%I #57 Mar 11 2024 18:05:30
%S 1,10,56,9196,9504,56160,121176,239096,354892,411264,555520,716040,
%T 804384,904704,1063348,1387386,1444352,1454112,1884800,2708640,
%U 3317248,3548920,4009824,4634784,6179712,6795360,7285248,14511744,16328466,28377216,29855232,31940280,37444736,42711552,49762944,52815744
%N Numbers k for which k * gcd(sigma(k), A276086(k)) is equal to sigma(k) * gcd(k, A276086(k)), where A276086 is the primorial base exp-function, and sigma gives the sum of divisors of its argument.
%C Numbers k such that k * A324644(k) = A000203(k) * A324198(k).
%C Numbers k such that gcd(A064987(k), A324580(k)) = gcd(A064987(k), A351252(k)).
%C Numbers k such that their abundancy index [sigma(k)/k] is equal to A324644(k)/A324198(k). See A364286.
%C A324644 gives odd values for even numbers and for the odd squares. A324198 is odd on all arguments, therefore on odd squares the above equation reduces to odd * odd = odd * odd, and on odd nonsquares as odd * even = even * odd. It is an open question whether there are any odd terms after the initial a(1)=1.
%C If k is even, but not a multiple of 3, then A276086(k) is a multiple of 3, but not even (i.e., is an odd multiple of 3). If for such k also sigma(k) = 3*k, then A007949(A324644(k)) = min(A007949(sigma(k)), A007949(A276086(k))) = 1, while A007949(A324198(k)) = min(A007949(k), A007949(A276086(k))) = 0, therefore all such k's do occur in this sequence, for example, the two known terms of A005820 (3-perfect numbers) that are not multiples of three: 459818240, 51001180160, but also any hypothetical term of A005820 of the form 4u+2, where 2u+1 is not multiple of 3, and which by necessity is then also an odd perfect number.
%C Similarly, of the 65 known 5-multiperfect numbers (A046060), those 20 that are not multiples of five are included in this sequence. Note that all 65 are multiples of six.
%C It is conjectured that the intersection of this sequence with the multiperfect numbers (A007691) gives A323653, see comments in the latter.
%C For all even terms k of this sequence, A007814(A000203(k)) = A007814(k), sigma preserves the 2-adic valuation, and A007949(A000203(k)) >= A007949(k), i.e., does not decrease the 3-adic valuation. The condition is equivalence (=) when k is a multiple of 6. With odd terms, any hypothetical odd perfect number x would yield a one greater 2-adic valuation for sigma(x) than for x, but would satisfy the main condition of this sequence. - Corrected Feb 17 2022
%C If k is a nonsquare positive odd number (in A088828), then it must be a term of A191218. - _Antti Karttunen_, Mar 10 2024
%H Antti Karttunen, <a href="/A351458/b351458.txt">Table of n, a(n) for n = 1..132</a>
%H Antti Karttunen, <a href="https://oeis.org/plot2a?name1=A324644&name2=A324198&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawlines=true">Ratio A324644(n)/A324198(n) plotted with OEIS Plot2-tool</a>
%H <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%o (PARI)
%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o isA351458(n) = { my(s=sigma(n), z=A276086(n)); (n*gcd(s,z))==(s*gcd(n,z)); };
%o (PARI)
%o A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]); \\ Works OK with rationals also!
%o isA351458(n) = { my(orgn=n, s=sigma(n), abi=s/n, p=2, q=A006530(abi), d, e1, e2); while((1!=abi)&&(p<=q), d = n%p; e1 = min(d, valuation(s, p)); e2 = min(d, valuation(orgn, p)); d = e1-e2; if(valuation(abi,p)!=d, return(0), abi /= (p^d)); n = n\p; p = nextprime(1+p)); (abi==1); }; \\ (This implementation does not require the construction of largish intermediate numbers, A276086, but might still be slower and return a few false positives on the long run, so please check the results with the above program). - _Antti Karttunen_, Feb 19 2022
%Y Cf. A000203, A005820, A007691, A007814, A007949, A046060, A088828, A191218, A276086, A323653, A324198, A324644, A327938, A364286.
%Y Cf. A349745 for a similar sequence, also A351548 and A351549.
%K nonn
%O 1,2
%A _Antti Karttunen_, Feb 13 2022