A234644 -id:A234644 - cp's OEIS frontend

A234530 Primes p with q(p) + 1 also prime, where q(.) is the strict partition function (A000009).

2, 3, 11, 13, 29, 37, 47, 71, 79, 89, 103, 127, 131, 179, 181, 197, 233, 271, 331, 379, 499, 677, 691, 757, 887, 911, 1019, 1063, 1123, 1279, 1429, 1531, 1559, 1637, 2251, 2719, 3571, 4007, 4201, 4211, 4297, 4447, 4651, 4967, 5953, 6131, 7937, 8233, 8599, 8819, 9013, 11003, 11093, 11813, 12251, 12889, 12953, 13487, 13687, 15259

Offset: 1

Zhi-Wei Sun, Dec 27 2013

nonn

By the conjecture in A234514, this sequence should have infinitely many terms.
It seems that a(n+1) < a(n) + a(n-1) for all n > 4.
See A234366 for primes of the form q(p) + 1 with p prime.
See also A234644 for a similar sequence.

			a(1) = 2 since 2 and q(2) + 1 = 2 are both prime.
a(2) = 3 since 3 and q(3) + 1 = 3 are both prime.
a(3) = 11 since 11 and q(11) + 1 = 13 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..150

Cf. A000009, A000040, A233346, A233393, A234366, A234470, A234475, A234514, A234567, A234569, A234572, A234615, A234644, A234647

n=0;Do[If[PrimeQ[PartitionsQ[Prime[k]]+1],n=n+1;Print[n," ",Prime[k]]],{k,1,10^5}]
Select[Prime[Range[2000]],PrimeQ[PartitionsQ[#]+1]&] (* Harvey P. Dale, Apr 23 2017 *)

A234569 Primes p with P(p-1) also prime, where P(.) is the partition function (A000041).

Original entry on oeis.org

3, 5, 7, 37, 367, 499, 547, 659, 1087, 1297, 1579, 2137, 2503, 3169, 3343, 4457, 4663, 5003, 7459, 9293, 16249, 23203, 34667, 39971, 41381, 56383, 61751, 62987, 72661, 77213, 79697, 98893, 101771, 127081, 136193, 188843, 193811, 259627, 267187, 282913, 315467, 320563, 345923, 354833, 459029, 482837, 496477, 548039, 641419, 647189

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

By the conjecture in A234567, this sequence should have infinitely many terms. It seems that a(n+1) < a(n) + a(n-1) for all n > 5.
The b-file lists all terms not exceeding the 500000th prime 7368787. Note that P(a(113)-1) is a prime having 2999 decimal digits.
See also A234572 for primes of the form P(p-1) with p prime.

			a(1) = 3 since P(2-1) = 1 is not prime, but P(3-1) = 2 is prime.
a(2) = 5 since P(5-1) = 5 is prime.
a(3) = 7 since P(7-1) = 11 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..113
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A000041, A049575, A233346, A234470, A234475, A234514, A234530, A234567, A234572, A234615, A234644

n=0;Do[If[PrimeQ[PartitionsP[Prime[k]-1]],n=n+1;Print[n," ",Prime[k]]],{k,1,10^6}]

A234615 Number of ways to write n = k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) - 1 are both prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 7.
(ii) Any integer n > 7 not equal to 15 can be written as k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) + 1 are both prime.
(iii) Any integer n > 83 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a square. Also, each integer n > 45 can be written as k + m with k > 0 and m > 0 such that prime(k) + phi(m)/2 is a triangular number.
Clearly, part (i) of this conjecture implies that there are infinitely many primes p with q(p) - 1 also prime (cf. A234644).

			a(6) = 1 since 6 = 2 + 4 with prime(2) + phi(4) = 5 and q(5) - 1 = 2 both prime.
a(58) = 1 since 58 = 12 + 46 with prime(12) + phi(46) = 59 and q(59) - 1 = 9791 both prime.
a(526) = 1 since 526 = 389 + 137 with prime(389) + phi(137) = 2819 and q(2819) - 1 = 326033386646595458662191828888146112979 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..8000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000010, A000040, A234475, A234514, A234530, A234567, A234569, A234572, A234644

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

A235344 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

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 *)

A234647 Primes of the form q(p) - 1, where p is a prime and q(.) is the strict partition function (A000009).

Original entry on oeis.org

2, 11, 17, 37, 53, 103, 1259, 1609, 5119, 9791, 70487, 570077, 20792119, 281138047, 23515017983, 35692320959, 48626519093, 3626048321047, 27077619952639, 1651411233432319, 10743948315198451, 13378670620050079, 39413984631175423, 58553713102334907283, 145464242180631569963, 25408177717067357968543, 1374387931601409538722802926765483199, 20557774525717988142856527912112710143, 326033386646595458662191828888146112979, 27403889354101748193301659902924397784656229

Offset: 1

Zhi-Wei Sun, Dec 29 2013

nonn

Though the primes in this sequence are very rare, by the conjecture in A234615 there should be infinitely many such primes.

See A234644 for a list of known primes p with q(p) - 1 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100

Cf. A000009, A000040, A234366, A234470, A234475, A234514, A234530, A234567, A234569, A234572, A234615, A234644

a(1) = 2 since 2 = q(5) - 1 with 2 and 5 both prime.

p[n_]:=A234615(n)
Table[PartitionsQ[p[n]]-1,{n,1,30}]

a(n) = A000009(A234615(n)) - 1.

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 *)

A234900 Primes p with P(p+1) also prime, where P(.) is the partition function (A000041).

Original entry on oeis.org

2, 3, 5, 131, 167, 211, 439, 2731, 3167, 3541, 4261, 7457, 8447, 18289, 22669, 23201, 23557, 35401, 44507, 76781, 88721, 108131, 126097, 127079, 136319, 141359, 144139, 159169, 164089, 177487, 202627, 261757, 271181, 282911, 291971, 307067, 320561, 389219, 481589, 482627, 602867, 624259, 662107, 682361, 818887, 907657, 914189, 964267, 1040191, 1061689

Offset: 1

Zhi-Wei Sun, Jan 01 2014

nonn

It seems that this sequence contains infinitely many terms.

See also A234569 for a similar sequence.

			a(1) = 2 since P(2+1) = 3 is prime.
a(2) = 3 since P(3+1) = 5 is prime.
a(3) = 5 since P(5+1) = 11 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..60

Cf. A000040, A000041, A049575, A233346, A234470, A234514, A234530, A234567, A234569, A234615, A234644.

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

cp's OEIS Frontend

A234530 Primes p with q(p) + 1 also prime, where q(.) is the strict partition function (A000009).

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A234569 Primes p with P(p-1) also prime, where P(.) is the partition function (A000041).

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A234615 Number of ways to write n = k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and q(p) - 1 are both prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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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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A234647 Primes of the form q(p) - 1, where p is a prime and q(.) is the strict partition function (A000009).

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A236418 Primes p with A047967(p) also prime.

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A234900 Primes p with P(p+1) also prime, where P(.) is the partition function (A000041).

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