A233307 -id:A233307 - cp's OEIS frontend

A233360 Primes of the form L(k) + q(m) with k > 0 and m > 0, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 79, 83, 101, 103, 107, 127, 131, 149, 151, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 307, 337, 347, 349, 379, 397, 401, 419, 421, 449, 463, 487, 523, 541, 571, 643, 647, 661

Offset: 1

Zhi-Wei Sun, Dec 08 2013

nonn

Conjecture: The sequence has infinitely many terms.

This follows from the conjecture in A233359.

			a(1) = 2 since L(1) + q(1) = 1 + 1 = 2.
a(2) = 3 since L(1) + q(3) = 1 + 2 = 3.
a(3) = 5 since L(2) + q(3) = 3 + 2 = 5.

Zhi-Wei Sun, Table of n, a(n) for n = 1..270

Cf. A000009, A000040, A000032, A000204, A232504, A233307, A233346, A233359.

n=0
Do[Do[If[LucasL[j]>=Prime[m],Goto[aa],
Do[If[PartitionsQ[k]==Prime[m]-LucasL[j],
n=n+1;Print[n," ",Prime[m]];Goto[aa]];If[PartitionsQ[k]>Prime[m]-LucasL[j],Goto[bb]];Continue,{k,1,2*(Prime[m]-LucasL[j])}]];
Label[bb];Continue,{j,1,2*Log[2,Prime[m]]}];
Label[aa];Continue,{m,1,125}]

A233439 a(n) = |{0 < k < n: prime(k)^2 + 4*prime(n-k)^2 is prime}|.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 2, 1, 3, 4, 4, 8, 4, 6, 3, 1, 7, 3, 8, 5, 2, 9, 2, 11, 8, 7, 5, 4, 8, 7, 8, 8, 8, 7, 5, 9, 5, 10, 9, 7, 13, 9, 11, 10, 14, 5, 11, 10, 10, 11, 12, 7, 13, 10, 10, 8, 15, 11, 12, 11, 13, 14, 6, 12, 11, 22, 21, 5, 15, 7, 13, 15, 17, 15, 10, 16, 11, 13, 14, 12, 17, 12, 16, 16, 19, 22, 17, 12, 19, 17, 19, 17, 16, 17, 18, 20, 19, 17, 10, 16

Offset: 1

Zhi-Wei Sun, Dec 09 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 3.
(ii) For any integer n > 10, prime(j)^3 + 2*prime(n-j)^2 is prime for some 0 < j < n, and prime(k)^3 + 2*prime(n-k)^3 is prime for some 0 < k < n.
(iii) If n > 5, then prime(k)^3 + 2*p(n-k)^3 is prime for some 0 < k < n, where p(.) is the partition function (A000041). If n > 2, then prime(k)^3 + 2*q(n-k)^3 is prime for some 0 < k < n, where q(.) is the strict partition function (A000009).

			a(4) = 1 since prime(3)^2 + 4*prime(1)^2 = 5^2 + 4*2^2 = 41 is prime.
a(6) = 1 since prime(5)^2 + 4*prime(1)^2 = 11^2 + 4*2^2 = 137 is prime.
a(8) = 1 since prime(3)^2 + 4*prime(5)^2 = 5^2 + 4*11^2 = 509 is prime.
a(16) = 1 since prime(6)^2 + 4*prime(10)^2 = 13^2 + 4*29^2 = 3533 is prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
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], 2014-2016.

Cf. A000040, A002313, A220413, A232174, A232269, A232465, A232502, A233150, A233204, A233206, A233296, A233307, A233346.

a[n_]:=Sum[If[PrimeQ[Prime[k]^2+4*Prime[n-k]^2],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

A233360 Primes of the form L(k) + q(m) with k > 0 and m > 0, where L(k) is the k-th Lucas number (A000204), and q(.) is the strict partition function (A000009).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A233439 a(n) = |{0 < k < n: prime(k)^2 + 4*prime(n-k)^2 is prime}|.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica