

A347391


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


15



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

1,1


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


LINKS



EXAMPLE

Sigma(3) = 4 is located as the sibling of 3 in the Doudnatree (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 Doudnatree, thus 4 is included in this sequence.
Sigma(5) = 6 is located as the sibling of 5 in the Doudnatree, 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
/ \
/ \
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
/ \
/ \
945 240
/ \ / \
etc etc etc 480
/ \
etc 960
/ \
etc. 1920
/ \
etc. etc.


PROG

(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)};
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));


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



STATUS

approved



