A354194 - OEIS

A354194

Numbers k for which phi(A267099(k)) is equal to phi(k), but the number of 4m+1 and 4m+3 primes in the prime factorization of k (when counted with multiplicity) is not equal. Here A267099 is fully multiplicative involution swapping the positions of 4m+1 and 4m+3 primes, and phi is Euler totient function.

%I #11 May 28 2022 16:37:34

%S 69037,70807,76635,79577,81631,82425,88335,95025,138074,141614,149209,

%T 153270,153703,159154,163262,164850,171989,176670,177199,190050,

%U 276148,283228,298418,306540,307406,318308,326524,329700,343978,353340,354398,380100,552296,566456,596836,613080,614812,636616,653048,659400,687956

%N Numbers k for which phi(A267099(k)) is equal to phi(k), but the number of 4m+1 and 4m+3 primes in the prime factorization of k (when counted with multiplicity) is not equal. Here A267099 is fully multiplicative involution swapping the positions of 4m+1 and 4m+3 primes, and phi is Euler totient function.

%H Antti Karttunen, <a href="/A354194/b354194.txt">Table of n, a(n) for n = 1..377</a>

%e A354102(69037) = phi(A267099(69037)) = phi(70807) = phi(69037) = 62400, and 69037 = 17*31*131, therefore 69037 is included in this sequence, and likewise is 70807 = 11*41*157.

%o (PARI)

%o A354188(n) = (eulerphi(A267099(n)) == eulerphi(n)); \\ Uses the program given in A267099.

%o A342025(n) = {my(f = factor(n)); sum(k=1, #f~, ((f[k, 1] % 4)==1)*f[k, 2]) == sum(k=1, #f~, ((f[k, 1] % 4)==3)*f[k, 2]); }; \\ From isok function in A072202

%o isA354194(n) = (A354188(n) && !A342025(n));

%Y Setwise difference A354189 \ A072202.

%Y Cf. A002144, A002145, A342025, A354188.

%K nonn

%O 1,1

%A _Antti Karttunen_, May 20 2022