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A328934 Nonsquarefree N such that A328919(N) = 1; nonsquarefree N such that {sigma_t(N) mod N: t >= 1} is purely periodic, while {sigma_t(N) mod N: t >= 0} is not.

%I A328934 #18 Nov 04 2019 00:46:36
%S A328934 18,20,24,28,44,45,52,56,60,68,76,84,88,90,92,99,116,120,124,126,132,
%T A328934 140,148,152,153,156,164,168,172,180,184,188,198,204,207,212,220,228,
%U A328934 234,236,240,244,248,260,261,264,268,276,280,284,292,306,308,312,315,316,332,336
%N A328934 Nonsquarefree N such that A328919(N) = 1; nonsquarefree N such that {sigma_t(N) mod N: t >= 1} is purely periodic, while {sigma_t(N) mod N: t >= 0} is not.
%C A328934 If N is squarefree then A328919(N) = 1, but the converse is not true. While it is conjectured that 12 is the only N such that 0 = A328919(N) < A051903(N), there are infinitely many N such that 1 = A328919(N) < A051903(N).
%C A328934 Let p_1, p_2, ..., p_(k-1) be k-1 distinct odd primes, k >= 3. Let N = 2^k*p_1*p_2*...*p_(k-1), then N is here. It is easy to see that {sigma_t(N) mod 2^k: t >= k} and {sigma_t(N) mod p_i: t >= 1} are both purely periodic.
%C A328934 To show this, it is sufficient to show that:
%C A328934 (a) sigma_t(N) == sigma_(t+2^(k-2))(N) (mod 2^k) for all 1 <= t <= k-1, so {sigma_t(N) mod 2^k: t >= 1} is purely periodic;
%C A328934 Proof. Write N = 2^k*M, then sigma_(t+2^(k-2))(N) - sigma_t(N) = Sum_{d|M} (d^(t+2^(k-2)) + (2d)^(t+2^(k-2)) + ... + (2^k*t)^(t+2^(k-2)) - d^t - (2d)^t - ... - (2^k*d)^t). This is divisible by 2^k if and only if 2^k | Sum_{d|M} (2d)^t, or 2^(k-t) | sigma_t(M). This is obvious because sigma_t(p_i) is even, as sigma_t() is multiplicative, 2^(k-1) | sigma_t(M).
%C A328934 (b) there exists some i such that sigma_0(N) !== sigma_(p_i-1)(N) (mod p_i).
%C A328934 Proof. Write N = M*p_i, then sigma_(p_i-1)(N) - sigma_0(N) = Sum_{d|M} (d^(p_i-1) + (p_i*d)^(p_i-1) - d - 1) == -(Sum_{d|M} 1) = (k+1)*2^(k-1) (mod p_i). As max{p_1, p_2, ..., p_(k-1)} >= A000040(k) > k+1, there exists some prime p_i that does not divide p_i.
%C A328934 This shows that A051903(N) - A328919(N) is unbounded, and can take every natural number as its value for infinitely many times. It is conjectured that the smallest N such that A328919(N) = 1 and A051903(N) = k is N = 2^(k-1)*A002110(k-1), k >= 3.
%e A328934 For t > 0, sigma_t(24) == 12 (mod 24) if t is odd, sigma_t(24) == 10 (mod 24) if t is even. Note that sigma_0(24) = 8, so A328919(24) = 1, so 24 is a term.
%o A328934 (PARI) for(n=1, oo, if(A328919(n)==1 && !issquarefree(n), return(n))) \\ See A328919 for its program.
%Y A328934 Cf. A328919, A051903, A000040, A002110.
%K A328934 nonn
%O A328934 1,1
%A A328934 _Jianing Song_, Oct 31 2019