A082284 a(n) = smallest number k such that k - tau(k) = n, or 0 if no such number exists, where tau(n) = the number of divisors of n (A000005).

1, 3, 6, 5, 8, 7, 9, 0, 0, 11, 14, 13, 18, 0, 20, 17, 24, 19, 22, 0, 0, 23, 25, 27, 0, 0, 32, 29, 0, 31, 34, 35, 40, 0, 38, 37, 0, 0, 44, 41, 0, 43, 46, 0, 50, 47, 49, 51, 56, 0, 0, 53, 0, 57, 58, 0, 0, 59, 62, 61, 72, 65, 68, 0, 0, 67, 0, 0, 0, 71, 74, 73, 84, 77, 0, 0, 81, 79, 82, 0, 88

Offset: 0

Author

Amarnath Murthy, Apr 14 2003

Keywords

• nonn

Comments

a(p-2) = p for odd primes p.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..124340

Crossrefs

Column 1 of A265751.

Cf. A000005, A002182, A002183, A049820, A060990, A261100.

Cf. A262686 (the largest such number), A262511 (positions where these are equal and nonzero).

Cf. A266114 (same sequence sorted into ascending order, with zeros removed).

Cf. A266115 (positive numbers missing from this sequence).

Cf. A266110 (number of iterations before zero is reached), A266116 (final nonzero value reached).

Cf. also tree A263267 and its illustration.

Programs

Maple

N:= 1000: # to get a(0) .. a(N)
V:= Array(0..N):
for k from 1 to 2*(N+1) do
  v:= k - numtheory:-tau(k);
  if v <= N and V[v] = 0 then V[v]:= k fi
od:
seq(V[n],n=0..N); # Robert Israel, Dec 21 2015

Mathematica

Table[k = 1; While[k - DivisorSigma[0, k] != n && k <= 2 (n + 1), k++]; If[k > 2 (n + 1), 0, k], {n, 0, 80}] (* Michael De Vlieger, Dec 22 2015 *)

PARI

allocatemem(123456789);
uplim1 = 2162160 + 320; \\ = A002182(41) + A002183(41).
uplim2 = 2162160;
v082284 = vector(uplim1);
A082284 = n -> if(!n,1,v082284[n]);
for(n=1, uplim1, k = n-numdiv(n); if((0 == A082284(k)), v082284[k] = n));
for(n=0, 124340, write("b082284.txt", n, " ", A082284(n)));
\\ Antti Karttunen, Dec 21 2015

Scheme

(define (A082284 n) (if (zero? n) 1 (let ((u (+ n (A002183 (+ 2 (A261100 n)))))) (let loop ((k n)) (cond ((= (A049820 k) n) k) ((> k u) 0) (else (loop (+ 1 k))))))))
;; Antti Karttunen, Dec 21 2015

Formula

Other identities and observations. For all n >= 0:

a(n) <= A262686(n).

Extensions

More terms from David Wasserman, Aug 31 2004