A266110 -id:A266110 - 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).

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

Amarnath Murthy, Apr 14 2003

nonn

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

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

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.

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

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 *)

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

(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

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

a(n) <= A262686(n).

More terms from David Wasserman, Aug 31 2004

A266116 The last nonzero term on each row of A265751.

Original entry on oeis.org

7, 7, 13, 7, 8, 7, 13, 7, 8, 13, 20, 13, 25, 13, 20, 19, 24, 19, 25, 19, 20, 37, 25, 37, 24, 25, 40, 37, 28, 37, 50, 37, 40, 33, 50, 37, 36, 37, 50, 43, 40, 43, 49, 43, 50, 67, 49, 67, 56, 49, 50, 67, 52, 67, 68, 55, 56, 67, 68, 67, 136, 67, 68, 63, 64, 67, 66, 67, 68, 79, 74, 79, 136, 79, 74, 75, 103, 79, 98, 79, 88, 103, 98, 103, 136, 85

Offset: 0

Antti Karttunen, Dec 21 2015

nonn

Starting from j = n, search for a smallest number k such that k - d(k) = j, and if found such a number, replace j with k and repeat the procedure. When eventually such k is no longer found, then the (last such) j must be one of the terms of A045765, and it is set as the value of a(n).

			Starting from n = 21, we get the following chain: 21 -> 23 -> 27 -> 29 -> 31 -> 35 -> 37, with A082284 iterated 6 times before the final term 37 (for which A060990(37) = A082284(37) = 0) is encountered. Thus a(21) = 37.

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

Cf. A000005, A045765, A060990, A082284, A265751.
Cf. A266110 (gives the number of iterations of A082284 needed before a(n) is found).
Cf. also tree A263267 (and its illustration).

(definec (A266116 n) (cond ((A082284 n) => (lambda (lad) (if (zero? lad) n (A266116 lad))))))
;; Alternatively:
(define (A266116 n) (A265751bi n (A266110 n))) ;; Code for A265751bi given in A265751.

a(n) = A265751(n, A266110(n)).
If A060990(n) = 0, a(n) = n, otherwise a(n) = a(A082284(n)), where A082284(n) = smallest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).
Other identities and observations. For all n >= 0:
a(n) >= n.
A060990(a(n)) = 0. [All terms are in A045765.]

A266111 If A082284(n) = 0, a(n) = 1, otherwise a(n) = 1 + a(A082284(n)), where A082284(n) = smallest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

Original entry on oeis.org

5, 4, 5, 3, 2, 2, 4, 1, 1, 3, 3, 2, 4, 1, 2, 3, 2, 2, 3, 1, 1, 7, 2, 6, 1, 1, 3, 5, 1, 4, 5, 3, 2, 1, 4, 2, 1, 1, 3, 3, 1, 2, 3, 1, 2, 9, 2, 8, 2, 1, 1, 7, 1, 6, 4, 1, 1, 5, 3, 4, 8, 3, 2, 1, 1, 2, 1, 1, 1, 5, 2, 4, 7, 3, 1, 1, 9, 2, 5, 1, 2, 8, 4, 7, 6, 1, 3, 6, 1, 5, 13, 6, 2, 4, 12, 5, 5, 4, 1, 3, 1, 2, 11, 1, 4, 3, 10, 2, 1, 1, 2, 2, 1, 1, 9, 3, 1, 1, 8, 2, 3, 7

Offset: 0

Antti Karttunen, Dec 21 2015

nonn

			Starting from n = 21, we get the following chain: 21 -> 23 -> 27 -> 29 -> 31 -> 35 -> 37, with A082284 iterated 6 times before the final term 37 (for which A060990(37) = A082284(37) = 0) is encountered. Here we count the terms (not steps) in whole chain, thus a(21) = 7.

Antti Karttunen, Table of n, a(n) for n = 0..10000

One more than A266110.
Number of significant terms on row n of A265751 (without its trailing zeros).
Cf. tree A263267 (and its illustration).
Cf. A000005, A060990, A082284.
Cf. also A264971.

If A060990(n) = 0, a(n) = 1, otherwise a(n) = 1 + a(A082284(n)).
Other identities. For all n >= 0:
a(n) = 1 + A266110(n).

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).

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A266116 The last nonzero term on each row of A265751.

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A266111 If A082284(n) = 0, a(n) = 1, otherwise a(n) = 1 + a(A082284(n)), where A082284(n) = smallest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

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