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A347383 Odd composites k, not powers of primes, such that for all their nontrivial unitary divisors d it holds that A347381(d) > A347381(k).

%I A347383 #26 Jul 09 2025 10:16:01
%S A347383 189,1271,2125,9261,63767,133907,142859,161257,189209,226967,368063,
%T A347383 426373,777923,801727,925101,961193,1003043,4566661,5244091,5588327,
%U A347383 6031163,6064439,8135263,8639879,10074227,10150571,11234875,12489107,16016003,19765547,22635539
%N A347383 Odd composites k, not powers of primes, such that for all their nontrivial unitary divisors d it holds that A347381(d) > A347381(k).
%C A347383 Here nontrivial unitary divisor d of k means any divisor d|k, such that 1 < d < k and gcd(d, k/d) = 1.
%C A347383 Any hypothetical odd term x in A005820 (triperfect numbers) would also be a member of this sequence. Proof: such an odd number cannot be a prime power (although it must be a square), thus it must have at least two nontrivial unitary divisors (with A034444(x) >= 4). Because sigma(x) = 3*x, it must be a term of A347391. From the illustration given there, we see that any odd square y in that sequence (i.e. with A347381(y)=1) would have an abundancy index of at least three (sigma(y)/y >= 3). But because abundancy index is multiplicative and always > 1 for n > 1, any nontrivial unitary divisor d of an odd triperfect number x must have sigma(d)/d < 3, thus for all such d, A347381(d) <> 1. And neither such divisor d can be a term of A336702, because 3*x is odd, therefore we must have A347381(d) > 1 for all nontrivial unitary divisors d of such a hypothetical x.
%C A347383 Any odd term of A000396, i.e., an odd perfect number, if such a hypothetical number exists, must also be a term of this sequence, by reasoning similar to above. See also illustration in A347392.
%H A347383 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%H A347383 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A347383 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A347383 189 is a term, because A347381(189) = 1, and the only way to factor 189 into nontrivial unitary divisors is 7*27, and A347381(7) = A347381(27) = 3 > 1.
%e A347383 63767 = 11^2 * 17 * 31 is a term, as its nontrivial unitary divisors are [17, 31, 121, 527, 2057, 3751], at which points A347381 obtains values [6, 10, 5, 11, 6, 8], every one which is larger than A347381(63767) = 3.
%o A347383 (PARI) isA347383(n) = if((1==n)||!(n%2)||isprimepower(n),0,my(w=A347381(n)); fordiv(n,d,if((d>1)&&(d<n)&&(1==gcd(d,n/d)) && (A347381(d)<=w), return(0))); (1));
%Y A347383 Cf. A000396, A005820, A005940, A034444, A336702, A347381, A347391, A347392.
%Y A347383 Subsequence of A347390, which is a subsequence of A347384.
%K A347383 nonn,hard
%O A347383 1,1
%A A347383 _Antti Karttunen_, Sep 10 2021
%E A347383 a(29)-a(31) from _Jinyuan Wang_, Jul 09 2025