A235343 -id:A235343 - cp's OEIS frontend

A235344 Numbers m with m - 1, m + 1 and q(m) + 1 all prime, where q(.) is the strict partition function (A000009).

4, 6, 18, 42, 72, 102, 270, 282, 312, 618, 1032, 1062, 1320, 1950, 2082, 3528, 7350, 7488, 10332, 15138, 17388, 21600, 40038, 44700, 134922, 156258, 187908, 243708, 339138, 389568, 495360, 610920, 761712, 911292, 916218, 943800, 1013532, 1217472, 1312602

Offset: 1

Zhi-Wei Sun, Jan 06 2014

Clearly, any term after the first term 4 is a multiple of 6. By part (i) of the conjecture in A235343, this sequence should have infinitely many terms. The prime q(a(51)) + 1 = q(3235368) + 1 has 1412 decimal digits.
See A235356 for primes of the form q(m) + 1 with m - 1 and m + 1 both prime.
See also A235346 for a similar sequence.

			a(1) = 4 since 4 - 1, 4 + 1 and q(4) + 1 = 3 are all prime.
a(2) = 6 since 6 - 1, 6 + 1 and q(6) + 1 = 5 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..51
Zhi-Wei Sun, Twin primes and the strict partition function, a message to Number Theory List, Jan. 15, 2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000040, A014574, A234530, A234569, A234644, A235343, A235346, 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}]

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

Original entry on oeis.org

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 *)

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}]

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A235344 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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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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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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