Duffinian numbers

So called by Richard Duffy who introduced them, they are those composite numbers $n$

which have no prime factors in common with the sum of their divisors $\sigma(n)$ .

For example, 35 is Duffinian since it is relatively prime to the sum of its divisors 1 + 5 + 7 + 35 = 48.

It is easy to see that there are infinite such numbers. Indeed any number of the form $p^k$ where $p$ is prime and $k\ge2$ is Duffinian, since $\sigma(p^k)=1+p+p^2+\cdots+p^k$ cannot be divisible by $p$ .

P. Heichelheim proved that exists a run of 5 consecutive Duffinian numbers starting at 202605639573839041, and that cannot exist a longer such run.

Rose Mary Zbiek has proved that every even Duffinian number is either a square or twice a square.

The smallest 3 × 3 magic square whose entries are consecutive Duffinian numbers is

18649	18631	18643
18635	18641	18647
18639	18651	18633

The first Duffinian numbers are 4, 8, 9, 16, 21, 25, 27, 32, 35, 36, 39, 49, 50, 55, 57, 63, 64, 65, 75, 77, 81, 85, 93, 98, 100 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many Duffinian numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links OEIS, Sequence A003624

Duffinian numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 32 50 64 + 9999392 aban 16 21 25 + 9000999 abundant 36 100 144 + 9999392 Achilles 392 800 968 + 9999392 alt.fact. 326981 alternating 16 21 25 + 9898987 amenable 16 21 25 + 9999997 apocalyptic 243 245 247 + 29999 arithmetic 21 27 35 + 9999997 astonishing 27 3591 23287 + 2732353 automorphic 25 625 2890625 Bell 203 4213597 betrothed 75 1575 1925 + 2576945 binomial 21 35 36 + 9956953 brilliant 21 25 35 + 9999811 c.decagonal 361 451 551 + 9989911 c.heptagonal 253 841 1331 + 9966797 c.nonagonal 55 253 325 + 9956953 c.octagonal 25 49 81 + 9991921 c.pentagonal 16 391 1156 + 9955051 c.square 25 85 221 + 9994921 c.triangular 64 85 235 + 9988471 cake 64 93 299 + 9735503 Canada 125 8549 16999 Carmichael 1105 8911 15841 + 9613297 Carol 959 65023 261119 4190207 congruent 21 39 55 + 9999997 constructible 16 32 64 + 8388608 cube 27 64 125 + 9393931 Cullen 25 65 161 + 4718593 Cunningham 35 50 63 + 9998243 Curzon 21 50 65 + 9999921 cyclic 35 65 77 + 9999997 D-number 21 39 57 + 7043133 d-powerful 63 175 209 + 9999865 de Polignac 905 959 1207 + 9999997 decagonal 27 85 175 + 9968227 deceptive 259 451 481 + 9897121 deficient 16 21 25 + 9999997 dig.balanced 21 35 49 + 9999993 droll 800 5184 6272 + 5017600 eban 32 36 50 + 4032064 economical 16 21 25 + 9999803 emirpimes 39 49 85 + 9999967 enlightened 256 2048 2304 + 3720087 equidigital 16 21 25 + 9999625 eRAP 98 2255 7625 + 9960047 esthetic 21 32 65 + 9898987 Eulerian 57 247 1191 1048555 evil 27 36 39 + 9999993 fibodiv 75 183 305 + 8899945 Fibonacci 21 55 144 + 9227465 Friedman 25 121 125 + 999163 frugal 125 128 243 + 9999803 gapful 100 121 143 + 9999827 Gilda 49 977909 3242189 + 4848955 happy 32 49 100 + 9999983 Harshad 21 27 36 + 9999881 heptagonal 55 81 189 + 9967027 hex 169 217 469 + 9986401 hexagonal 325 1225 3655 + 9916831 highly composite 36 hoax 85 265 319 + 9999355 Hogben 21 57 111 + 9988761 house 32 155 1285 + 9716393 hungry 144 161449 712201 2401519 hyperperfect 21 301 325 + 9699181 iban 21 27 77 + 777773 iccanobiF 39 1053 12815 + 792517 idoneal 16 21 25 + 385 impolite 16 32 64 + 8388608 inconsummate 63 65 75 + 999993 insolite 111 interprime 21 39 50 + 9999755 Jacobsthal 21 85 171 + 1398101 Jordan-Polya 16 32 36 + 9437184 junction 111 115 305 + 9999959 Kaprekar 55 999 7777 + 5072059 katadrome 21 32 50 + 9876541 Kynea 4223 16639 1050623 4198399 Lehmer 85 133 247 + 9996205 Leyland 32 57 100 + 8389137 lonely 1343 31429 31431 + 4652429 Lucas 15127 39603 64079 + 4870847 lucky 21 25 49 + 9999997 Lynch-Bell 36 128 175 + 1382976 magic 65 111 175 + 9951391 magnanimous 16 21 25 + 8844449 metadrome 16 25 27 + 2456789 modest 39 49 111 + 9998991 Moran 21 27 63 + 9999881 Motzkin 21 323 835 narcissistic 371 407 92727 1741725 nialpdrome 21 32 50 + 9999997 nonagonal 75 111 325 + 9885961 nude 36 55 77 + 9977373 O'Halloran 36 oban 16 25 27 + 999 octagonal 21 65 133 + 9988225 odious 16 21 25 + 9999997 palindromic 55 77 111 + 9998999 pancake 16 121 301 + 9997157 panconsummate 21 36 39 + 3097 pandigital 21 75 201 + 9998163 partition 77 385 1255 + 9289091 pentagonal 35 247 425 + 9991051 pernicious 21 25 35 + 9999997 Perrin 39 119 209 + 2240877 plaindrome 16 25 27 + 8999999 Poulet 341 1105 1387 + 9995671 power 16 25 27 + 9991921 powerful 16 25 27 + 9999392 practical 16 32 36 + 9999392 prim.abundant 1575 1660725 2553525 + 9819225 primeval 1037 1079 1379 + 1234679 pronic 2450 2827442 Proth 25 49 57 + 9998337 pseudoperfect 36 100 144 + 1000000 rare 65 repdigit 55 77 111 + 7777777 repfigit 75 1537 7385 + 7913837 repunit 21 57 63 + 9988761 Rhonda 12985 15625 19435 + 8519579 Ruth-Aaron 16 25 49 + 9997417 Saint-Exupery 202500 Sastry 183 715 self 64 75 121 + 9999965 semiprime 21 25 35 + 9999997 sliding 25 65 133 + 9868025 Smith 27 85 121 + 9999355 sphenic 385 399 595 + 9999983 square 16 25 36 + 9991921 star 121 253 793 + 9992341 straight-line 111 333 543 + 7777777 strobogrammatic 111 689 1111 + 6998669 subfactorial 265 super Niven 36 50 100 + 9000000 super-d 81 119 169 + 9999919 superabundant 36 tau 36 128 225 + 9903609 taxicab 20683 195841 327763 + 9443761 tetrahedral 35 1771 6545 + 9290431 tetranacci 20569 283953 547337 triangular 21 36 55 + 9956953 tribonacci 81 927 1705 + 1389537 trimorphic 25 49 75 + 7890625 uban 16 21 25 + 9000099 Ulam 16 36 57 + 9999977 undulating 121 161 171 + 9797979 unprimeable 324 325 512 + 9999965 untouchable 324 576 784 + 984064 upside-down 55 64 1379 + 9995111 vampire 2187 117067 124483 + 536539 wasteful 36 39 50 + 9999997 Wieferich 3279 7651 10533 + 3837523 Woodall 63 323 2047 + 9961471 Zeisel 4505 5719 24211 + 9773731 Zuckerman 36 111 115 + 7111111 Zumkeller 1575 14175 39375 + 96075 zygodrome 55 77 111 + 9999977