cp's OEIS Frontend

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

%I A336317 #43 Jul 22 2020 13:26:12
%S A336317 1,6,40,126,154,204,1716,1914,2772,8580,11264,12090,12540,50960,62790,
%T A336317 64350,77748,83200,104720,152320,186116,193440,331890,382720,432768,
%U A336317 518364,648788,684684,753480,817344,895356,1083852,1113840,1619352,1675044,1743588,1759680,1991340,2060322,2360484,2492028,2621080,2932800
%N A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).
%C A336317 Numbers k for which A336314(k) = A323173(k).
%C A336317 Sequence A122111(A001599(n)), n >= 1, sorted into ascending order. Positions of zeros in A323174 (corresponding to perfect numbers similarly mapped) is a subsequence.
%C A336317 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.
%C A336317 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.
%H A336317 Antti Karttunen, <a href="/A336317/b336317.txt">Table of n, a(n) for n = 1..61</a>
%H A336317 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%o A336317 (PARI)
%o A336317 isA001599(n) = !((sigma(n,0)*n)%sigma(n,1));
%o A336317 isA336317(n) = isA001599(A122111(n)); \\ Program for A122111 given under that entry.
%o A336317 (PARI)
%o A336317 \\ Standalone program:
%o A336317 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));
%Y A336317 Cf. A001599, A088902, A102750, A122111, A323173, A323174, A336314, A336315, A336353, A336397 (the intersection with A001599).
%K A336317 nonn
%O A336317 1,2
%A A336317 _Antti Karttunen_, Jul 19 2020