A060990 - OEIS

A060990

Number of solutions to x - d(x) = n, where d(n) is the number of divisors of n (A000005).

2, 2, 1, 1, 1, 1, 3, 0, 0, 1, 1, 3, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 4, 1, 0, 0, 1, 2, 0, 2, 1, 1, 1, 0, 2, 2, 0, 0, 2, 2, 0, 1, 1, 0, 1, 1, 3, 1, 2, 0, 0, 2, 0, 1, 1, 0, 0, 3, 2, 1, 1, 1, 2, 0, 0, 2, 0, 0, 0, 2, 4, 1, 1, 1, 0, 0, 1, 1, 2, 0, 1, 2, 1, 1, 1, 0, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 0, 1, 0, 1, 3, 0, 1, 1

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

OFFSET

0,1

COMMENTS

If x-d(x) is never equal to n, then n is in A045765 and a(n) = 0.

Number of solutions to A049820(x) = n. - Jaroslav Krizek, Feb 09 2014

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..110880

FORMULA

a(0) = 2; for n >= 1, a(n) = Sum_{k = n .. n+A002183(2+A261100(n))} [A049820(k) = n]. (Here [...] denotes the Iverson bracket, resulting 1 when A049820(k) is n and 0 otherwise.) - Antti Karttunen, Sep 25 2015, corrected Oct 12 2015.

a(n) = Sum_{k = A082284(n) .. A262686(n)} [A049820(k) = n] (when tacitly assuming that A049820(0) = 0.) - Antti Karttunen, Oct 12 2015

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

a(A045765(n)) = 0. a(A236562(n)) > 0. - Jaroslav Krizek, Feb 09 2014

EXAMPLE

a(11) = 3 because three numbers satisfy equation x-d(x)=11, namely {13,15,16} with {2,4,5} divisors respectively.

MATHEMATICA

lim = 105; s = Table[n - DivisorSigma[0, n], {n, 2 lim + 3}]; Length@ Position[s, #] & /@ Range[0, lim] (* Michael De Vlieger, Sep 29 2015, after Wesley Ivan Hurt at A049820 *)

PROG

(PARI)

allocatemem(123456789);

uplim = 2162160; \\ = A002182(41).

v060990 = vector(uplim);

for(n=3, uplim, v060990[n-numdiv(n)]++);

A060990 = n -> if(!n, 2, v060990[n]);

uplim2 = 110880; \\ = A002182(30).

for(n=0, uplim2, write("b060990.txt", n, " ", A060990(n)));

\\ Antti Karttunen, Sep 25 2015

(Scheme)

(define (A060990 n) (if (zero? n) 2 (add (lambda (k) (if (= (A049820 k) n) 1 0)) n (+ n (A002183 (+ 2 (A261100 n)))))))

;; Auxiliary function add implements sum_{i=lowlim..uplim} intfun(i)

(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

;; Proof-of-concept code for the given formula, by Antti Karttunen, Sep 25 2015

CROSSREFS

Cf. A000005, A002183, A049820, A049816, A082284, A155043, A236561, A236565, A259934, A261100, A262507, A262513, A262686.

Cf. A045765 (positions of zeros), A236562 (positions of nonzeros), A262511 (positions of ones).

Cf. A263087 (computed for squares).

Sequence in context: A358469 A156072 A215788 * A276309 A165031 A286634

Adjacent sequences: A060987 A060988 A060989 * A060991 A060992 A060993

KEYWORD

nonn

AUTHOR

Labos Elemer, May 11 2001

EXTENSIONS

Offset corrected by Jaroslav Krizek, Feb 09 2014

STATUS

approved