A234530 -id:A234530 - cp's OEIS frontend

A234514 Number of ways to write n = k + m with k > 0 and m > 0 such that p = k + phi(m)/2 and q(p) + 1 are both prime, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).

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

Offset: 1

Zhi-Wei Sun, Dec 27 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 12.
(ii) For any integer n > 4, there is a prime p < n - 2 such that q(p + phi(n-p)/2) + 1 is prime.
Clearly, part (i) of the conjecture implies that there are infinitely many primes p with q(p) + 1 prime (cf. A234530).
We have verified part (i) for n up to 10^5.

			a(11) = 1 since 11 = 1 + 10 with 1 + phi(10)/2 = 3 and q(3) + 1 = 3 both prime.
a(27) = 1 since 27 = 7 + 20 with 7 + phi(20)/2 = 11 and q(11) + 1 = 13 both prime.
a(30) = 1 since 30 = 8 + 22 with 8 + phi(22)/2 = 13 and q(13) + 1 = 19 both prime.
a(38) = 1 since 38 = 21 + 17 with 21 + phi(17)/2 = 29 and q(29) + 1 = 257 both prime.
a(572) = 1 since 572 = 77 + 495 with 77 + phi(495)/2 = 197 and q(197) + 1 = 406072423 both prime.
a(860) = 1 since 860 = 523 + 337 with 523 + phi(337)/2 = 691 and q(691) + 1 = 712827068077888961 both prime.

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

Cf. A000009, A000010, A000040, A229835, A233307, A233390, A233417, A234451, A234470, A234475, A234503, A234504, A234530

f[n_,k_]:=k+EulerPhi[n-k]/2
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}]

A234475 Number of ways to write n = k + m with 2 < k <= m such that q(phi(k)*phi(m)/4) + 1 is 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, 0, 1, 1, 2, 2, 3, 3, 4, 4, 3, 4, 5, 5, 4, 7, 7, 6, 5, 5, 7, 3, 6, 7, 7, 5, 7, 4, 8, 4, 7, 7, 8, 7, 4, 5, 5, 4, 4, 5, 5, 6, 5, 4, 5, 3, 5, 4, 6, 6, 4, 6, 5, 4, 3, 6, 4, 9, 4, 8, 6, 7, 6, 8, 4, 7, 4, 7, 8, 9, 2, 3, 1, 8, 6, 9, 6, 6, 6, 6, 4, 7, 5, 8, 8, 4, 5, 5, 9, 7, 10, 4, 10, 3, 7, 8, 6

Offset: 1

Zhi-Wei Sun, Dec 26 2013

nonn

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

This implies that there are infinitely many primes p with p - 1 a term of A000009.

			a(6) = 1 since 6 = 3 + 3 with q(phi(3)*phi(3)/4) + 1 = q(1) + 1 = 2 prime.
a(76) = 1 since 76 = 18 + 58 with q(phi(18)*phi(58)/4) + 1 = q(42) + 1 = 1427 prime.
a(197) = 1 since 197 = 4 + 193 with q(phi(4)*phi(193)/4) + 1 = q(96) + 1 = 317789.
a(356) = 1 since 356 = 88 + 268 with q(phi(88)*phi(268)/4) + 1 = q(1320) + 1 = 35940172290335689735986241 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2525
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000010, A000040, A232504, A233307, A233346, A233547, A233390, A233393, A234309, A234310, A234337, A234344, A234347, A234359, A234360, A234361, A234451, A234470, A234514, A234530, A234567, A234569

f[n_,k_]:=PartitionsQ[EulerPhi[k]*EulerPhi[n-k]/4]+1
a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,3,n/2}]
Table[a[n],{n,1,100}]

A234567 Number of ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 + 1 and P(p-1) are both prime, where phi(.) is Euler's totient function and P(.) is the partition function (A000041).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 727.
(ii) For the strict partition function q(.) (cf. A000009), any n > 93 can be written as k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 + 1 and q(p-1) - 1 are both prime.
(iii) If n > 75 is not equal to 391, then n can be written as k + m with k > 0 and m > 0 such that f(k,m) - 1, f(k,m) + 1 and q(f(k,m)) + 1 are all prime, where f(k,m) = phi(k) + phi(m)/2.
Part (i) of the conjecture implies that there are infinitely many primes p with P(p-1) prime.

			a(21) = 1 since 21 = 6 + 15 with  phi(6) + phi(15)/2 + 1 = 7 and P(6) = 11 both prime.
a(700) = 1 since 700 = 247 + 453 with phi(247) + phi(453)/2 + 1 = 367 and P(366) = 790738119649411319 both prime.
a(945) = 1 since 945 = 687 + 258 with phi(687) + phi(258)/2 + 1 = 499 and P(498) = 2058791472042884901563 both 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, A000041, A234470, A234475, A234514, A234530.

f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2
q[n_,k_]:=PrimeQ[f[n,k]+1]&&PrimeQ[PartitionsP[f[n,k]]]
a[n_]:=Sum[If[q[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

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}]

A234644 Primes p with q(p) - 1 also prime, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

5, 11, 13, 17, 19, 23, 41, 43, 53, 59, 79, 103, 151, 191, 269, 277, 283, 373, 419, 521, 571, 577, 607, 829, 859, 1039, 2503, 2657, 2819, 3533, 3671, 4079, 4153, 4243, 4517, 4951, 4987, 5689, 5737, 5783, 7723, 8101, 9137, 9173, 9241, 9539, 11467, 12323, 12697, 15017, 15277, 15427, 15803, 16057, 17959, 18661

Offset: 1

Zhi-Wei Sun, Dec 29 2013

nonn

By the conjecture in A234615, this sequence should have infinitely many terms.
See A234647 for primes of the form q(p) - 1 with p prime.
See also A234530 for a similar sequence.

			a(1) = 5 since neither q(2) - 1 = 0 nor q(3) - 1 = 1 is prime, but q(5) - 1 = 2 is prime.
a(2) = 11 since q(7) - 1 = 4 is composite, but q(11) - 1 = 11 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..140
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000009, A000040, A234470, A234475, A234514, A234530, A234567, A234569, A234572, A234615, A234647.

q[k_]:=q[k]=PrimeQ[PartitionsQ[Prime[k]]-1]
n=0;Do[If[q[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10^5}]

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 *)

A234572 Primes of the form P(p-1), where p is a prime and P(.) is the partition function (A000041).

Original entry on oeis.org

2, 5, 11, 17977, 790738119649411319, 2058791472042884901563, 27833079238879849385687, 8121368081058512888507057, 675004412390512738195023734124239, 1398703012615213588677365804960180341, 16193798232344933888778097136641377589301, 204931453786129197483756438132982529754356479553, 3019564607799532159016586951616642980389816614848623, 22757918197082858017617136646280039394687006502870793231847, 1078734573992480956821414895441907729656949308800686938161281

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

Though the primes in this sequence are very rare, by the conjecture in A234567 there should be infinitely many such primes.

See A234569 for a list of known primes p with P(p-1) also prime.

			a(1) = 2 since 2 = P(3-1) with 2 and 3 both prime.
a(2) = 5 since 5 = P(5-1) with 5 prime.
a(3) = 11 since 11 = P(7-1) with 7 and 11 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..50

Cf. A000040, A000041, A049575, A233346, A234470, A234475, A234514, A234530, A234567, A234569

p[n_]:= A234569(n)
Table[PartitionsP[p[n]-1],{n,1,15}]

a(n) = A000041(A234569(n)-1).

A234366 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, 3, 13, 19, 257, 761, 2591, 32993, 70489, 173683, 570079, 3725411, 5010689, 132535703, 150473569, 406072423, 3328423937, 26114971541, 519999315041, 4743946406977, 704890732521793, 445433800804233383, 712827068077888961

Offset: 1

Zhi-Wei Sun, Dec 28 2013

nonn

Though the primes in this sequence are very rare, by the conjecture in A234514 there should be infinitely many such primes.

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

Zhi-Wei Sun, Table of n, a(n) for n = 1..120

Cf. A000009, A000040, A234514, A234530

p[n_]:=A234530(n)
Table[PartitionsQ[p[n]]+1,{n,1,35}]

a(n) = A000009(A234530(n)) + 1.

cp's OEIS Frontend

A234514 Number of ways to write n = k + m with k > 0 and m > 0 such that p = k + phi(m)/2 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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A234475 Number of ways to write n = k + m with 2 < k <= m such that q(phi(k)*phi(m)/4) + 1 is prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).

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A234567 Number of ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 + 1 and P(p-1) are both prime, where phi(.) is Euler's totient function and P(.) is the partition function (A000041).

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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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A234644 Primes p with q(p) - 1 also prime, where 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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