A378737 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, 1496, 1506, 1542, 1578, 1614, 1626, 1662, 1686, 1698, 1758, 1842

Offset

1,1

Comments

For most of these numbers k, A378664(k) = 6. Note that A003961(6) = A003961(2*3) = 3*5 = 15 > 2*6, while sigma(6) = 12, making 6 non-abundant. Note that there seems to be only a finite amount (namely 13, see A337372) of such "stopper semiprimes" that would prevent of A378736 obtaining value 1.

Other possible values that A378664 obtains on these numbers are for example 68, 136, 170, 256, 290, 370, 410, 430, 470, 530, 646, 682, 754, 1276, 1292, 1364, 1508, 1628, 1804, 1892, 2068, 2332, 4756, 6844, 10846, 15334, etc. See A378740, which contains some of these.

Links

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

Index entries for sequences related to prime indices in the factorization of n.

Index entries for sequences related to sigma(n).

Formula

{A005101(i) for such indices i where A378735(i) > A378736(i)}.

Example

66 is a term as A378665(66) = 33, but A378664(66) = 6.
748 is a term as A378665(748) = 374, but A378664(748) = 68.
1866 is a term as A378665(1866) = 933, but A378664(1866) = 6.
1870 is a term as A378665(1870) = 935, but A378664(1870) = 170.

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A294935(n) = (sigma(n)<=(2*n));
A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
A378664(n) = { fordiv(n, d, if(A341612(n/d), return(n/d))); (1); };
A378665(n) = { fordiv(n, d, if(A294935(n/d), return(n/d))); (1); };
is_A378737(n) = (!A294935(n) && (A378664(n)!=A378665(n)));

Crossrefs

Subsequence of A005101.

Cf. A378738 (subsequence).

Cf. A337372, A378664, A378665, A378735, A378736, A378740.

Sequence in context: A109759 A194196 A281937 * A378738 A109141 A350197

Adjacent sequences: A378734 A378735 A378736 * A378738 A378739 A378740

Keyword

nonn,new

Author

Antti Karttunen, Dec 06 2024

Status

approved