A263107 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2/2) + pi(3*m^2/2), where pi(x) denotes the number of primes not exceeding x.

1, 1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 1, 4, 5, 1, 6, 3, 4, 5, 2, 4, 5, 2, 5, 5, 3, 4, 5, 4, 6, 2, 4, 4, 6, 3, 4, 5, 2, 6, 8, 1, 5, 6, 2, 4, 7, 3, 6, 5, 2, 6, 7, 1, 4, 7, 3, 6, 5, 3, 7, 4, 5, 5, 6, 5, 5, 4, 2, 6, 10, 3, 4, 6, 5, 3, 6, 5, 7, 3

Offset

1,4

Comments

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

We have verified this for n up to 3*10^5.

It seems that a(n) = 1 only for n = 1, 2, 3, 6, 19, 22, 48, 60, 396, 1076, 3033, 3889, 4741, 6804, 7919, 9604, 16938, 19169, 32533, 59903, 100407.

Links

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

Example

a(1) = 1 since 1 = 1 + 0 = pi(2^2/2) + pi(3*1^2/2).
a(2) = 1 since 2 = 2 + 0 = pi(3^2/2) + pi(3*1^2/2).
a(3) = 1 since 3 = 0 + 3 = pi(1^2/2) + pi(3*2^2/2).
a(6) = 1 since 6 = 0 + 6 = pi(1^2/2) + pi(3*3^2/2).
a(19) = 1 since 19 = 7 + 12 = pi(6^2/2) + pi(3*5^2/2).
a(22) = 1 since 22 = 1 + 21 = pi(2^2/2) + pi(3*7^2/2).
a(48) = 1 since 48 = 1 + 47 = pi(2^2/2) + pi(3*12^2/2).
a(60) = 1 since 60 = 25 + 35 = pi(14^2/2) + pi(3*10^2/2).
a(396) = 1 since 396 = 334 + 62 = pi(67^2/2) + pi(3*14^2/2).
a(1076) = 1 since 1076 = 47 + 1029 = pi(21^2/2) + pi(3*74^2/2).
a(3033) = 1 since 3033 = 7 + 3026 = pi(6^2/2) + pi(3*136^2/2).
a(3889) = 1 since 3889 = 1808 + 2081 = pi(176^2/2) + pi(3*110^2/2).
a(4741) = 1 since 4741 = 4699 + 42 = pi(301^2/2) + pi(3*11^2/2).
a(6804) = 1 since 6804 = 6047 + 757 = pi(346^2/2) + pi(3*62^2/2).
a(7919) = 1 since 7919 = 4049 + 3870 = pi(277^2/2) + pi(3*156^2/2).
a(9604) = 1 since 9604 = 4754 + 4850 = pi(303^2/2) + pi(3*177^2/2).
a(16938) = 1 since 16938 = 2223 + 14715 = pi(198^2/2) + pi(3*327^2/2).
a(19169) = 1 since 19169 = 6510 + 12659 = pi(361^2/2) + pi(3*301^2/2).
a(32533) = 1 since 32533 = 1768 + 30765 = pi(174^2/2) + pi(3*490^2/2).
a(59903) = 1 since 59903 = 59210 + 693 = pi(1213^2/2) + pi(3*59^2/2).
a(100407) = 1 since 100407 = 7554 + 92853 = pi(392^2/2) + pi(3*894^2/2).

Mathematica

s[n_]:=s[n]=PrimePi[3n^2/2]
t[n_]:=t[n]=PrimePi[n^2/2]
Do[r=0; Do[If[s[k]>n, Goto[bb]]; Do[If[t[j]>n-s[k], Goto[aa]]; If[t[j]==n-s[k], r=r+1]; Continue, {j, 1, n-s[k]+1}]; Label[aa]; Continue, {k, 1, n}]; Label[bb]; Print[n, " ", r]; Continue, {n, 1, 100}]

Crossrefs

Cf. A000290, A000720, A262995, A262999, A263001, A263020, A263100.

Sequence in context: A316842 A329466 A347627 * A286528 A112200 A112221

Adjacent sequences: A263104 A263105 A263106 * A263108 A263109 A263110

Keyword

nonn

Author

Zhi-Wei Sun, Oct 09 2015

Status

approved