A274010 - OEIS

%I #19 Mar 01 2020 12:13:16

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

%T 6,6,11,9,4,6,10,8,8,8,6,10,8,4,10,11,7,8,8,6,8,10,10,10,6,4,12,12,4,

%U 8,11,9,10,8,6,8,10,8,12,12,4,8,10,8,10,8,10

%N Boris Stechkin function: a(n) is the number of m with 2 <= m <= n and floor(n(m-1)/m) divisible by m-1.

%C Stechkin proves:

%C n-1 is prime iff a(n) = A000005(n).

%C n-1 and n+1 are twin primes, i.e., n is in A014574, iff a(n)+a(n+1) = 2*A000005(n).

%C If p < q are odd primes, then Sum_{k=p+1..q} (-1)^k a(k) = 0.

%D R. K. Guy, Unsolved Problems in Number Theory, Springer 2013, sec. A17.

%H Robert Israel, <a href="/A274010/b274010.txt">Table of n, a(n) for n = 0..10000</a>

%F Conjecture: a(n) = tau(n) + tau(n-1) - 2, for n>=2. - _Ridouane Oudra_, Feb 28 2020

%e For n = 6, the values of m are 2,3,5,6 so a(6) = 4.

%p N:= 1000: # to get a(0) to a(N)

%p A:= Vector(N):

%p for m from 2 to N do

%p   L:= [seq(seq(k*m+j,j=0..1),k=1..N/m)];

%p   if L[-1] > N then L:= L[1..-2] fi;

%p   A[L]:= map(`+`,A[L],1);

%p od:

%p 0, seq(A[i],i=1..N);

%t a[n_] := Sum[Boole[Divisible[Floor[n(m-1)/m], m-1]], {m, 2, n}];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Apr 29 2019 *)

%o (PARI) a(n)=sum(m=2,n,n*(m-1)\m%(m-1)==0) \\ _Charles R Greathouse IV_, Jun 08 2016

%Y Cf. A000005, A014574, A055004.

%K nonn

%O 0,4

%A _Robert Israel_, Jun 06 2016