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A193417
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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).
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1
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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
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OFFSET
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1,1
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COMMENTS
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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)
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LINKS
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EXAMPLE
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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}.
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MATHEMATICA
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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
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PROG
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(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<<valuation(v[2], 2)) \\ Jianing Song, Oct 19 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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