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 A060990 Number of solutions to x - d(x) = n, where d(n) is the number of divisors of n (A000005). 38
 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: A157896 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

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Last modified April 10 19:18 EDT 2021. Contains 342853 sequences. (Running on oeis4.)