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A344770 Ordinal transform of A011772.

%I A344770 #16 Dec 21 2021 12:03:22
%S A344770 1,1,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,1,2,2,1,1,3,1,2,1,3,1,4,1,1,2,2,
%T A344770 1,4,1,1,3,5,1,1,1,1,1,1,1,2,1,2,1,1,1,1,2,2,2,2,1,6,1,2,2,1,1,3,1,3,
%U A344770 2,2,1,2,1,2,3,1,1,4,1,1,1,2,1,3,1,1,1,3,1,1,1,3,2,1,2,3,1,4,1,4,1,1,1,2,2
%N A344770 Ordinal transform of A011772.
%C A344770 Number of values of k, 1 <= k <= n, with A011772(k) = A011772(n).
%C A344770 a(n) = 1 for all powers of primes, A000961. Ones occur on some other positions as well: 15, 22, 35, 38, 42, 44, 45, 46, 51, 52, 54, 65, ...
%H A344770 Antti Karttunen, <a href="/A344770/b344770.txt">Table of n, a(n) for n = 1..65537</a>
%F A344770 a(n) >= A344590(n).
%t A344770 A011772[n_] := For[m = 1, True, m++, If[Divisible[m(m+1)/2, n], Return[m]]];
%t A344770 b[_] = 0;
%t A344770 a[n_] := a[n] = With[{t = A011772[n]}, b[t] = b[t]+1];
%t A344770 Array[a, 105] (* _Jean-François Alcover_, Dec 21 2021 *)
%o A344770 (PARI)
%o A344770 up_to = 65537;
%o A344770 ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A344770 A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
%o A344770 v344770 = ordinal_transform(vector(up_to,n,A011772(n)));
%o A344770 A344770(n) = v344770[n];
%Y A344770 Cf. A000961, A011772, A344590.
%K A344770 nonn,look
%O A344770 1,6
%A A344770 _Antti Karttunen_, May 30 2021