login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A263100 Number of ordered pairs (k, m) with k > 0 and m > 0 such that n = pi(k^2) + pi(m^2/2), where pi(x) denotes the number of primes not exceeding x. 2

%I #12 Oct 09 2015 09:42:07

%S 1,2,1,3,2,3,3,2,4,2,6,2,5,2,5,4,4,4,4,5,3,5,5,4,4,6,6,1,7,4,6,4,4,7,

%T 6,4,5,5,5,6,6,4,6,3,7,6,5,6,6,6,5,5,6,4,7,8,4,3,10,2,6,6,6,6,7,5,5,9,

%U 3,6,8,6,7,5,5,6,7,7,8,3,9,3,10,2,7,9,7,2,7,8,5,8,4,6,9,5,7,6,5,7

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

%C Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 28.

%C (ii) Any integer n > 0 can be written as pi(k^2) + pi((m^2+1)/2) with k and m positive integers.

%C (iii) Each n = 1,2,3,... can be written as pi(k^2/2) + pi((m^2+1)/2) with k and m positive integers.

%C See also A262995, A262999, A263001 and A263020 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A263100/b263100.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 1 since 1 = 0 + 1 = pi(1^2) + pi(2^2/2).

%e a(3) = 1 since 3 = 2 + 1 = pi(2^2) + pi(2^2/2).

%e a(28) = 1 since 28 = 11 + 17 = pi(6^2) + pi(11^2/2).

%t s[n_]:=s[n]=PrimePi[n^2]

%t t[n_]:=t[n]=PrimePi[n^2/2]

%t 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}];

%t Label[bb]; Print[n, " ", r]; Continue, {n,1,100}]

%Y Cf. A000290, A000720, A038107, A262995, A262999, A263001, A263020.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Oct 09 2015

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)