A235344 -id:A235344 - cp's OEIS frontend

A235346 Numbers m with m - 1, m + 1 and q(m) - 1 all prime, where q(.) is the strict partition function (A000009).

6, 240, 420, 1032, 1062, 1278, 2238, 4020, 12612, 15972, 19890, 22110, 34500, 44772, 134370, 141768, 145602, 191142, 217368, 290658, 436482, 454578, 464382, 618030, 668202, 849348, 888870, 964260, 1179150, 1364970, 1446900, 1593498, 1737102, 1866438, 2291802, 3237432

Offset: 1

Zhi-Wei Sun, Jan 06 2014

Clearly, each term is a multiple of 6. By the conjecture in A235358 (which is part (ii) of the conjecture in A235343), this sequence should have infinitely many terms. q(a(36)) - 1 = q(3237432) - 1 is a prime having 1412 decimal digits.
See A235357 for primes of the form q(m) - 1 with m - 1 and m + 1 both prime.
See also A235344 for a similar sequence.

			a(1) = 6 since q(4) - 1 = 1 is not a prime, and 6 - 1, 6 + 1 and q(6) - 1 = 3 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..36
Zhi-Wei Sun, Twin primes and the strict partition function, a message to Number Theory List, Jan. 15, 2014.

Cf. A000009, A000040, A014574, A234530, A234569, A234644, A235343, A235344, A235356, A235357, A235358.

f[k_]:=PartitionsQ[Prime[k]+1]-1
n=0;Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[f[k]],n=n+1;Print[n," ",Prime[k]+1]],{k,1,10000}]
Select[Mean/@Select[Partition[Prime[Range[10000]],2,1],#[[2]]-#[[1]] == 2&],PrimeQ[PartitionsQ[#]-1]&] (* The program generates the first 14 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Feb 01 2022 *)

A235343 a(n) = |{0 < k < n: f(n,k) - 1, f(n,k) + 1 and q(f(n,k)) + 1 are all prime with f(n,k) = phi(k) + phi(n-k)/4}|, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 06 2014

nonn

Conjecture: (i) a(n) > 0 for all n >= 60.
(ii) For any integer n > 1234, there is a positive integer k < n such that g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime, where g(n,k) = phi(k) + phi(n-k)/8.
Clearly, part (i) implies that there are infinitely many primes of the form q(m) + 1 with m - 1 and m + 1 also prime, and part (ii) implies that there are infinitely many primes of the form q(m) - 1 with m - 1 and m + 1 also prime. As log q(m) is asymptotically equivalent to  pi*sqrt(m/3), the conjecture is much stronger than the twin prime conjecture.
We have verified parts (i) and (ii) for n up to 100000 and 60000 respectively.

			a(50) = 1 since phi(34) + phi(16)/4 = 18 with 18 - 1, 18 + 1 and q(18) + 1 = 47 all prime.
a(215) = 1 since phi(87) + phi(128)/4 = 72 with 72 - 1, 72 + 1 and q(72) + 1 = 36353 all prime.
a(645) = 1 since phi(365) + phi(280)/4 = 312 with 312 - 1, 312 + 1 and q(312) + 1 = 207839472391 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. A000009, A000010, A000040, A001359, A006512, A014574, A234514, A234567, A234615, A235344, A235346, A235356, A235357, A235358.

f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/4
p[n_,k_]:=PrimeQ[f[n,k]-1]&&PrimeQ[f[n,k]+1]&&PrimeQ[PartitionsQ[f[n,k]]+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A235356 Primes of the form q(m) + 1 with m - 1 and m + 1 both prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

3, 5, 47, 1427, 36353, 525017, 24782061071, 46193897033, 207839472391, 58195383726460417, 20964758762885249107969, 47573613463034233651201, 35940172290335689735986241, 39297101749677990678763409480449, 538442167350331131544523981355841

Offset: 1

Zhi-Wei Sun, Jan 07 2014

nonn

Though the primes in this sequence are very rare, by part (i) of the conjecture in A235343 there should be infinitely many such primes.
See A235344 for a list of known numbers m with m - 1, m + 1 and q(m) + 1 all prime.
See also A235357 for a similar sequence.

			a(1) = 3 since 3 = q(4) + 1 with 4 - 1 and 4 + 1 both prime.
a(2) = 5 since 5 = q(6) + 1 with 6 - 1 and 6 + 1 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..30

Cf. A000009, A000040, A014574, A235343, A235344, A235346, A235357.

f[n_]:=A235344(n)
Table[PartitionsQ[f[n]]+1,{n,1,15}]

a(n) = A000009(A235344(n)) + 1.

A235357 Primes of the form q(m) - 1 with m - 1 and m + 1 both prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

3, 4919887991, 28253252977151, 20964758762885249107967, 47573613463034233651199, 12796446358667905839216959, 10712934162879755412803989317623807, 33014011446550388413724585366558782455972162239

Offset: 1

Zhi-Wei Sun, Jan 07 2014

Though the primes in this sequence are very rare, by part (ii) of the conjecture in A235343, there should be infinitely many such primes.
See A235346 for a list of known numbers m with m - 1, m + 1 and q(m) - 1 all prime.
See also A235356 for a similar sequence.

			a(1) = 3 since 3 = q(6) - 1 with 6 - 1 and 6 + 1 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..25

Cf. A000009, A000040, A014574, A235343, A235344, A235346, A235356.

g[n_]:=A235346(n)
Table[PartitionsQ[g[n]]-1,{n,1,10}]

a(n) = A000009(A235346(n)) - 1.

A235358 a(n) = |{0 < k < n: g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime with g(n,k) = phi(k) + phi(n-k)/8}|, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jan 07 2014

nonn

Conjecture: a(n) > 0 for all n > 1234.
See also part (ii) of the conjecture in A235343.
We have verified the conjecture for n up to 100000.

			a(50) = 1 since phi(10) + phi(40)/4 = 6 with 6 - 1, 6 + 1 and q(6) - 1 = 3 all prime.

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

Cf. A000009, A000010, A000040, A001359, A006512, A014574, A234514, A234567, A234615, A235343, A235344, A235346, A235356, A235357.

f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/8
p[n_,k_]:=PrimeQ[f[n,k]-1]&&PrimeQ[f[n,k]+1]&&PrimeQ[PartitionsQ[f[n,k]]-1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A236418 Primes p with A047967(p) also prime.

Original entry on oeis.org

13, 23, 43, 53, 71, 83, 107, 257, 269, 313, 1093, 2659, 2851, 3527, 8243, 20173, 20717, 24329, 26161, 26237, 31583, 53611, 60719, 74717, 83401, 118259, 118369, 130817, 133811, 145109, 152381, 169111, 178613, 183397, 205963

Offset: 1

Zhi-Wei Sun, Jan 25 2014

nonn

According to the conjecture in A236417, this sequence should have infinitely many terms.

			a(1) = 13 with 13 and A047967(13) = 83 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..42

Cf. A000040, A047967, A234530, A234569, A234644, A235344, A235346, A236413, A236417, A236419, A236440.

pq[n_]:=PrimeQ[n]&&PrimeQ[PartitionsP[n]-PartitionsQ[n]]
n=0;Do[If[pq[m],n=n+1;Print[n," ",m]],{m,1,10000}]
Select[Prime[Range[20000]],PrimeQ[PartitionsP[#]-PartitionsQ[#]]&] (* Harvey P. Dale, Jan 02 2022 *)

cp's OEIS Frontend

A235346 Numbers m with m - 1, m + 1 and q(m) - 1 all prime, where q(.) is the strict partition function (A000009).

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A235343 a(n) = |{0 < k < n: f(n,k) - 1, f(n,k) + 1 and q(f(n,k)) + 1 are all prime with f(n,k) = phi(k) + phi(n-k)/4}|, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

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A235356 Primes of the form q(m) + 1 with m - 1 and m + 1 both prime, where q(.) is the strict partition function (A000009).

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A235357 Primes of the form q(m) - 1 with m - 1 and m + 1 both prime, where q(.) is the strict partition function (A000009).

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A235358 a(n) = |{0 < k < n: g(n,k) - 1, g(n,k) + 1 and q(g(n,k)) - 1 are all prime with g(n,k) = phi(k) + phi(n-k)/8}|, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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A236418 Primes p with A047967(p) also prime.

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