A347391 - OEIS

%I #75 Jul 15 2024 10:24:12

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

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

%C 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.

%C 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.

%C .

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

%C              .............../ \...............

%C             /                                 \

%C            /                                   \

%C           x                                     2m

%C          / \                                   / \

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

%C                   / \                  / \             / \

%C                etc.  etc.           etc.  \           /  etc.

%C                                            \         /

%C                                            6x       9x = sigma(2x)

%C                                            / \     / \

%C                                          etc. \ etc.  etc.

%C                                                \

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

%C .

%C 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.

%C 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.

%C 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.

%C From _Antti Karttunen_, Jul 10 2024: (Start)

%C 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).

%C 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.

%C 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.

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

%C (End)

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

%C Conjecture: sequence is finite.

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

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%e 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.

%e 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.

%e Sigma(5) = 6 is located as the sibling of 5 in the Doudna-tree, thus 5 is included in this sequence.

%e 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:

%e .

%e              40

%e             /  \

%e    A003961 /    \ *2

%e           /      \

%e         189       80

%e         / \      / \

%e      etc   etc etc  160

%e                    / \

%e                  etc  320

%e                      / \

%e                    etc. etc.

%e .

%e 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:

%e             120

%e             /  \

%e    A003961 /    \ *2

%e           /      \

%e         945       240

%e         / \      / \

%e      etc   etc  etc  480

%e                    / \

%e                  etc  960

%e                      / \

%e                    etc. 1920

%e                         / \

%e                      etc. etc.

%o (PARI) isA347391(n) = (1==A347381(n));

%o (PARI)

%o 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)};

%o A252463(n) = if(!(n%2),n/2,A064989(n));

%o 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));

%Y Positions of 1's in A347381.

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

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

%K nonn,hard,more

%O 1,1

%A _Antti Karttunen_, Aug 30 2021