A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

1, 6, 40, 126, 154, 204, 1716, 1914, 2772, 8580, 11264, 12090, 12540, 50960, 62790, 64350, 77748, 83200, 104720, 152320, 186116, 193440, 331890, 382720, 432768, 518364, 648788, 684684, 753480, 817344, 895356, 1083852, 1113840, 1619352, 1675044, 1743588, 1759680, 1991340, 2060322, 2360484, 2492028, 2621080, 2932800

Offset: 1

Author

Antti Karttunen, Jul 19 2020

Keywords

• nonn

Comments

Numbers k for which A336314(k) = A323173(k).

Sequence A122111(A001599(n)), n >= 1, sorted into ascending order. Positions of zeros in A323174 (corresponding to perfect numbers similarly mapped) is a subsequence.

Note that all terms after 1 seem to be present in A102750. This observation is equal to Ore's conjecture that there are no odd Harmonic numbers larger than one.

Also, all terms after 1 seem to be even, which would imply that apart from its initial 1, A001599 were a subsequence of A102750. However, this is false, as there are terms of A001599 not in A102750, for example 8011798098793361832960 found by David A. Corneth. Note that A122111(8011798098793361832960) = 96922193555635754403846044921625, which is thus an odd term of this sequence.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..61
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A001599, A088902, A102750, A122111, A323173, A323174, A336314, A336315, A336353, A336397 (the intersection with A001599).

Programs

PARI

isA001599(n) = !((sigma(n,0)*n)%sigma(n,1));
isA336317(n) = isA001599(A122111(n)); \\ Program for A122111 given under that entry.

PARI

\\ Standalone program:
isA336317(n) = if(1==n,1,my(f=factor(n),es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,d=1,s=1,x=1,p,e); for(i=1, #es, pri += es[i]; p = prime(pri); e = 1+is[i]-is[1+i]; d *= e; s *= ((p^e)-1)/(p-1); x *= (p^(e-1))); !((x*d)%s));