A347392 Numbers k such that nearest common ancestor of k and sigma(k) in Doudna tree (A347879) is the grandparent of k.

8, 9, 12, 13, 24, 35, 160, 455, 42550, 127650, 8041950, 22469750, 58506250, 67409250, 175518750, 394055550, 4246782750

Offset

1,1

Comments

Note how 13 * 35 = 455.

If there exists any odd perfect numbers x, with sigma(x) = 2x, then 2*x would be a term of this sequence, as then sigma(2*x) = 6*x would be situated as a descendant under the other branch of the grandparent of 2*x (a parent of x), which is m = A064989(x), with m in A005101. Opn x itself would be a term of A336702. Furthermore, if such x is not a multiple of 3 (in which case m is odd and in A005231), then also 3x would be a term of this sequence as sigma(3*x) = 4*sigma(x) = 8*x would be situated as a grandchild of 2x, with 2x being a first cousin of 3x. Also, in that case, 6*x would be located in A336702 (particularly, in A027687) because then sigma(6*x) = 12*sigma(x) = 24*x = 4*(6*x).

.

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

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

/ \

/ \

/ \

x 2m

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

/ \ / \ / \

etc. \ etc. \ etc. etc.

\ \

4x sigma(2x) = 6x

/ \ / \

etc \ etc. \

\ \

8x = sigma(3x) 12x

if m odd \

\

24x = sigma(6x) if m odd.

.

Furthermore, if there were any hypothetical odd terms y in A005820 (triperfect numbers), then 2y would be a term of this sequence. See the diagram in A347391.

If it exists, a(18) > 2^33.

Links

Table of n, a(n) for n=1..17.

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

Example

455 is included in the sequence as sigma(455) = 672, and the nearest common ancestor of 455 and 672 in Doudna tree is 42, which is the grandparent of 455 [as 455 = A003961(A003961(42))] and the grand-grand-grand-parent of 672 [as 672 = (2^4)*42].

Prog

(PARI) isA347392(n) = (2==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));
isA347391_or_A347392(n) = if(1==n, 0, my(m=A252463(A252463(n)), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));
isA347392(n) = (isA347391_or_A347392(n) && !isA347391(n));

Crossrefs

Positions of 2's in A347381.

Cf. A000203, A005101, A005231, A005820, A005940, A027687, A336702, A347391, A347393, A347394, A347879, A348041.

Cf. also A336834, A353365.

Sequence in context: A168415 A199635 A131864 * A179443 A264828 A080756

Adjacent sequences: A347389 A347390 A347391 * A347393 A347394 A347395

Keyword

nonn,hard,more

Author

Antti Karttunen, Aug 30 2021

Status

approved