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

189, 1271, 2125, 9261, 63767, 133907, 142859, 161257, 189209, 226967, 368063, 426373, 777923, 801727, 925101, 961193, 1003043, 4566661, 5244091, 5588327, 6031163, 6064439, 8135263, 8639879, 10074227, 10150571, 11234875, 12489107, 16016003, 19765547, 22635539

Offset: 1

Author

Antti Karttunen, Sep 10 2021

Keywords

• nonn
• hard

Comments

Here nontrivial unitary divisor d of k means any divisor d|k, such that 1 < d < k and gcd(d, k/d) = 1.

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.

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.

Examples

			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.
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.
		

Links

• Index entries for sequences where odd perfect numbers must occur, if they exist at all
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000396, A005820, A005940, A034444, A336702, A347381, A347391, A347392.

Subsequence of A347390, which is a subsequence of A347384.

Programs

PARI

isA347383(n) = if((1==n)||!(n%2)||isprimepower(n),0,my(w=A347381(n)); fordiv(n,d,if((d>1)&&(dA347381(d)<=w), return(0))); (1));

Extensions

a(29)-a(31) from Jinyuan Wang, Jul 09 2025