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A262507 a(n) = number of times n occurs in A155043.

%I A262507 #16 Nov 27 2015 00:17:26
%S A262507 1,2,3,5,4,5,6,4,4,4,8,4,4,5,8,7,7,7,7,8,5,6,6,8,10,7,8,7,7,5,5,6,6,8,
%T A262507 6,7,7,7,4,5,5,6,6,8,7,5,5,6,7,11,5,4,5,8,12,7,9,5,8,8,9,10,14,11,12,
%U A262507 11,9,11,13,12,12,11,11,11,12,12,10,9,9,9,8,6,10,9,10,8,7,7,8,11,10,10,12,9,7,6,5,5,5,5,4,7,8,6,7,9,7,5,11,13,13,8,10,12,13,10,12,16,9,8,12
%N A262507 a(n) = number of times n occurs in A155043.
%C A262507 Records are: 1, 2, 3, 5, 6, 8, 10, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 26, 27, 31, 35, 39, 44, ... and they occur at positions: 0, 1, 2, 3, 6, 10, 24, 49, 54, 62, 117, 236, 445, 484, 892, 893, 1022, 1784, 1911, 1912, 1913, 20600, 50822, ...
%C A262507 a(n) gives the length of each row of irregular table A263265.
%H A262507 Antti Karttunen, <a href="/A262507/b262507.txt">Table of n, a(n) for n = 0..110880</a>
%F A262507 a(n) = Sum_{k=n..A262502(2+n)} [A155043(k) == n]. (Here [...] denotes the Iverson bracket, resulting 1 when A155043(k) is n and 0 otherwise.)
%F A262507 Other identities. For all n >= 0:
%F A262507 a(n) = A263279(n) + A263280(n).
%o A262507 (PARI)
%o A262507 allocatemem(123456789);
%o A262507 uplim = 2162160; \\ = A002182(41).
%o A262507 v155043 = vector(uplim);
%o A262507 v155043[1] = 1; v155043[2] = 1;
%o A262507 for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]);
%o A262507 uplim2 = 110880; \\ = A002182(30).
%o A262507 v262507 = vector(uplim2);
%o A262507 for(i=1, uplim, if(v155043[i] <= uplim2, v262507[v155043[i]]++));
%o A262507 A262507 = n -> if(!n,1,v262507[n]);
%o A262507 for(n=0, uplim2, write("b262507.txt", n, " ", A262507(n)));
%o A262507 (Scheme)
%o A262507 (define (A262507 n) (add (lambda (k) (if (= (A155043 k) n) 1 0)) n (A262502 (+ 2 n))))
%o A262507 ;; Auxiliary function add implements sum_{i=lowlim..uplim} intfun(i)
%o A262507 (define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
%Y A262507 Cf. A000005, A060990, A155043, A262502, A262505, A263265, A263270, A263279, A263280.
%Y A262507 Cf. A261089, A262503.
%Y A262507 Cf. A262508 (positions of ones).
%Y A262507 Cf. A263260 (partial sums).
%K A262507 nonn
%O A262507 0,2
%A A262507 _Antti Karttunen_, Sep 25 2015