A349284 Numbers k such that A051378(k) > 2*k and A333926(k) <= 2*k.

126720, 134400, 149760, 188160, 195840, 456960, 510720, 549120, 618240, 718080, 748800, 779520, 802560, 833280, 940800, 979200, 994560, 1094400, 1102080, 1155840, 1263360, 1324800, 1393920, 1424640, 1585920, 1639680, 1670400, 1785600, 1800960, 1908480, 1946880

Offset: 1

Author

Amiram Eldar, Nov 13 2021

Keywords

• nonn

Comments

(1+e)-abundant numbers are numbers k such that A051378(k) > 2*k, i.e., numbers k whose sum of (1+e)-divisors exceeds 2*k.

Since all the recursive divisors (see A282446) of a number are also its (1+e)-divisors, the sequence of (1+e)-abundant numbers includes all the recursive abundant numbers (A333928). The first 21387 (1+e)-abundant numbers are also recursive abundant numbers. Therefore, this sequence includes only the (1+e)-abundant numbers that are not recursive abundant numbers.

Examples

			126720 is a term since A051378(126720) = 261144 > 2*126720 = 253440 and A333926(126720) = 246168 < 253440.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A049599, A049603, A051378, A282446, A333926, A333928.

Programs

Mathematica

oesigma[1] = 1; oesigma[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recsigma[1] = 1; recsigma[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^6], oesigma[#] > 2*# && recsigma[#] <= 2*# &]