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A262503 a(n) = largest k such that A155043(k) = n.

%I A262503 #22 Oct 02 2015 12:10:16
%S A262503 0,2,6,12,18,22,30,34,42,48,60,72,84,96,108,120,132,140,112,116,126,
%T A262503 124,130,138,150,156,168,180,176,184,192,204,216,228,240,248,264,280,
%U A262503 250,258,270,288,296,312,306,320,328,340,352,364,372,354,358,368,384,396,420,402,414,418,432,450,468,480,504,520,540,560,572,580,594,612,610,618,622,628,648,672,592
%N A262503 a(n) = largest k such that A155043(k) = n.
%C A262503 The first odd terms occur as a(121) = 1089, a(123) = 1093, a(349) = 3253, a(717) = 7581, a(807) = 8685, a(1225) = 13689, etc.
%H A262503 Antti Karttunen, <a href="/A262503/b262503.txt">Table of n, a(n) for n = 0..110880</a>
%H A262503 A. Karttunen, <a href="https://oeis.org/plot2a?name1=A262502&amp;name2=A262503&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=-2&amp;radiop1=ratio&amp;drawlines=true">Ratio A262502(n+2)/a(n) drawn with the help of OEIS Plot2-script</a>
%H A262503 A. Karttunen, <a href="https://oeis.org/plot2a?name1=A262503&amp;name2=A261089&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Ratio a(n)/A261089(n) drawn with the help of OEIS Plot2-script</a>
%F A262503 Other identities and observations. For all n >= 0:
%F A262503 A262502(n+2) > a(n). [Not rigorously proved, but empirical evidence and common sense agrees.]
%t A262503 lim = 80; a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; t = Table[a@ n, {n, 0, 12 lim}]; Last@ Flatten@ Position[t, #] - 1 & /@ Range[0, lim] (* Uses the product of a limit and an arbitrary coefficient (12) based on observation of output for low values (n < 500). This might need to be adjusted for large n to give correct values of a(n). - _Michael De Vlieger_, Sep 29 2015 *) (* Note: one really should use a general safe limit, like A262502(n+2) I use in my Scheme-program. - _Antti Karttunen_, Sep 29 2015 *)
%o A262503 (PARI)
%o A262503 allocatemem(123456789);
%o A262503 uplim = 2162160; \\ = A002182(41).
%o A262503 v155043 = vector(uplim);
%o A262503 v155043[1] = 1; v155043[2] = 1;
%o A262503 for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]);
%o A262503 A155043 = n -> if(!n,n,v155043[n]);
%o A262503 uplim2 = 110880; \\ = A002182(30).
%o A262503 v262503 = vector(uplim2);
%o A262503 for(i=1, uplim, if(v155043[i] <= uplim2, v262503[v155043[i]] = i));
%o A262503 A262503 = n -> if(!n,n,v262503[n]);
%o A262503 for(n=0, uplim2, write("b262503.txt", n, " ", A262503(n)));
%o A262503 (Scheme)
%o A262503 (define (A262503 n) (let loop ((k (A262502 (+ 2 n)))) (if (= (A155043 k) n) k (loop (- k 1)))))
%Y A262503 Cf. A261089 (gives the first occurrence of n in A155043).
%Y A262503 Cf. A262507 (gives the number of times n occurs in A155043).
%Y A262503 Cf. A259934, A262502.
%K A262503 nonn
%O A262503 0,2
%A A262503 _Antti Karttunen_, Sep 24 2015