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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
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

(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.

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

LINKS

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

EXAMPLE

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

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

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

MATHEMATICA

s[n_]:=s[n]=PrimePi[n^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, A038107, A262995, A262999, A263001, A263020.

Sequence in context: A317963 A219609 A087825 * A261388 A253281 A029206

Adjacent sequences:  A263097 A263098 A263099 * A263101 A263102 A263103

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 09 2015

STATUS

approved

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Last modified June 22 21:29 EDT 2021. Contains 345393 sequences. (Running on oeis4.)