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A347391
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Numbers k such that sigma(k) is either their sibling in Doudna tree (A005940), or one of the sibling's descendants.
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15
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3, 4, 5, 15, 20, 189, 945, 2125, 6375, 9261, 46305, 401625, 19679625
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OFFSET
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1,1
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COMMENTS
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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.
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.............../ \...............
/ \
/ \
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.
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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.
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LINKS
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EXAMPLE
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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:
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40
/ \
/ \
189 80
/ \ / \
etc etc etc 160
/ \
etc 320
/ \
etc. etc.
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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.
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PROG
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(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));
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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