A274010 Boris Stechkin function: a(n) is the number of m with 2 <= m <= n and floor(n(m-1)/m) divisible by m-1.

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

Offset: 0

Author

Robert Israel, Jun 06 2016

Keywords

• nonn

Comments

Stechkin proves:

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

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

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

Examples

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

References

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

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000005, A014574, A055004.

Programs

Maple

N:= 1000: # to get a(0) to a(N)
A:= Vector(N):
for m from 2 to N do
  L:= [seq(seq(k*m+j,j=0..1),k=1..N/m)];
  if L[-1] > N then L:= L[1..-2] fi;
  A[L]:= map(`+`,A[L],1);
od:
0, seq(A[i],i=1..N);

Mathematica

a[n_] := Sum[Boole[Divisible[Floor[n(m-1)/m], m-1]], {m, 2, n}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 29 2019 *)

PARI

a(n)=sum(m=2,n,n*(m-1)\m%(m-1)==0) \\ Charles R Greathouse IV, Jun 08 2016

Formula

Conjecture: a(n) = tau(n) + tau(n-1) - 2, for n>=2. - Ridouane Oudra, Feb 28 2020