A228428 -id:A228428 - cp's OEIS frontend

A231201 Number of ways to write n = x + y (x, y > 0) with 2^x + y prime.

0, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 2, 1, 2, 4, 4, 4, 5, 3, 2, 4, 1, 2, 1, 4, 4, 4, 2, 3, 4, 4, 4, 3, 2, 5, 4, 4, 4, 3, 5, 4, 5, 3, 4, 7, 6, 5, 2, 5, 3, 5, 7, 1, 3, 5, 5, 4, 6, 5, 4, 4, 5, 3, 1, 4, 7, 6, 5, 5, 4, 5, 7, 4, 5, 3, 5, 6, 8, 3, 4, 4, 6, 3, 5, 2, 2, 3, 6, 6, 4, 5, 6, 5, 5, 8, 6, 4, 7, 5, 4

Offset: 1

Zhi-Wei Sun, Nov 10 2013

Conjecture: (i) a(n) > 0 for all n > 1. Moreover, any integer n > 7 can be written as x + y with 0 < x < y such that 2^x + y is prime.
(ii) Every n = 2, 3, ... can be written as x + y (x, y > 0) with 2^x + y*(y+1)/2 prime.
(iii) Each integer n > 1 can be written as x + y (x, y > 0) with 2^x + y^2 - 1 prime. Also, any integer n > 1 not equal to 16 can be written as x + y (x, y > 0) with 2^x + y^4 - 1 prime.
We have verified part (i) of the conjecture for n up to 1.6*10^6. For example, 421801 = 149536 + 272265 with 2^149536 + 272265 prime.
We have extended our verification of part (i) of the conjecture for n up to 2*10^6. For example, 1657977 = 205494 + 1452483 with 2^205494 + 1452483 prime.  - Zhi-Wei Sun, Aug 30 2015
The verification of part (i) of the conjecture has been made for n up to 7.29*10^6. For example, we find that 5120132 = 250851 + 4869281 with 2^250851 + 4869281 a prime of 75514 decimal digits. - Zhi-Wei Sun, Nov 16 2015
We have finished the verification of part (i) of the conjecture for n up to 10^7. For example, we find that 9302003 = 311468 + 8990535 with 2^311468 + 8990535 a prime of 93762 decimal digits. - Zhi-Wei Sun, Jul 28 2016
In a paper published in 2017, the author announced a USD $1000 prize for the first solution to his conjecture that a(n) > 0 for all n > 1. - Zhi-Wei Sun, Dec 03 2017

			a(8) = 1 since 8 = 3 + 5 with 2^3 + 5 = 13 prime.
a(53) = 1 since 53 = 20 + 33 with 2^{20} + 33 = 1048609 prime.
a(64) = 1 since 64 = 13 + 51 with 2^{13} + 51 = 8243 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Write n = k + m with 2^k + m prime, a message to Number Theory List, Nov. 16, 2013.
Z.-W. Sun, On a^n+ bn modulo m, arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2015.
Z.-W. Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT].)

Cf. A000040, A000079, A228425, A228428, A228429, A228430, A228431, A231516, A231555, A231557, A231561, A231577, A231631.

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

A231555 Number of ways to write n = x + y (x, y > 0) with x*(x + 1) + F(y) prime, where F(y) denotes the y-th Fibonacci number (A000045).

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Nov 10 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Also, any integer n > 1 can be written as x + y (x, y > 0) with x + F(y) prime.
(ii) Each positive integer n not among 1, 7, 55 can be written as x + y (x, y > 0) with x*(x+1)/2 + F(y) prime. Also, any positive integer n not among 1, 10, 13, 20, 255 can be written as x + y (x, y > 0) with x^2 + F(y) prime.
We also have similar conjectures involving some second-order recurrences other than the Fibonacci sequence.

			a(19) = 1 since 19 = 17 + 2 with 17*18 + F(2) = 307 prime.
a(30) = 1 since 30 = 8 + 22 with 8*9 + F(22) = 17783 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Cf. A000040, A000045, A231201, A228425, A228428, A228429, A228430, A228431.

a[n_]:=Sum[If[PrimeQ[x(x+1)+Fibonacci[n-x]],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]

A228429 Number of ways to write n = x + y (x, y > 0) with p(39, x) + p(40, y) prime, where p(m, k) denotes the m-gonal number (m-2)k(k-1)/2 + k.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 4, 2, 2, 4, 1, 3, 1, 2, 5, 5, 1, 2, 3, 4, 3, 11, 4, 4, 2, 3, 4, 9, 6, 3, 5, 6, 3, 5, 4, 3, 9, 6, 3, 4, 7, 5, 13, 8, 3, 5, 5, 6, 13, 9, 9, 6, 3, 4, 6, 7, 3, 5, 5, 8, 5, 11, 8, 11, 8, 5, 10, 9, 5, 13, 9, 10, 11, 4, 7, 14, 12, 6, 11, 9, 5, 9, 15, 5, 14, 11, 6, 7, 8, 13, 14

Offset: 1

Zhi-Wei Sun, Nov 10 2013

nonn

By a conjecture in A228425, we should have a(n) > 0 for all n > 1.

Conjecture: For each m = 3, 4, ..., any sufficiently large integer n can be written as x + y (x, y > 0) with p(m, x) + p(m+1, y) prime.

			a(9) = 1 since 9 = 5 + 4 with p(39, 5) + p(40, 4) = 607 prime.
a(26) = 1 since 26 = 19 + 7 with p(39, 19) + p (40, 7) = 7151 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000040, A228425, A228428, A228430, A228431.

p[m_,x_]:=(m-2)x(x-1)/2+x
a[n_]:=Sum[If[PrimeQ[p[39,x]+p[40,n-x]],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]

A228430 Number of ways to write n = x + y (x, y > 0) with x^4 + y*(y+1)/2 prime.

Original entry on oeis.org

0, 1, 1, 2, 2, 0, 2, 3, 2, 3, 2, 3, 4, 1, 2, 1, 4, 3, 1, 1, 6, 4, 2, 4, 4, 4, 1, 5, 3, 5, 6, 4, 6, 3, 5, 5, 6, 3, 3, 5, 5, 5, 9, 3, 3, 11, 6, 7, 4, 8, 7, 12, 7, 5, 10, 4, 3, 8, 8, 3, 11, 6, 5, 10, 4, 6, 14, 6, 3, 9, 3, 12, 12, 9, 3, 11, 6, 10, 15, 7, 7, 8, 3, 6, 11, 8, 11, 10, 7, 3, 11, 10, 7, 11, 4, 6, 13, 11, 9, 8

Offset: 1

Zhi-Wei Sun, Nov 10 2013

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 6.

(ii) For any positive integer n not among 1, 3, 14, 25, there are positive integers x and y with x + y = n such that x^3 + y*(y+1)/2 is prime.

			a(14) = 1 since 14 = 4 + 10 with 4^4 + 10*11/2 = 311 prime.
a(27) = 1 since 27 = 22 + 5 with 22^4 + 5*6/2 = 234271 prime.

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

Cf. A000040, A000217, A000583, A228425, A228428, A228429.

a[n_]:=Sum[If[PrimeQ[x^4+(n-x)(n-x+1)/2],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]

A228431 Number of ordered ways to write n = x + y (x, y > 0) with p(3, x) + p(6, y) prime, where p(3, k) denotes the triangular number k(k+1)/2 and p(6, k) denotes the hexagonal number k(2k-1) = p(3, 2k-1).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 4, 4, 1, 4, 3, 1, 2, 3, 2, 5, 5, 3, 2, 3, 2, 4, 3, 3, 8, 4, 1, 3, 2, 2, 11, 5, 1, 5, 5, 4, 4, 5, 4, 7, 4, 3, 7, 6, 3, 9, 4, 2, 5, 4, 3, 12, 7, 2, 4, 10, 1, 7, 8, 4, 10, 7, 3, 10, 9, 5, 8, 5, 4, 10, 9, 5, 10, 9, 3, 12, 13, 4, 4, 9, 4, 11, 10, 5, 11, 16, 5, 10, 8, 5, 16, 8, 3, 11, 15

Offset: 1

Zhi-Wei Sun, Nov 10 2013

nonn

Conjecture: a(n) > 0 for all n > 1.
This implies that there are infinitely many primes each of which can be written as a sum of a triangular number and a hexagonal number.
See also A228425, A228428, A228429 and A228430 for more similar conjectures.

			a(14) = 1 since 14 = 10 + 4 with p(3, 10) + p(6, 4) = 83 prime.
a(38) = 1 since 38 = 31 + 7 with p(3, 31) + p (6, 7) = 587 prime.

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

Cf. A000040, A000217, A000384, A228425, A228428, A228429, A228430.

p[m_,x_]:=(m-2)x(x-1)/2+x
a[n_]:=Sum[If[PrimeQ[p[3,x]+p[6,n-x]],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A231201 Number of ways to write n = x + y (x, y > 0) with 2^x + y prime.

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A231555 Number of ways to write n = x + y (x, y > 0) with x*(x + 1) + F(y) prime, where F(y) denotes the y-th Fibonacci number (A000045).

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A228429 Number of ways to write n = x + y (x, y > 0) with p(39, x) + p(40, y) prime, where p(m, k) denotes the m-gonal number (m-2)*k*(k-1)/2 + k.

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A228430 Number of ways to write n = x + y (x, y > 0) with x^4 + y*(y+1)/2 prime.

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A228431 Number of ordered ways to write n = x + y (x, y > 0) with p(3, x) + p(6, y) prime, where p(3, k) denotes the triangular number k*(k+1)/2 and p(6, k) denotes the hexagonal number k*(2*k-1) = p(3, 2*k-1).

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A228429 Number of ways to write n = x + y (x, y > 0) with p(39, x) + p(40, y) prime, where p(m, k) denotes the m-gonal number (m-2)k(k-1)/2 + k.

A228431 Number of ordered ways to write n = x + y (x, y > 0) with p(3, x) + p(6, y) prime, where p(3, k) denotes the triangular number k(k+1)/2 and p(6, k) denotes the hexagonal number k(2k-1) = p(3, 2k-1).