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%I #35 Feb 22 2024 20:35:30
%S 63,91,117,126,133,171,182,189,217,234,247,252,259,266,273,275,279,
%T 301,315,333,341,342,351,364,378,387,399,403,427,434,441,451,455,468,
%U 469,481,494,504,511,513,518,532,546,549,550,553,558,559,567,585,589,602
%N Numbers n such that the number of the odd divisors of phi(n) is different from the number of the odd divisors of lambda(n).
%C n such that A193453(n) is different of A193169(n).
%C Numbers n such that A000265(lambda(n)) < A000265(phi(n)), where A000265(m) is the odd part (largest odd divisor) of m. - _Amiram Eldar_ and _Thomas Ordowski_, Feb 04 2019
%C From _Jianing Song_, Oct 19 2021: (Start)
%C Let G = (Z/kZ)* be the multiplicative group of integers modulo k and G_2 be the Sylow 2-subgroup of G. Sequence lists k such that G/G_2 is not cyclic; equivalently, decompose G as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j, then k is a term if and only if m > 1 and k_{m-1} is not a power of 2.
%C Numbers k such that there exists an odd prime p such that the p-rank of G is greater than 1. (The rank of a finitely generated group rank(G) is defined to be the size of the minimal generating sets of G, and the p-rank of G is the rank of the Sylow p-subgroup of G.)
%C k is a term if and only if k satisfies at least one of the two conditions: (a) there exists an odd prime p such that k has two distinct prime factors congruent to 1 modulo p (for example 91 = 7 * 13, 7 == 13 == 1 (mod 3)); (b) there exists an odd prime p such that k has a prime factor congruent to 1 modulo p and that k is divisible by p^2 (for example 275 = 11 * 5^2, 11 == 1 (mod 5)). (End)
%H Jinyuan Wang, <a href="/A193417/b193417.txt">Table of n, a(n) for n = 1..10000</a>
%e 63 is in the sequence because phi(63) = 36 with 3 odd divisors {1, 3, 9} and lambda(63) = 6 with only 2 odd divisors {1, 3}.
%t f[n_] := Block[{d = Divisors[EulerPhi[n]]}, Count[OddQ[d], True]]; Table[f[n], {n, 500}]; g[n_] := Block[{e = Divisors[CarmichaelLambda[n]]}, Count[OddQ[e], True]]; Table[g[n], {n, 500}]; a={};Do[If[ f[n] != g[n], AppendTo[a,n]], {n, 500}];a
%o (PARI) is(n) = my(cp = eulerphi(n), cn=cp>>valuation(cp,2), cl=lcm(znstar(n)[2])); cl = cl >> valuation(cl, 2); numdiv(cl) != numdiv(cn) \\ _David A. Corneth_, Feb 18 2019
%o (PARI) isA193417(n) = my(v=znstar(n)[2]); (#v<=1) || (v[2]==1<<valuation(v[2], 2)) \\ _Jianing Song_, Oct 19 2021
%Y Cf. A000010, A000265, A002322, A053575.
%K nonn
%O 1,1
%A _Michel Lagneau_, Jul 26 2011
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