A233417 -id:A233417 - 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}]

A233390 a(n) = |{0 < k < n: 2^k - 1 + q(n-k) is prime}|, where q(.) is the strict partition function (A000009).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 4, 4, 2, 3, 2, 3, 3, 7, 4, 4, 5, 3, 4, 5, 5, 5, 6, 7, 6, 5, 4, 4, 9, 3, 6, 6, 5, 4, 7, 1, 4, 5, 6, 9, 6, 8, 6, 8, 4, 5, 8, 7, 4, 3, 4, 7, 6, 6, 3, 6, 5, 6, 4, 6, 8, 7, 8, 4, 5, 3, 6, 7, 7, 3, 10, 7, 5, 6, 10, 4, 8, 4, 6, 7, 6, 8, 10, 4, 6, 8, 9, 5, 6, 5, 7, 13, 5, 5, 6

Offset: 1

Zhi-Wei Sun, Dec 08 2013

nonn

Conjecture: a(n) > 0 for all n > 1.

We have verified this for n up to 150000. For n = 124669, the least positive integer k with 2^k - 1 + q(n-k) prime is 13413.

			a(6) = 1 since 2^2 - 1 + q(4) = 3 + 2 = 5 is prime.
a(10) = 1 since 2^4 - 1 + q(6) = 15 + 4 = 19 is prime.
a(41) = 1 since 2^{16} - 1 + q(25) = 65535 + 142 = 65677 is prime.
a(127) = 1 since 2^{21} - 1 + q(106) = 2097151 + 728260 = 2825411 is prime.
a(153) = 1 since 2^{70} - 1 + q(83) = 1180591620717411303423 + 101698 = 1180591620717411405121 is prime.
a(164) = 1 since 2^{26} - 1 + q(138) = 67108863 + 8334326 = 75443189 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..8000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2017.

Cf. A000009, A000040, A000225, A233307, A233346, A233359, A233360, A233393, A233417, A233439.

a[n_]:=Sum[If[PrimeQ[2^k-1+PartitionsQ[n-k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A238516 a(n) = |{0 < k < n: (p(k)+1)*p(n) + 1 is prime}|, where p(.) is the partition function (A000041).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 3, 4, 5, 5, 2, 4, 6, 5, 3, 3, 3, 4, 1, 7, 7, 2, 6, 3, 8, 7, 4, 1, 6, 3, 4, 5, 8, 4, 4, 2, 2, 4, 9, 7, 6, 3, 6, 4, 2, 6, 6, 3, 8, 5, 6, 4, 7, 7, 4, 8, 7, 9, 1, 6, 7, 7, 3, 3, 7, 2, 5, 4, 10, 8, 5, 1, 8, 9, 1, 4, 6, 7, 12, 3, 2, 4, 10, 4, 4

Offset: 1

Zhi-Wei Sun, Feb 28 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Also, for any integer n > 4 there is a positive integer k < n with (p(k)-1)*p(n) - 1 prime.
(ii) Let q(.) be the strict partition function (A000009). If n > 5, then p(n)*q(k) + 1 is prime for some 3 < k < n. If n > 6, then p(n)*q(k) - 1 is prime for some 0 < k < n. If n > 1, then q(n)*q(k) + 1 is prime for some 0 < k < n. If n > 3, then q(n)*q(k) - 1 is prime for some 0 < k < n.
We have verified that a(n) > 0 for all n = 2, 3, ..., 60000.

			a(4) = 1 since (p(1)+1)*p(4) + 1 = 2*5 + 1 = 11 is prime.
a(20) = 1 since (p(12)+1)*p(20) + 1 = 78*627 + 1 = 48907 is prime.
a(246) = 1 since (p(45)+1)*p(246) + 1 = 89135*169296722391554 + 1 = 15090263350371165791 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000009, A000040, A000041, A233417, A238457, A238509.

p[n_,k_]:=PrimeQ[PartitionsP[n]*(PartitionsP[k]+1)+1]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A229835 Number of ways to write n = (p - 1)/6 + q, where p is a prime, and q is a term of the sequence A000009.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 5, 5, 4, 6, 5, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 7, 10, 9, 6, 8, 6, 10, 8, 9, 7, 7, 10, 10, 9, 8, 7, 10, 7, 10, 3, 7, 12, 8, 10, 6, 8, 9, 6, 10, 8, 11, 7, 11, 8, 7, 9, 8, 12, 10, 8, 12, 7, 9, 10, 10, 8, 11, 10, 7, 10, 9, 14, 9, 9, 9, 8, 10, 10, 9, 7, 8, 9, 9, 8, 10, 9, 10, 10, 9, 7, 8, 7, 12, 8

Offset: 1

Zhi-Wei Sun, Dec 19 2013

nonn

Conjecture: a(n) > 0 for all n > 1. Also, any integer n > 1 can be written as (p + 1)/6 + q, where p is a prime and q is a term of A000009.

We have verified this for n up to 2*10^8. Note that 26128189 cannot be written as (p - 1)/4 + q with p a prime and q a term of A000009. Also, 65152682 cannot be written as (p + 1)/4 + q with p a prime and q a term of A000009.

			a(2) = 1 since 2 = (7 - 1)/ 6 + 1 with 7 prime, and 1 = A000009(i) for i = 0, 1, 2.
a(3) = 2 since 3 = (7 - 1 )/6 + 2 with 7 prime and 2 = A000009(3) = A000009(4), and 3 = (13 - 1 )/6 + 1 with 13 prime and 1 = A000009(i) for i = 0, 1, 2.

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

Cf. A000009, A000040, A232398, A232463, A233307, A233346, A233390, A233393, A233417, A233359.

Do[m=0;Do[If[PartitionsQ[k]>=n,Goto[aa]];If[k>1&&PartitionsQ[k]==PartitionsQ[k-1],Goto[bb]];
If[PrimeQ[6(n-PartitionsQ[k])+1],m=m+1];Label[bb];Continue,{k,1,2n}];
Label[aa];Print[n," ",m];Continue,{n,1,100}]

A236442 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that A000009(k) + A047967(m) is prime.

Original entry on oeis.org

0, 0, 1, 3, 3, 4, 3, 4, 3, 5, 2, 3, 4, 3, 1, 4, 4, 1, 2, 4, 4, 2, 4, 4, 3, 5, 8, 5, 4, 5, 7, 4, 3, 5, 2, 7, 5, 3, 5, 4, 5, 9, 4, 5, 5, 5, 8, 6, 7, 7, 8, 9, 5, 9, 7, 8, 13, 5, 4, 8, 4, 8, 3, 9, 9, 6, 7, 8, 6, 9, 7, 7, 4, 10, 7, 6, 8, 8, 5, 9, 6, 10, 5, 10, 12, 6, 11, 5, 5, 9, 8, 8, 4, 4, 11, 8, 8, 12, 6, 8

Offset: 1

Zhi-Wei Sun, Jan 26 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 2.
(ii) If n > 2 is neither 18 nor 30, then n can be written as k + m with k > 0 and m > 0 such that A000009(k)^2 + A047967(m)^2 is prime.
(iii) Any integer n > 4 can be written as k + m with k > 0 and m > 0 such that A000009(k)*A047967(m) - 1 (or A000009(k)*A047967(m) + 1) is prime.

			a(15) = 1 since 15 = 13 + 2 with A000009(13) + A047967(13) = 18 + 1 = 19 prime.
a(18) = 1 since 18 = 3 + 15 with A000009(3) + A047967(15) = 2 + 149 = 151 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, A000040, A047967, A232504, A233307, A233417, A233390.

p[n_,k_]:=PrimeQ[PartitionsQ[k]+(PartitionsP[n-k]-PartitionsQ[n-k])]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

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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A233390 a(n) = |{0 < k < n: 2^k - 1 + q(n-k) is prime}|, where q(.) is the strict partition function (A000009).

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A238516 a(n) = |{0 < k < n: (p(k)+1)*p(n) + 1 is prime}|, where p(.) is the partition function (A000041).

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A229835 Number of ways to write n = (p - 1)/6 + q, where p is a prime, and q is a term of the sequence A000009.

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A236442 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that A000009(k) + A047967(m) is prime.

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