:=PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]]&&PrimeQ[f[f[f[Prime[k]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]

A236074 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/6 + 1, prime(2p) - 2prime(p) and prime(p) - 2*prime((p-1)/2) are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 1, 1, 2, 0, 1, 0, 0, 0, 0, 2, 2, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 4, 0, 0, 1, 2, 0, 1, 2, 1, 0, 3, 4, 0, 0, 0, 2, 1, 3, 1, 2, 0

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 116.
(ii) For any integer n > 196, there is a positive integer k < n such that p = phi(k) + phi(n-k)/6 + 1, prime(2*p) - 2*prime(p) and prime(p-1) - 2*prime((p-1)/2) are all prime.
Clearly, part (i) (or part (ii))  implies that there are infinitely many odd primes p with prime(2*p) - 2*prime(p) and prime(p) - 2*prime((p-1)/2) (or prime(p-1) - 2*prime((p-1)/2), resp.) both prime.

			a(30) = 1 since phi(4) + phi(26)/6 + 1 = 5, prime(2*5) - 2*prime(5) = 29 - 2*11 = 7 and prime(5) - 2*prime((5-1)/2) = 11 - 2*3 = 5 are all prime.
a(204) = 1 since phi(159) + phi(45)/6 + 1 = 109, prime(2*109) - 2*prime(109) = 1361 - 2*599 = 163, and prime(109) - 2*prime((109-1)/2) = 599 - 2*251 = 97 are all prime.

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

Cf. A000010, A000040, A234694, A235924, A236075.

PQ[n_]:=PQ[n]=n>0&&PrimeQ[n]
p[n_]:=PQ[n]&&PQ[Prime[2n]-2Prime[n]]&&PQ[Prime[n]-2*Prime[(n-1)/2]]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/6+1
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236138 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/3 - 1, prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) are all prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 19 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 73.
(ii) For any integer n > 472, there is a positive integer k < n such that p = phi(k) + phi(n-k)/4 - 1, prime(p) - 2*prime((p-1)/2) and prime(p) - 2*prime((p+1)/2) are all prime.
Clearly, part (i) of the conjecture implies that there are infinitely many odd primes p with prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) both prime, and part (ii) implies that there are infinitely many odd primes p with prime(p) - 2*prime((p-1)/2) and prime(p) - 2*prime((p+1)/2) both prime.

			a(20) = 1 since phi(11) + phi(9)/3 - 1 = 11, prime(10) - 10 = 29 - 10 = 19 and prime(10) - 2*prime(5) = 29 - 2*11 = 7 are all prime.
a(293) = 1 since phi(267) + phi(26)/3 - 1 = 176 + 12/3 - 1 = 179, prime(178) - 178 = 1061 - 178 = 883 and prime(178) - 2*prime(89) = 1061 - 2*461 = 139 are all prime.

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

Cf. A000010, A000040, A234694, A235924, A236074, A236097.

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

A235984 Primes p with f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))), f(f(f(f(f(p))))) all prime, where f(n) = prime(n) - n + 1.

Original entry on oeis.org

2, 3, 501187, 560029, 2076881, 2836003, 2907011, 8254787, 8822347, 10322189, 11329181, 11354641, 12307693, 14528069, 15801601, 17757427, 19023091, 24995669, 25871971

Offset: 1

Zhi-Wei Sun, Jan 17 2014

nonn

By the general conjecture in A235925, this sequence should have infinitely many terms.

			a(3) = 501187 with 501187, f(501187) = 6886357, f(6886357) = 113948711, f(113948711) = 2224096873, f(2224096873) =  50351471977 and f(50351471977) = 1303792228393 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..19
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A234694, A234695, A235924, A235925, A235934, A235935.

f[n_]:=Prime[n]-n+1
p[k_]:=PrimeQ[f[Prime[k]]]&&PrimeQ[f[f[Prime[k]]]]&&PrimeQ[f[f[f[Prime[k]]]]]&&PrimeQ[f[f[f[f[Prime[k]]]]]]&&PrimeQ[f[f[f[f[f[Prime[k]]]]]]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10^7}]

cp's OEIS Frontend

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

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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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A235934 Primes p with f(p), f(f(p)) and f(f(f(p))) all prime, where f(n) = prime(n) - n + 1.

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A236074 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/6 + 1, prime(2*p) - 2*prime(p) and prime(p) - 2*prime((p-1)/2) are all prime}|, where phi(.) is Euler's totient function.

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Mathematica

A236138 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/3 - 1, prime(p-1) - (p-1) and prime(p-1) - 2*prime((p-1)/2) are all prime}|, where phi(.) is Euler's totient function.

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A235984 Primes p with f(p), f(f(p)), f(f(f(p))), f(f(f(f(p)))), f(f(f(f(f(p))))) all prime, where f(n) = prime(n) - n + 1.

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Mathematica

A236074 a(n) = |{0 < k < n: p = phi(k) + phi(n-k)/6 + 1, prime(2p) - 2prime(p) and prime(p) - 2*prime((p-1)/2) are all prime}|, where phi(.) is Euler's totient function.