A237127 -id:A237127 - cp's OEIS frontend

A218829 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(m)) + 2 are both prime.

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

Offset: 1

Zhi-Wei Sun, Feb 05 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 22, 25, 38, 101, 273.
(ii) Each n = 2, 3, ... can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1 and prime(prime(m)) + 2 are all prime.
(iii) Any integer n > 5 can be written as k + m with k > 0 and m > 0 such that phi(k) - 1, phi(k) + 1 and prime(prime(m)) + 2 are all prime, where phi(.) is Euler's totient function.
(iv) If n > 2 is neither 10 nor 31, then n can be written as k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(prime(m))) + 2 are both prime.
(v) If n > 1 is not equal to 133, then n can be written as k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1 and prime(prime(prime(m))) + 2 are all prime.
Clearly, each part of the conjecture implies the twin prime conjecture.
We have verified part (i) for n up to 10^9. See the comments in A237348 for an extension of this part.

			a(3) = 1 since 3 = 2 + 1 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(1)) + 2 = prime(2) + 2 = 5 both prime.
a(22) = 1 since 22 = 20 + 2 with prime(20) + 2 = 71 + 2 = 73 and prime(prime(2)) + 2 = prime(3) + 2 = 5 + 2 = 7 both prime.
a(25) = 1 since 25 = 2 + 23 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(23)) + 2 = prime(83) + 2 = 431 + 2 = 433 both prime.
a(38) = 1 since 38 = 35 + 3 with prime(35) + 2 = 149 + 2 = 151 and prime(prime(3)) + 2 = prime(5) + 2 = 11 + 2 = 13 both prime.
a(101) = 1 since 101 = 98 + 3 with prime(98) + 2 = 521 + 2 = 523 and prime(prime(3)) + 2 = prime(5) + 2 = 11 + 2 = 13 both prime.
a(273) = 1 since 273 = 2 + 271 with prime(2) + 2 = 3 + 2 = 5 and prime(prime(271)) + 2 = prime(1741) + 2 = 14867 + 2 = 14869 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Andrei-Lucian Dragoi, The "Vertical" Generalization of the Binary Goldbach's Conjecture as Applied on "Iterative" Primes with (Recursive) Prime Indexes (i-primeths), Journal of Advances in Mathematics and Computer Science (2017), Vol. 25, No. 2, pp. 1-32.
Zhi-Wei Sun, Unification of Goldbach's conjecture and the twin prime conjecture, a message to Number Theory List, Jan. 29, 2014.
Zhi-Wei Sun, Super Twin Prime Conjecture, a message to Number Theory List, Feb. 6, 2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A001359, A002822, A006512, A072281, A182662, A199920, A236531, A236566, A236831, A236968, A237127, A237130, A237168, A237253, A237259, A237260, A237348.

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

A237130 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that both {3k - 1, 3k + 1} and {phi(m) - 1, phi(m) + 1} are twin prime pairs, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 04 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 8.
(ii) Any integer n > 6 can be written as k + m with k > 0 and m > 0 such that both {prime(k), prime(k) + 2} and {phi(m) - 1, phi(m) + 1} are twin prime pairs.
(iii) Each n = 12, 13, ... can be written as p + q (q > 0) with p, p + 6, phi(q) - 1 and phi(q) + 1 all prime.
(iv) If n > 2 is neither 10 nor 430, then n can be written as k + m with k > 0 and m > 0 such that both {3k - 1, 3*k + 1} and {6*m - 1, 6*m + 1} are twin prime pairs.
Note that each part of the above conjecture implies the twin prime conjecture.

			a(7) = 1 since 7 = 2 + 5 with 3*2 - 1 = 5, 3*2 + 1 =7, phi(5) - 1 = 3 and phi(5) + 1 = 5 all prime.
a(140) = 1 since 140 = 104 + 36 with 3*104 - 1 = 311, 3*104 + 1 = 313, phi(36) - 1 = 11 and phi(36) + 1 = 13 all prime.

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

Cf. A000010, A000040, A001359, A006512, A014574, A072281, A182662, A236531, A236566, A236831, A236968, A237127.

PQ[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[PrimeQ[3k-1]&&PrimeQ[3k+1]&&PQ[n-k],1,0],{k,1,n-1}]
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}]

A237253 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that phi(k) - 1, phi(k) + 1 and prime(prime(prime(m))) - 2 are all prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 05 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 7.
(ii) Any integer n > 22 can be written as k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(prime(m))) - 2 are both prime.
Note that either part of the conjecture implies the twin prime conjecture.

			 a(12) = 1 since 12 = 9 + 3 with phi(9) - 1 = 5, phi(9) + 1 = 7 and prime(prime(prime(3))) - 2 = prime(prime(5)) - 2 = prime(11) - 2 = 29 all prime.
a(103) = 1 since 103 = 73 + 30 with phi(73) - 1 = 71, phi(73) + 1 = 73 and prime(prime(prime(30))) - 2 = prime(prime(113)) - 2 = prime(617) - 2 = 4547 all prime.

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

Cf. A000010, A000040, A001359, A006512, A072281, A218829, A236531, A236566, A237127, A237130, A237168.

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

A237183 Primes p with phi(p+1) - 1 and phi(p+1) + 1 both prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

7, 11, 13, 17, 37, 41, 53, 61, 97, 151, 181, 197, 227, 233, 251, 269, 277, 397, 433, 457, 487, 541, 557, 571, 593, 619, 631, 719, 743, 769, 839, 857, 929, 941, 947, 953, 1013, 1021, 1049, 1061, 1063, 1201, 1237, 1277, 1307, 1321, 1367, 1481, 1511, 1549

Offset: 1

Zhi-Wei Sun, Feb 04 2014

nonn

According to part (i) of the conjecture in A237168, this sequence should have infinitely many terms.

			a(1) = 7 since 7, phi(7+1) - 1 = 3 and phi(7+1) + 1 = 5 are all prime, but phi(2+1) - 1 = phi(3+1) - 1 = phi(5+1) - 1 = 1 is not prime.

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

Cf. A000010, A000040, A001359, A006512, A072281, A237127, A237130, A237168.

PQ[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
n=0;Do[If[PQ[Prime[k]+1],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]
Select[Prime[Range[300]],And@@PrimeQ[EulerPhi[#+1]+{1,-1}]&] (* Harvey P. Dale, Mar 06 2014 *)

s=[]; forprime(p=2, 2000, if(isprime(eulerphi(p+1)-1) && isprime(eulerphi(p+1)+1), s=concat(s, p))); s \\ Colin Barker, Feb 04 2014

A237184 Number of ordered ways to write n = (1+(n mod 2))*p + q with p, q, phi(p+1) - 1 and phi(q-1) + 1 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 04 2014

nonn

Conjecture: a(n) > 0 for all n > 23.
This is stronger than Goldbach's conjecture and Lemoine's conjecture (cf. A046927).
We have verified the conjecture for n up to 3*10^6.

			a(10) = 1 since 10 = 7 + 3 with 7, 3, phi(7+1) - 1 = 3 and phi(3-1) + 1 = 2 all prime.
a(499) = 1 since 499 = 2*199 + 101 with 199, 101, phi(199+1) - 1 = 79 and phi(101-1) + 1 = 41 all prime.
a(869) = 1 since 869 = 2*433 + 3 with 433, 3, phi(433+1) - 1 = 179 and phi(3-1) + 1 = 2 all prime.

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

Cf. A000040, A002372, A002375, A039698, A046927, A078892, A237127, A237130, A237168, A237183.

pq[n_]:=PrimeQ[n]&&PrimeQ[EulerPhi[n+1]-1]
PQ[n_]:=PrimeQ[n]&&PrimeQ[EulerPhi[n-1]+1]
a[n_]:=Sum[If[pq[k]&&PQ[n-(1+Mod[n,2])k],1,0],{k,1,(n-1)/(1+Mod[n,2])}]
Table[a[n],{n,1,80}]

A237531 a(n) = |{0 < k < n/2: phi(k(n-k)) - 1 and phi(k(n-k)) + 1 are both prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 09 2014

nonn

Conjecture: a(n) > 0 for all n > 5.

Clearly, this implies the twin prime conjecture.

			a(12) = 1 since 12 = 3 + 9 with phi(3*9) - 1 = 17 and phi(3*9) + 1 = 19 both prime.
a(19) = 1 since 19 = 1 + 18 with phi(1*18) - 1 = 5 and phi(1*18) + 1 = 7 both prime.
a(86) = 1 since 86 = 8 + 78 with phi(8*78) - 1 = 191 and phi(8*78) + 1 = 193 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000010, A000040, A001359, A006512, A014574, A072281, A233547, A234200, A237127, A237130, A237168, A237523.

p[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[p[k(n-k)],1,0],{k,1,(n-1)/2}]
Table[a[n],{n,1,80}]

cp's OEIS Frontend

A218829 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that prime(k) + 2 and prime(prime(m)) + 2 are both prime.

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A237130 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that both {3*k - 1, 3*k + 1} and {phi(m) - 1, phi(m) + 1} are twin prime pairs, where phi(.) is Euler's totient function.

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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.

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A237253 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that phi(k) - 1, phi(k) + 1 and prime(prime(prime(m))) - 2 are all prime, where phi(.) is Euler's totient function.

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A237183 Primes p with phi(p+1) - 1 and phi(p+1) + 1 both prime, where phi(.) is Euler's totient function.

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A237184 Number of ordered ways to write n = (1+(n mod 2))*p + q with p, q, phi(p+1) - 1 and phi(q-1) + 1 all prime.

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A237531 a(n) = |{0 < k < n/2: phi(k*(n-k)) - 1 and phi(k*(n-k)) + 1 are both prime}|, where phi(.) is Euler's totient function.

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A237130 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that both {3k - 1, 3k + 1} and {phi(m) - 1, phi(m) + 1} are twin prime pairs, where phi(.) is Euler's totient function.

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.

A237531 a(n) = |{0 < k < n/2: phi(k(n-k)) - 1 and phi(k(n-k)) + 1 are both prime}|, where phi(.) is Euler's totient function.