Digit Permutations of Numbers

I've written about permutations of the digits of a number in many posts. For example, in my recent post titled Determinants of Circulant Matrices (11th of August 2024) I looked at circulant matrices that have determinants with digits that are permutations of the number's digits. In this current post, I'll be looking at some new types, specifically where the digits of the number are equal to:

• a permutation of the digits of the aliquot sum of the divisors of the number
• a permutation of the digits of the sum of the divisors of the number
• a permutation of the digits of the totient of the number

SUM OF PROPER DIVISORS

There are 37 numbers in the range up to 40,000 that have the same digits as the sum of their proper divisors. These numbers are listed below together with the permutation. Notice that perfect numbers remain unchanged. Permalink.

(6, 6), (28, 28), (411, 141), (496, 496), (604, 460), (1305, 1035), (3664, 3466), (4086, 4806), (4672, 4726), (4896, 9846), (5046, 5406), (7785, 5787), (8128, 8128), (8739, 3897), (9331, 1933), (14535, 13545), (16012, 12016), (18342, 21438), (18585, 18855), (19648, 19468), (20634, 23046), (21534, 23154), (21628, 16228), (22365, 22563), (25911, 11529), (27568, 25876), (28108, 21088), (29160, 69210), (29188, 21898), (31185, 38511), (32091, 13029), (32271, 12273), (34956, 53496), (35898, 38598), (35925, 23595), (36172, 27136), (37698, 39678)

These numbers constitute OEIS A085844 and they are:

6, 28, 411, 496, 604, 1305, 3664, 4086, 4672, 4896, 5046, 7785, 8128, 8739, 9331, 14535, 16012, 18342, 18585, 19648, 20634, 21534, 21628, 22365, 25911, 27568, 28108, 29160, 29188, 31185, 32091, 32271, 34956, 35898, 35925, 36172, 37698

SUM OF DIVISORS

There are 45 numbers in the range up to 40,000 that have the same digits as the sum of their divisors including the number itself this time. These numbers are listed below together with the permutations. Permalink.

(1, 1), (69, 96), (258, 528), (270, 720), (276, 672), (609, 960), (639, 936), (2391, 3192), (2556, 6552), (2931, 3912), (3409, 3904), (3678, 7368), (3679, 3976), (4291, 4912), (5092, 9520), (6937, 7936), (8251, 8512), (10231, 11032), (12087, 18720), (12931, 13192), (15480, 51480), (16387, 18736), (20850, 52080), (22644, 62244), (22893, 32928), (24369, 32496), (26145, 52416), (26442, 62244), (27846, 78624), (28764, 78624), (29880, 98280), (29958, 59928), (30823, 33208), (31812, 81312), (32658, 65328), (34207, 34720), (34758, 75348), (34909, 39904), (36045, 65340), (36249, 49632), (36729, 67392), (36978, 73968), (36990, 99360), (38491, 39184), (38538, 83538)

These numbers constitute OEIS A115920 and they are:

1, 69, 258, 270, 276, 609, 639, 2391, 2556, 2931, 3409, 3678, 3679, 4291, 5092, 6937, 8251, 10231, 12087, 12931, 15480, 16387, 20850, 22644, 22893, 24369, 26145, 26442, 27846, 28764, 29880, 29958, 30823, 31812, 32658, 34207, 34758, 34909, 36045, 36249, 36729, 36978, 36990, 38491, 38538

TOTIENT

There are 36 numbers in the range up to 40,000 that have the same digits as their totients. These numbers are listed below together with the permutations. Permalink.

(1, 1), (21, 12), (63, 36), (291, 192), (502, 250), (2518, 1258), (2817, 1872), (2991, 1992), (4435, 3544), (5229, 2952), (5367, 3576), (5637, 3756), (6102, 2016), (6174, 1764), (6543, 4356), (6822, 2268), (7236, 2376), (7422, 2472), (8022, 2280), (8541, 5184), (8982, 2988), (17631, 11736), (18231, 11832), (18261, 12168), (20301, 13200), (20518, 10258), (20617, 20176), (21058, 10528), (22471, 21472), (22851, 15228), (25196, 12596), (25918, 12958), (27615, 12576), (29817, 19872), (34816, 16384), (35683, 33568)

These numbers constitute OEIS A115921 and all except 1 are deficient. They are:

1, 21, 63, 291, 502, 2518, 2817, 2991, 4435, 5229, 5367, 5637, 6102, 6174, 6543, 6822, 7236, 7422, 8022, 8541, 8982, 17631, 18231, 18261, 20301, 20518, 20617, 21058, 22471, 22851, 25196, 25918, 27615, 29817, 34816, 35683