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

Author

Antti Karttunen, Dec 06 2024

Keywords

• nonn

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.

Examples

			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.
		

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).

Crossrefs

Subsequence of A005101.

Cf. A378738 (subsequence).

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

Programs

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)));

Formula

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