A082284 - OEIS

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

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OFFSET

0,2

COMMENTS

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

LINKS

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

FORMULA

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

a(n) <= A262686(n).

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 *)

PROG

(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

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.

Sequence in context: A330525 A201418 A123688 * A241474 A259556 A063520

Adjacent sequences: A082281 A082282 A082283 * A082285 A082286 A082287

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Apr 14 2003

EXTENSIONS

More terms from David Wasserman, Aug 31 2004

STATUS

approved