A193417 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).

63, 91, 117, 126, 133, 171, 182, 189, 217, 234, 247, 252, 259, 266, 273, 275, 279, 301, 315, 333, 341, 342, 351, 364, 378, 387, 399, 403, 427, 434, 441, 451, 455, 468, 469, 481, 494, 504, 511, 513, 518, 532, 546, 549, 550, 553, 558, 559, 567, 585, 589, 602

Offset: 1

Author

Michel Lagneau, Jul 26 2011

Keywords

• nonn

Comments

n such that A193453(n) is different of A193169(n).

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

From Jianing Song, Oct 19 2021: (Start)

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.

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.)

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)

Examples

			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}.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000265, A002322, A053575.

Programs

Mathematica

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

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

PARI

isA193417(n) = my(v=znstar(n)[2]); (#v<=1) || (v[2]==1<Jianing Song, Oct 19 2021