Abundant number

An abundant number is a number [math]\displaystyle{ n }[/math] for which its sum of divisors, excluding itself, is greater than [math]\displaystyle{ 2n }[/math]. 12 is an abundant number because [math]\displaystyle{ 1+2+3+4+6+12\gt 24 }[/math]. 18 is also an abundant number because [math]\displaystyle{ 1+2+3+6+9+18\gt 36 }[/math]. The series is in the Online Encyclopedia of Integer Sequences (Sloane) as A005101. 120 is considered a super abundant number since the sum of its divisors equals exactly twice the number. 945 is the first odd abundant number. The smallest abundant number to end in 1 is 81,081.

If a number is not abundant, it is called either deficient or perfect.

List of abundant numbers

List of the first 100 abundant numbers:

12	18	20	24	30	36	40	42	48	54
56	60	66	72	78	80	84	88	90	96
100	102	104	108	112	114	120	126	132	138
140	144	150	156	160	162	168	174	176	180
186	192	196	198	200	204	208	210	216	220
222	224	228	234	240	246	252	258	260	264
270	272	276	280	282	288	294	300	304	306
308	312	318	320	324	330	336	340	342	348
350	352	354	360	364	366	368	372	378	380
384	390	392	396	400	402	408	414	416	420

Abundant Number Media

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  Demonstration, with Cuisenaire rods, of the abundance of the number 12

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  Let a(n) be the number of abundant numbers not exceeding n. Plot of a(n)/n for n < 10^6 (with n log-scaled)