cp's OEIS Frontend

A378738 Primitively abundant numbers k for which A378665(k) > A378664(k).

%I A378738 #30 Jan 29 2025 07:55:34
%S A378738 66,102,174,186,222,246,258,282,318,354,366,402,426,438,474,498,534,
%T A378738 582,606,618,642,654,678,748,762,786,822,834,894,906,942,978,1002,
%U A378738 1038,1074,1086,1146,1158,1182,1194,1266,1338,1362,1374,1398,1434,1446,1506,1542,1578,1614,1626,1662,1686,1698,1758,1842,1866
%N A378738 Primitively abundant numbers k for which A378665(k) > A378664(k).
%C A378738 Subsequence of A378737: 1496 is its first term that does not occur here.
%C A378738 Equal to primitively abundant numbers k such that A032742(k) > A378664(k), because for primitively abundant numbers the greatest non-abundant divisor is the largest proper divisor, A378665(k) = A032742(k).
%C A378738 Question: What is the asymptotic density of these numbers among A091191? Does it tend to 1?
%C A378738 Conjecture: A001222(a(n)) = 3 <=> 3|a(n).
%H A378738 Antti Karttunen, <a href="/A378738/b378738.txt">Table of n, a(n) for n = 1..20000</a>
%F A378738 {k such that A294930(k) = 1 and A032742(k) > A378664(k)}.
%e A378738 Examples given in A378737 for 66, 748, 1866, and 1870 all work also here, because those four numbers are all in A091191.
%o A378738 (PARI)
%o A378738 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A378738 A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
%o A378738 A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };
%o A378738 A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
%o A378738 is_A091191(n) = if(sigma(n)<=2*n, 0, fordiv(n,d,if(d<n && sigma(d)>2*d, return(0))); (1));
%o A378738 is_A378738(n) = (is_A091191(n) && (A378664(n)!=A032742(n)));
%Y A378738 Intersection of A091191 and A378737.
%Y A378738 Cf. A001222, A032742, A294930, A337372, A341612, A341614, A378664, A378665, A378735, A378736, A378739 [= A378664(a(n))], A378741 (subsequence), A378742 (subsequence after its initial term).
%K A378738 nonn
%O A378738 1,1
%A A378738 _Antti Karttunen_, Dec 07 2024