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Number of ordered ways to write n = p + q (q > 0) with p, 2*p^2 - 1 and 2*q^2 - 1 all prime.
+10
8
0, 0, 0, 1, 2, 2, 1, 1, 3, 3, 2, 1, 4, 3, 4, 2, 4, 3, 4, 5, 4, 2, 3, 6, 3, 3, 3, 5, 2, 3, 3, 3, 1, 2, 4, 2, 2, 3, 3, 1, 5, 2, 3, 3, 7, 3, 5, 4, 6, 3, 5, 6, 5, 5, 3, 6, 2, 5, 5, 3, 4, 5, 6, 2, 6, 6, 5, 1, 5, 3, 3, 3, 2, 2, 5, 6, 5, 1, 5, 6, 4, 4, 6, 6, 1, 5, 5, 4, 3, 4, 3, 3, 6, 5, 4, 1, 5, 7, 2, 4
COMMENTS
Conjecture: a(n) > 0 for all n > 3.
We have verified this for n up to 2*10^7.
EXAMPLE
a(7) = 1 since 7 = 3 + 4 with 3, 2*3^2 - 1 = 17, 2*4^2 - 1 = 31 all prime.
a(40) = 1 since 40 = 2 + 38, and 2, 2*2^2 - 1 = 7 , 2*38^2 - 1 = 2887 are all prime.
a(68) = 1 since 68 = 43 + 25, and all the three numbers 43, 2*43^2 - 1 = 3697 and 2*25^2 - 1 = 1249 are prime.
MATHEMATICA
a[n_]:=Sum[If[PrimeQ[2Prime[i]^2-1]&&PrimeQ[2(n-Prime[i])^2-1], 1, 0], {i, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 100}]
Number of ordered ways to write n = x*(x+1)/2 + y with y*(y+1)/2 + 1 prime, where x and y are nonnegative integers.
+10
5
1, 1, 1, 3, 1, 1, 3, 2, 2, 2, 3, 2, 3, 3, 1, 2, 3, 3, 3, 2, 2, 5, 3, 2, 2, 4, 2, 2, 4, 2, 2, 2, 3, 1, 3, 2, 3, 2, 2, 4, 3, 1, 3, 5, 2, 3, 4, 5, 2, 4, 2, 3, 3, 2, 3, 5, 4, 2, 4, 1, 4, 3, 5, 4, 3, 5, 3, 4, 3, 3, 6, 4, 2, 5, 4, 3, 4, 5, 5, 2, 4, 4, 2, 3, 6, 4, 2, 3, 5, 4, 3, 5, 1, 4, 3, 6, 3, 5, 7, 3
COMMENTS
Conjecture: a(n) > 0 for all n > 0. Moreover, if n > 0 is not among 1, 3, 60, then there are positive integers x and y with x*(x+1)/2 + y = n such that y*(y+1)/2 + 1 is prime.
EXAMPLE
a(6) = 1 since 6 = 2*3/2 + 3 with 3*4/2 + 1 = 7 prime.
a(60) = 1 since 60 = 0*1/2 + 60 with 60*61/2 + 1 = 1831 prime.
MATHEMATICA
T[n_]:=n(n+1)/2
a[n_]:=Sum[If[PrimeQ[T[n-T[i]]+1], 1, 0], {i, 0, (Sqrt[8n+1]-1)/2}]
Table[a[n], {n, 1, 100}]
Number of ways to write n = p + q with p and (p+1)*q/2 + 1 both prime.
+10
4
0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 2, 1, 4, 1, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 4, 2, 2, 8, 1, 6, 6, 2, 3, 2, 3, 5, 5, 5, 1, 5, 3, 7, 5, 1, 7, 10, 1, 3, 4, 8, 5, 3, 3, 3, 5, 8, 4, 10, 2, 9, 3, 3, 4, 7, 5, 9, 5, 4, 3, 15, 4, 12, 7, 4, 5, 9, 3, 11, 4, 6, 5, 9, 5, 6, 12, 6, 5, 8, 1, 4, 8, 5, 13, 9, 2, 6, 5, 8, 4
COMMENTS
Conjecture: a(n) > 0 for all n > 3.
We have verified this for n up to 10^8.
We also have some similar conjectures, for example, any integer n > 3 not equal to 17 or 66 can be written as p + q with p and (p+1)*q/2 - 1 both prime.
EXAMPLE
a(15) = 1 since 15 = 5 + 10 with 5 and (5+1)*10/2+1 = 31 both prime.
a(30) = 1 since 30 = 2 + 28 with 2 and (2+1)*28/2+1 = 43 both prime.
MATHEMATICA
a[n_]:=Sum[If[PrimeQ[(Prime[i]+1)(n-Prime[i])/2+1], 1, 0], {i, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 100}]
Number of ways to write 2*n - 1 = p + q with p, p + 6 and q^4 + 1 all prime, where q is a positive integer.
+10
3
0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 4, 3, 3, 4, 4, 3, 3, 4, 1, 5, 4, 3, 5, 5, 5, 4, 6, 4, 5, 5, 3, 3, 5, 4, 4, 2, 6, 8, 5, 4, 6, 7, 5, 5, 7, 6, 5, 7, 4, 6, 6, 3, 6, 5, 7, 6, 4, 6, 7, 6, 2, 7, 6, 2, 5, 5, 3, 7, 7, 5, 7, 9, 6, 7, 4, 6, 6, 4, 3, 9, 7, 4, 9, 9, 6, 5, 10, 8, 5, 9, 6, 7, 8, 4
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 3. Also, any odd number greater than 6 can be written as p + q (q > 0) with p, p + 6 and q^2 + 1 all prime.
(ii) Any integer n > 1 can be written as x + y (x, y > 0) with x^4 + 1 and y^2 + y + 1 both prime.
(iii) Each integer n > 2 can be expressed as x + y (x, y > 0) with 4*x^2 + 3 and 4*y^2 -3 both prime.
Either of parts (i) and (ii) implies that there are infinitely many primes of the form x^4 + 1.
EXAMPLE
a(6) = 2 since 2*6-1 = 5 + 6 = 7 + 4, and 5, 5+6 = 11, 7, 7+6 = 13, 6^4+1 = 1297 and 4^4+1 = 257 are all prime.
a(25) = 1 since 2*25-1 = 47 + 2, and 47, 47+6 = 53, 2^4+1 = 17 are all prime.
MATHEMATICA
a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[(2n-1-Prime[i])^4+1], 1, 0], {i, 1, PrimePi[2n-2]}]
Table[a[n], {n, 1, 100}]
PROG
(PARI) a(n)=my(s, p=5, q=7); forprime(r=11, 2*n+4, if(r-p==6&&isprime((2*n-1-p)^4+1), s++); if(r-q==6&&isprime((2*n-1-q)^4+1), s++); p=q; q=r); s \\ Charles R Greathouse IV, Oct 14 2013
Number of ways to write n = a + b + c with 0 < a <= b <= c such that {a^2+a-1, a^2+a+1}, {b^2+b-1, b^2+b+1}, {c^2+c-1, c^2+c+1} are twin prime pairs.
+10
2
0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 3, 2, 2, 2, 2, 3, 4, 4, 3, 5, 5, 4, 6, 6, 4, 4, 4, 3, 3, 4, 1, 2, 3, 4, 4, 5, 6, 6, 7, 6, 6, 7, 6, 4, 3, 5, 4, 4, 3, 5, 5, 6, 8, 6, 7, 11, 7, 6, 9, 8, 4, 8, 6, 5, 7, 5, 4, 8, 10, 5, 7, 9, 6, 10, 6, 7, 7, 7, 4, 4, 8, 5, 5, 4, 6, 9, 7, 7, 7, 7, 7, 8
COMMENTS
Conjecture: a(n) > 0 for all n > 5.
This implies that there are infinitely many twin prime pairs of the form {x^2 + x - 1, x^2 + x + 1}.
See also A230514 for a similar conjecture.
EXAMPLE
a(8) = 1 since 8 = 2 + 3 + 3, and {2*3 - 1, 2*3 + 1} = {5, 7} and {3*4 - 1, 3*4 + 1} = {11, 13} are twin prime pairs.
a(39) = 1 since 39 = 3 + 15 + 21, and {3*4 - 1, 3*4 + 1} = {11, 13}, {15*16 - 1, 15*16 + 1} = {239, 241}, {21*22 - 1, 21*22 + 1} = {461, 463} are twin prime pairs.
MATHEMATICA
pp[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[n(n+1)+1]
a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j], 1, 0], {i, 1, n/3}, {j, i, (n-i)/2}]
Table[a[n], {n, 1, 100}]
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