A347391 Numbers k such that sigma(k) is either their sibling in Doudna tree (A005940), or one of the sibling's descendants.

3, 4, 5, 15, 20, 189, 945, 2125, 6375, 9261, 46305, 401625, 19679625

Offset: 1

Author

Antti Karttunen, Aug 30 2021

Keywords

• nonn
• hard
• more

Comments

Numbers k > 1 such that nearest common ancestor of k and sigma(k) in Doudna tree is the parent of k, and sigma(k) is not a descendant of k.

Any hypothetical odd term x in A005820 (triperfect numbers) would also be a member of this sequence. This is illustrated in the following diagram which shows how the neighborhood of such x would look like in the Doudna tree (A005940). If m (the parent of x, x = A003961(m), m = A064989(x)) is even, then x is a multiple of 3, while if m is odd, then 3 does not divide x. Because the abundancy index decreases when traversing leftwards in the Doudna tree, m must be a term of A068403. Both x and m would also need to be squares, by necessity.

.

<--A003961-- m ---(*2)--->

.............../ \...............

/ \

/ \

x 2m

/ \ / \

etc.../ \.....2x sigma(x) = 3x..../ \.....4m

/ \ / \ / \

etc. etc. etc. \ / etc.

\ /

6x 9x = sigma(2x)

/ \ / \

etc. \ etc. etc.

\

12x = sigma(3x) if m odd.

.

From the diagram we also see that 2x would then need to be a term of A347392 (as well as that of A159907 and also in A074388, thus sqrt(x) should be a term of A097023), and furthermore, if x is not a multiple of 3 (i.e., when m is odd), then sigma(3*x) = 4*sigma(x) = 4*(3*x), thus 3*x = sigma(x) would be a term of A336702 (particularly, in A027687) and x would be a term of A323653.

Moreover, any odd square x in this sequence (for which sigma(x) would also be odd), would have an abundancy index of at least three (sigma(x)/x >= 3). See comments in A347383.

Note how 401625 = 6375 * 63 = 945 * 425, 46305 = 945 * 49, 9261 = 189 * 49, 6375 = 2125 * 3, 945 = 189 * 5 = 15 * 63 and 9261*2125 = 19679625. It seems that when the multiplicands are coprime, then they are both terms of this sequence, e.g. 2125 and 3, 189 and 5, 2125 and 9261.

From Antti Karttunen, Jul 10 2024: (Start)

Regarding the observation above, for two coprime odd numbers x, y, if both are included here because sigma(x) = 2^a * A064989(x) and sigma(y) = 2^b * A064989(y), then also their product x*y is included because in that case sigma(x*y) = 2^(a+b) * A064989(x*y).

Also, for two coprime odd numbers x, y, if both are included here because sigma(x) = A065119(i) * x and sigma(y) = A065119(j) * y, then also their product x*y is included because sigma(x*y) = A065119(k) * x*y, where A065119(k) = A065119(i)*A065119(j). The existence of such numbers (that would include odd triperfect and odd 6-perfect numbers, see A046061) is so far hypothetical, none is known.

It is not possible that the odd x is in this sequence if sigma(x) = k*A003961^e(x) and e = A061395(k)-2 >= 1.

Note that all odd terms < 2^33 here are some of the exponentially odd divisors of 19679625 (see A374199, also A374463 and A374464).

(End)

Question: from a(6) = 189 onward, are the rest of terms all in A347390?

Conjecture: sequence is finite.

If it exists, a(14) > 2^33.

Examples

			Sigma(3) = 4 is located as the sibling of 3 in the Doudna-tree (see the illustration in A005940), thus 3 is included in this sequence.
Sigma(4) = 7 is located as a grandchild of 3 (which is the sibling of 4) in the Doudna-tree, thus 4 is included in this sequence.
Sigma(5) = 6 is located as the sibling of 5 in the Doudna-tree, thus 5 is included in this sequence.
189 (= 3^3 * 7) is a term, as sigma(189) = 320, and 320 occurs as a descendant of 80 (which is the right sibling of 189) in the Doudna tree, as illustrated below:
.
             40
            /  \
   A003961 /    \ *2
          /      \
        189       80
        / \      / \
     etc   etc etc  160
                   / \
                 etc  320
                     / \
                   etc. etc.
.
945 (= 3^3 * 5 * 7) is a term, as sigma(945) = 1920, and 1920 occurs as a descendant of 240, which is the right sibling of 945 in the Doudna tree, as illustrated below:
            120
            /  \
   A003961 /    \ *2
          /      \
        945       240
        / \      / \
     etc   etc  etc  480
                   / \
                 etc  960
                     / \
                   etc. 1920
                        / \
                     etc. etc.
		

Links

• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Positions of 1's in A347381.

Cf. A000203, A003961, A005940, A005820, A065119, A068403, A074388, A097023, A159907, A374199, A374463, A374464.

Cf. also A027687, A292583, A323653, A336702, A347383, A347390, A347392.

Programs

PARI

isA347391(n) = (1==A347381(n));

PARI

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA347391(n) = if(1==n,0,my(m=A252463(n), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));