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A336560 Numbers k at which points A336456(k) appears multiplicative, but A051027(k) does not.

%I A336560 #7 Jul 27 2020 21:10:03
%S A336560 15,39,51,60,78,87,95,111,123,143,159,183,204,215,219,222,231,240,247,
%T A336560 267,291,303,312,323,327,330,335,339,348,366,380,399,407,411,438,444,
%U A336560 447,455,471,494,506,519,543,559,579,582,591,624,636,654,671,687,695,699,703,714,723,731,732,767,771,779,798,803,807
%N A336560 Numbers k at which points A336456(k) appears multiplicative, but A051027(k) does not.
%C A336560 Numbers in A336557 but not in A336547.
%C A336560 Note that if A051027(k) = Product_{p^e|k} A051027(p^e) then also A336456(n) = Product_{p^e|n} A336456(p^e), because A336456(n) = A335915(A051027(n)) and A335915 is fully multiplicative, thus A336547 is a subsequence of A336557.
%H A336560 Antti Karttunen, <a href="/A336560/b336560.txt">Table of n, a(n) for n = 1..25000</a>
%o A336560 (PARI)
%o A336560 is_fun_mult_on_n(fun,n) = { my(f=factor(n)); prod(k=1,#f~,fun(f[k,1]^f[k,2]))==fun(n); };
%o A336560 A051027(n) = sigma(sigma(n));
%o A336560 A000265(n) = (n>>valuation(n,2));
%o A336560 A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265((f[k,1]^2)-1)^f[k,2]))); };
%o A336560 A336456(n) = A335915(A051027(n));
%o A336560 A336546(n) = is_fun_mult_on_n(A051027,n);
%o A336560 A336556(n) = is_fun_mult_on_n(A336456,n);
%o A336560 isA336560(n) = (A336546(n)<A336556(n));
%Y A336560 Cf. A051027, A335915, A336456.
%Y A336560 Setwise difference of A336557 and A336547. Equally, setwise difference of A336559 and A336549. Subsequence of A336548.
%Y A336560 Cf. also A336561.
%K A336560 nonn
%O A336560 1,1
%A A336560 _Antti Karttunen_, Jul 25 2020