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A344771 Ordinal transform of A342915, where A342915(n) = gcd(1+n, psi(n)).

%I A344771 #14 Dec 22 2021 10:30:14
%S A344771 1,1,1,2,1,3,1,2,1,4,1,5,1,3,2,6,1,7,1,4,2,8,1,9,3,5,2,10,1,11,1,6,4,
%T A344771 12,2,13,1,7,3,14,1,15,1,1,5,16,1,17,6,8,3,18,1,19,4,9,7,20,1,21,1,10,
%U A344771 2,22,2,23,1,11,8,24,1,25,1,12,4,26,3,27,1,2,9,28,1,29,10,13,5,30,1,31,5,14,11,32,2,33,1,15
%N A344771 Ordinal transform of A342915, where A342915(n) = gcd(1+n, psi(n)).
%C A344771 Number of values of k, 1 <= k <= n, with A342915(k) = A342915(n).
%C A344771 a(p) = 1 for all primes p (and for some other numbers as well).
%H A344771 Antti Karttunen, <a href="/A344771/b344771.txt">Table of n, a(n) for n = 1..65537</a>
%F A344771 a(n) <= A344773(n).
%t A344771 psi[n_] := If[n= 1, 1, Times@@((#1+1)*#1^(#2-1)& @@@ FactorInteger[n])];
%t A344771 A342915[n_] := GCD[n+1, psi[n]];
%t A344771 b[_] = 0;
%t A344771 a[n_] := a[n] = With[{t = A342915[n]}, b[t] = b[t]+1];
%t A344771 Array[a, 105] (* _Jean-François Alcover_, Dec 22 2021 *)
%o A344771 (PARI)
%o A344771 up_to = 65537;
%o A344771 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 A344771 A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
%o A344771 A342915(n) = gcd(1+n,A001615(n));
%o A344771 v344771 = ordinal_transform(vector(up_to,n,A342915(n)));
%o A344771 A344771(n) = v344771[n];
%Y A344771 Cf. A001615, A342915, A344773.
%Y A344771 Differs from A339914 for the first time at n=44, where a(44) = 1, while A339914(44) = 8.
%K A344771 nonn
%O A344771 1,4
%A A344771 _Antti Karttunen_, May 31 2021