Yesterday I turned 25446 days and this number was identified by Numbers Aplenty as an admirable number, defined as a number \(n\) for which there exists a divisor \(d\) of \(n\) such that \(2n = \sigma(n)-2d\). In other words, \(n\) is equal to the sum of its proper divisors, where one of them has a minus sign.
For 25446, the divisors are: 1, 2, 3, 6, 4241, 8482, 12723, 25446 and the sum of these divisors is 50904. However, 50904 - 2 x 6 = 50892 = 2 x 25446 and here the divisor 6 has been assigned the minus sign. The modified divisors (1, 2, 3, -6, 4241, 8482, 12723) now add to 25446. The previously mentioned website goes on to say that:
Clearly, admirable numbers are a subset of abundant numbers and they are infinite because, for example, all the numbers 6\(p\), with \(p\)>3 prime, are admirable. The largest number that cannot be written as a sum of admirable numbers is 1003. Pairs of consecutive admirable numbers are rarer than pairs of consecutive abundant numbers. Up to \(10^{12}\), there are only two such pairs, namely 29691198404, 29691198405 and 478012798575, 478012798576.
On the other hand, pairs of admirable numbers that differ by two are more common but still sparse. There are 72 such pairs up to 27000.
The smallest 3 x 3 magic square made up of admirable numbers is shown in Figure 1.
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| Figure 1: smallest possible magic square made from admirable numbers |
OEIS A111592 lists the initial admirable numbers:
12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222, 224, 234, 246, 258, 270, 282, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 650, 654, 672, 678, 762, 786, 812, ...
These numbers are related to Zumkellar, Half-Zumkellar and pseudoperfect numbers in that they all involve the divisors of the number. See my blog post on these sorts of numbers.
In the OEIS comments, we read that "the concept of admirable numbers was developed by educator Jerome Michael Sachs (1914-2012) for a television in-service training course in mathematics for elementary school teachers." Here is the link to the article that he wrote in The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295. However, in that article he allows more than one of the divisors of a number to be negative. For example, he writes 24 as being equal to the following algebraic sum of its divisors: 4+6+8+12-1-2-3. However, this is the same as 1+2+3+4+8+12-6 so it's not clear whether a sum involving multiple negative divisors is always equivalent to another sum involving a single negative divisor.
Sachs also introduces the notion of a compatible number pair as an extension or relaxation of the concept of an amicable number pair. For example, 220 and 284 are an amicable pair because the proper divisors of each add to the other number. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 and these add to 284. The proper divisors of 284 are 1, 2, 4, 71 and 142 and these add to 220. In such cases, the smaller number is abundant and the larger number deficient.
Sach's proposal for a compatible number pair is two numbers such that the algebraic sums of their divisors each leads to the other number. For example:
- 30 has divisors of 1, 2, 3, 5, 6, 10, 15
- 40 has divisors of 1, 2, 4, 5, 8, 10 and 20
- 40 = 2 + 3 + 5 + 6 + 10 + 15 - 1
- 30 = 1 + 2 + 4 + 5 + 8 + 20 - 10
So he defines 30 and 40 as compatible numbers.
The smaller members of such pairs are listed in OEIS A109797 while the larger members are listed in OEIS A109798.
Here is the SageMath code to generate the admirable numbers between 25000 and 26000
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