



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

1,5


COMMENTS

Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 4, 6, 11, 21, 54, 253, 325.
This is slightly stronger than part (ii) of the conjecture in A262439.
I have verified the conjecture for n up to 10^5.  ZhiWei Sun, Sep 27 2015


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. (Cf. Section C6 on addition chains.)
ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.


LINKS



EXAMPLE

a(4) = 1 since pi(4*5/2+1) = pi(11) = 5 = 1 + 4 = pi(2) + pi(7) = pi(1*2/2+1) + pi(3*4/2+1).
a(6) = 1 since pi(6*7/2+1) = pi(22) = 8 = 2 + 6 = pi(4) + pi(16) = pi(2*3/2+1) + pi(5*6/2+1).
a(11) = 1 since pi(11*12/2+1) = pi(67) = 19 = 5 + 14 = pi(11) + pi(46) = pi(4*5/2+1) + pi(9*10/2+1).
a(21) = 1 since pi(21*22/2+1) = pi(232) = 50 = 14 + 36 = pi(46) + pi(154) = pi(9*10/2+1) + pi(17*18/2+1).
a(54) = 1 since pi(54*55/2+1) = pi(1486) = 235 = 30 + 205 = pi(121) + pi(1276) = pi(15*16/2+1) + pi(50*51/2+1).
a(253) = 1 since pi(253*254/2+1) = pi(32132) = 3447 = 747 + 2700 = pi(5672) + pi(24311) = pi(106*107/2+1) + pi(220*221/2+1).
a(325) = 1 since pi(325*326/2+1) = pi(52976) = 5406 = 1446 + 3960 = pi(12091) + pi(37402) = pi(155*156/2+1) + pi(37402*37403/2+1).


MATHEMATICA

f[n_]:=PrimePi[n(n+1)/2+1]
T[n_]:=Table[f[k], {k, 1, n}]
Do[r=0; Do[If[2*f[k]>=f[n], Goto[aa]]; If[MemberQ[T[n], f[n]f[k]], r=r+1]; Continue, {k, 1, n1}]; Label[aa]; Print[n, " ", r]; Continue, {n, 1, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



