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 odd abundant numbers that end in 1, 3, 7, and 9, are 81081, 153153, 207207, and 189189, respectively.

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

List of abundant numbers

List of the first 120 abundant numbers:

12	18	20	24	30	36	40	42	48	54
56	60	66	70	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	426	432	438	440	444	448	450	456	460
462	464	468	474	480	486	490	492	498	500

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)