A296087 - OEIS

%I #20 Nov 24 2018 00:46:40

%S 15265,27962,30217,30530,45795,50541,54379,54905,57598,60434,61060,

%T 64255,66526,72357,72713,89585,90651,91590,101082,101949,108758,

%U 109810,120868,122120,128510,136555,137385,137883,138761,144714,145426,149739,151085,152633,161386,163137,164715,166315,179170,181302,181543,182942

%N Numbers n such that there is k < n for which A003557(k) = A003557(n), A048250(k) = A048250(n) and A173557(k) = A173557(n).

%C Because Euler phi(n) = A000010(n) = A003557(n) * A173557(n), Dedekind psi(n) = A001615(n) = A003557(n) * A048250(n), and because also sigma(n)  (A000203) can be computed from those three elements (see A291750), these numbers form also a subset of the positions of such duplicated occurrences of values computed for those functions. See for example A069822 and A296214.

%C a(11) = 61060 is the first term that is not squarefree.

%H Antti Karttunen, <a href="/A296087/b296087.txt">Table of n, a(n) for n = 1..2876</a>

%e 15265 is a term because A003557(15265) = 1 = A003557(15169), A048250(15265) = 19008 = A048250(15169), A173557(15265) = 11760 = A173557(15169).

%e 27962 is a term because A003557(27962) = 1 = A003557(26355), A048250(27962) = 48384 = A048250(26355), A173557(27962) = 12000 = A173557(26355).

%o (PARI)

%o search_up_to = (2^23);

%o A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };

%o A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));

%o A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from _Michel Marcus_, Oct 31 2017

%o Anotsubmitted1(n) = (1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n));

%o Akaikki3(n) = (1/2)*(2 + ((A048250(n)+Anotsubmitted1(n))^2) - A048250(n) - 3*Anotsubmitted1(n));

%o om = Map(); m = 0; i=0; for(n = 1, search_up_to, k = Akaikki3(n); if(!mapisdefined(om,k), mapput(om,k,n), i++; write("b296087.txt", i, " ", n)));

%Y Cf. A000010, A001615, A003557, A048250, A173557.

%Y Subsequence of A069822 and of A296214.

%K nonn

%O 1,1

%A _Antti Karttunen_, Dec 08 2017