

A263001


Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k*(k+1)) + pi(m*(m+1)/2), where pi(x) denotes the number of primes not exceeding x.


5



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

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 1, 4, 6.
We have verified this for n up to 10^5.
See also A262995, A262999 and A263020 for similar conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


EXAMPLE

a(1) = 1 since 1 = pi(1*2) + pi(1*2/2).
a(4) = 1 since 4 = pi(1*2) + pi(3*4/2).
a(6) = 1 since 6 = pi(2*3) + pi(3*4/2).


MATHEMATICA

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


CROSSREFS

Cf. A000217, A000720, A002378, A111208, A262995, A262999, A263020.
Sequence in context: A321887 A046051 A025812 * A109698 A029231 A025808
Adjacent sequences: A262998 A262999 A263000 * A263002 A263003 A263004


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 07 2015


STATUS

approved



