A036431 a(n) = number of positive integers b which, when added to the number of their divisors, tau(b), gives n.

0, 1, 0, 1, 1, 0, 2, 0, 1, 1, 0, 2, 1, 1, 1, 0, 0, 2, 2, 0, 2, 0, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 1, 1, 0, 0, 1, 3, 2, 0, 0, 1, 2, 0, 2, 0, 0, 1, 1, 3, 1, 1, 0, 0, 2, 1, 0, 2, 1, 0, 2, 2, 1, 1, 0, 1, 0, 0, 3, 0, 1, 1, 2, 2, 1, 0, 0, 2, 0, 0, 3, 1, 0, 1, 1, 3, 0, 0, 1, 2, 2, 0, 0, 0, 1, 2, 1, 2, 2, 0, 0, 1, 1, 1, 2

Offset: 1

Author

Simon Colton (simonco(AT)cs.york.ac.uk)

Keywords

• nonn

Comments

Invented by the HR concept formation program.

Examples

			a(7) = 2 because (i) 4+tau(4)=7 and (ii) 5+tau(5)=7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
• S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Crossrefs

Cf. A036432.

Programs

Maple

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
for b from 1 to N do
  v:= b + numtheory:-tau(b);
  if v <= N then A[v]:= A[v]+1 fi
od:
convert(A,list); # Robert Israel, Jun 10 2018

PARI

a(n) = sum(i=1, n, i+numdiv(i) == n); \\ Michel Marcus, Oct 01 2021

Formula

a(n) = |{b in N : b + tau(b) = n}|

Extensions

a(87)=0 corrected by Michel Marcus, Aug 31 2013