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

Author

Antti Karttunen, Aug 30 2021

Keywords

• nonn
• hard
• more

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.

Examples

			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].
		

Links

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

Programs

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));