cp's OEIS Frontend

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).

%I A082284 #28 Jun 03 2025 05:04:29
%S A082284 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,
%T A082284 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,
%U A082284 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
%N 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).
%C A082284 a(p-2) = p for odd primes p.
%H A082284 Antti Karttunen, <a href="/A082284/b082284.txt">Table of n, a(n) for n = 0..124340</a>
%F A082284 Other identities and observations. For all n >= 0:
%F A082284 a(n) <= A262686(n).
%p A082284 N:= 1000: # to get a(0) .. a(N)
%p A082284 V:= Array(0..N):
%p A082284 for k from 1 to 2*(N+1) do
%p A082284   v:= k - numtheory:-tau(k);
%p A082284   if v <= N and V[v] = 0 then V[v]:= k fi
%p A082284 od:
%p A082284 seq(V[n],n=0..N); # _Robert Israel_, Dec 21 2015
%t A082284 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 *)
%o A082284 (PARI)
%o A082284 allocatemem(123456789);
%o A082284 uplim1 = 2162160 + 320; \\ = A002182(41) + A002183(41).
%o A082284 uplim2 = 2162160;
%o A082284 v082284 = vector(uplim1);
%o A082284 A082284 = n -> if(!n,1,v082284[n]);
%o A082284 for(n=1, uplim1, k = n-numdiv(n); if((0 == A082284(k)), v082284[k] = n));
%o A082284 for(n=0, 124340, write("b082284.txt", n, " ", A082284(n)));
%o A082284 \\ _Antti Karttunen_, Dec 21 2015
%o A082284 (Scheme)
%o A082284 (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))))))))
%o A082284 ;; _Antti Karttunen_, Dec 21 2015
%Y A082284 Column 1 of A265751.
%Y A082284 Cf. A000005, A002182, A002183, A049820, A060990, A261100.
%Y A082284 Cf. A262686 (the largest such number), A262511 (positions where these are equal and nonzero).
%Y A082284 Cf. A266114 (same sequence sorted into ascending order, with zeros removed).
%Y A082284 Cf. A266115 (positive numbers missing from this sequence).
%Y A082284 Cf. A266110 (number of iterations before zero is reached), A266116 (final nonzero value reached).
%Y A082284 Cf. also tree A263267 and its illustration.
%K A082284 nonn
%O A082284 0,2
%A A082284 _Amarnath Murthy_, Apr 14 2003
%E A082284 More terms from _David Wasserman_, Aug 31 2004