A262403 - OEIS

A262403

Number of ways to write pi(T(n)) = pi(T(k)) + pi(T(m)) with 1 < k < m < n, where T(x) is the triangular number x*(x+1)/2, and pi(x) is the number of primes not exceeding x.

0, 0, 0, 0, 1, 1, 1, 2, 2, 1, 2, 1, 3, 4, 4, 4, 3, 3, 3, 3, 5, 4, 3, 4, 6, 4, 5, 2, 3, 6, 4, 1, 5, 8, 3, 2, 6, 1, 4, 5, 4, 2, 7, 2, 4, 5, 5, 5, 3, 4, 9, 9, 4, 5, 4, 8, 7, 6, 9, 4, 7, 5, 6, 2, 5, 9, 3, 8, 5, 6, 8, 5, 4, 3, 8, 4, 8, 7, 8, 5, 7, 8, 7, 4, 6, 2, 7, 7, 8, 7, 4, 5, 6, 4, 6, 4, 6, 4, 6, 6

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

OFFSET

1,8

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 4, and a(n) = 1 only for n = 5, 6, 7, 10, 12, 32, 38, 445, 727.

(ii) All those numbers pi(T(n)) (n = 1,2,3,...) are pairwise distinct. Moreover, if sum_{i=j,...,k}1/pi(T(i)) and sum_{r=s,...,t}1/pi(T(r)) with 1 < j <= k and j <= s <= t have the same fractional part but the ordered pairs (j,k) and (s,t) are different, then j = 2, k = 5 and s = t = 4.

Clearly, part (i) is related to addition chains, and the first assertion in part (ii) is an analog of Legendre's conjecture that pi(n^2) < pi((n+1)^2) for all n = 1,2,3,....

See also A262408 and A262409 for related conjectures involving powers.

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. (Cf. Section C6 on addition chains.)

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

LINKS

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

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

EXAMPLE

a(5) = 1 since pi(T(5)) = pi(15) = 6 = 2 + 4 = pi(3) + pi(10) = pi(T(2)) + pi(T(4)).

a(6) = 1 since pi(T(6)) = pi(21) = 8 = 2 + 6 = pi(3) + pi(15) = pi(T(2)) + pi(T(5)).

a(7) = 1 since pi(T(7)) = pi(28) = 9 = 3 + 6 = pi(6) + pi(15) = pi(T(3)) + pi(T(5)).

a(10) = 1 since pi(T(10)) = pi(55) = 16 = 2 + 14 = pi(3) + pi(45) = pi(T(2)) + pi(T(9)).

a(12) = 1 since pi(T(12)) = pi(78) = 21 = 3 + 18 = pi(6) + pi(66) = pi(T(3)) + pi(T(11)).

a(32) = 1 since pi(T(32)) = pi(528) = 99 = 9 + 90 = pi(28) + pi(465) = pi(T(7)) + pi(T(30)).

a(38) = 1 since pi(T(38)) = pi(741) = 131 = 32 + 99 = pi(136) + pi(528) = pi(T(16)) + pi(T(32)).

a(445) = 1 since pi(T(445)) = pi(99235) = 9526 = 2963 + 6563 = pi(27028) + pi(65703) = pi(T(232)) + pi(T(362)).

a(727) = 1 since pi(T(727)) = pi(264628) = 23197 = 10031 + 13166 = pi(105111) + pi(141778) = pi(T(458)) + pi(T(532)).

MATHEMATICA

f[n_]:=PrimePi[n(n+1)/2]

T[m_, n_]:=Table[f[k], {k, m, n}]

Do[r=0; Do[If[MemberQ[T[k+1, n-1], f[n]-f[k]], r=r+1]; Continue, {k, 2, n-2}]; Print[n, " ", r]; Continue, {n, 1, 100}]

CROSSREFS

Cf. A000217, A000720, A111208, A262408, A262409, A262439, A262446.

Sequence in context: A340057 A303476 A187201 * A343491 A326952 A109909

Adjacent sequences: A262400 A262401 A262402 * A262404 A262405 A262406

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Sep 21 2015

STATUS

approved