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

66, 102, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 748, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 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

Offset

1,1

Comments

Subsequence of A378737: 1496 is its first term that does not occur here.

Equal to primitively abundant numbers k such that A032742(k) > A378664(k), because for for primitively abundant numbers the greatest non-abundant divisor is the largest proper divisor, A378665(k) = A032742(k).

Question: What is the asymptotic density of these numbers among A091191? Does it tend to 1?

Conjecture: A001222(a(n)) = 3 <=> 3|a(n).

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

{k such that A294930(k) = 1 and A032742(k) > A378664(k)}.

Example

Examples given in A378737 for 66, 748, 1866, and 1870 all work also here, because those four numbers are all in A091191.

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378664(n) = { fordiv(n, d, if(A341612(n/d), return(n/d))); (1); };
A032742(n) = if(1==n, n, n/vecmin(factor(n)[, 1]));
is_A091191(n) = if(sigma(n)<=2*n, 0, fordiv(n, d, if(d<n && sigma(d)>2*d, return(0))); (1));
is_A378738(n) = (is_A091191(n) && (A378664(n)!=A032742(n)));

Crossrefs

Intersection of A091191 and A378737.

Cf. A001222, A032742, A294930, A337372, A341612, A341614, A378664, A378665, A378735, A378736, A378739 [= A378664(a(n))], A378741 (subsequence), A378742 (subsequence after its initial term).

Sequence in context: A194196 A281937 A378737 * A109141 A350197 A350204

Adjacent sequences: A378735 A378736 A378737 * A378739 A378740 A378741

Keyword

nonn,new

Author

Antti Karttunen, Dec 07 2024

Status

approved