A266110 -id:A266110

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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).

+10
22

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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Apr 14 2003

EXTENSIONS

More terms from David Wasserman, Aug 31 2004

STATUS

approved

A266116

The last nonzero term on each row of A265751.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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).

LINKS

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

FORMULA

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.]

EXAMPLE

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.

PROG

(Scheme)

(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.

CROSSREFS

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).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 21 2015

STATUS

approved

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).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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

FORMULA

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).

EXAMPLE

PROG

(Scheme, with memoization-macro definec)

(definec (A266111 n) (cond ((A082284 n) => (lambda (lad) (if (zero? lad) 1 (+ 1 (A266111 lad)))))))

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 21 2015

STATUS

approved

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