Hogben numbers

The

-th Hogben number $H_n$

is equal to $n^2-n+1$

In a spiral arrangment of the integers, Hogben numbers appear on the main diagonal (see picture aside).

Hogben numbers are often called central polygonal numbers.

$H_n$ is also the maximal number of ones that a $\{0,1\}$ $n\times n$ matrix can contain and still be invertible.

The first Hogben numbers are 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381 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 Hogben numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links OEIS, Sequence A002061
R.Munafo, Hogben's Centered Polygonal Numbers

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

a-pointer 13 24181 121453 + 9487636621 aban 13 21 31 + 999999000001 Achilles 7906143973 alternating 21 43 183 + 981036363 amenable 13 21 57 + 999856021 apocalyptic 157 871 931 + 29757 arithmetic 13 21 31 + 9995083 balanced p. 157 211 1123 + 9945973171 binomial 21 91 703 + 4734390674091 brilliant 21 703 871 + 999413383 c.decagonal 31 211 9901 + 729500152756861 c.heptagonal 43 463 38221 + 300797044280503 c.nonagonal 91 703 104653 + 160829915470003 c.pentagonal 31 601 43891 + 131602780302451 c.square 13 421 14281 + 745519146510613 c.triangular 31 2971 291061 + 26835295698121 Carmichael 5310721 2278677961 9593125081 29859667201 Chen 13 31 157 + 99790111 congruent 13 21 31 + 9969807 cube 343 Cunningham 31 8191 Curzon 21 273 813 + 199953741 cyclic 13 31 43 + 9982441 D-number 21 57 111 + 6993381 d-powerful 43 463 2353 + 9526483 de Polignac 757 1807 17557 + 99930013 deceptive 91 703 50851 + 30244862011 deficient 13 21 31 + 9995083 dig.balanced 21 601 651 + 199812361 Duffinian 21 57 111 + 9988761 economical 13 21 31 + 19994313 emirp 13 31 73 + 199162657 emirpimes 183 381 553 + 99810091 equidigital 13 21 31 + 19994313 eRAP 21058218111 esthetic 21 43 343 10101 Eulerian 57 1191 evil 43 57 111 + 999792781 fibodiv 183 87321 Fibonacci 13 21 Friedman 343 326613 453603 frugal 343 105301 179353 + 891589741 gapful 12883 16003 17823 + 99992242441 good prime 307 3907 6007 + 196770757 happy 13 31 91 + 9988761 Harshad 21 111 133 + 9988303423 heptagonal 18361 191407 1342123 + 29093062258603 hex 91 1261 17557 + 346332937450507 hexagonal 91 703 104653 + 160829915470003 hoax 44733 190533 198471 + 99351057 Honaker 5701 8011 98911 + 975906361 hyperperfect 21 1333 16513 + 1951956868501 iban 21 43 73 + 777043 iccanobiF 13 idoneal 13 21 57 + 273 inconsummate 381 553 993 + 995007 insolite 111 interprime 21 111 381 + 99450757 Jacobsthal 21 43 junction 111 307 507 + 99870043 Kaprekar 703 katadrome 21 31 43 + 87321 Lehmer 91 133 703 + 984588837961 Leyland 57 lonely 211 lucky 13 21 31 + 9976123 m-pointer 1123 magic 111 magnanimous 21 43 601 6007 metadrome 13 57 157 modest 13 111 133 + 1024992241 Moran 21 111 133 + 99390931 Motzkin 21 nialpdrome 21 31 43 + 998844421 nonagonal 111 651 99541 + 423236211589731 nude 111 oban 13 57 73 + 993 octagonal 21 133 4033 + 36485471742721 odious 13 21 31 + 999982507 Ormiston 37831 530713 7681213 + 1997598331 palindromic 111 343 757 + 755971313179557 palprime 757 30103 pancake 211 7141 242557 + 372759573255307 panconsummate 21 31 43 + 211 pandigital 21 217623 228963 + 15401701 pentagonal 651 5551 6247501 + 576009576039801 pernicious 13 21 31 + 9988761 persistent 10158926473 10865247933 12859673401 + 79865303421 Pierpont 13 73 plaindrome 13 57 111 + 44444455555557 Poulet 4033 25761 65281 + 421925923038061 power 343 powerful 343 7906143973 prime 13 31 43 + 999999000001 primeval 13 Proth 13 57 241 + 274877382657 repdigit 111 repunit 13 21 31 + 999999993568953 Ruth-Aaron 154843 1532291881 6847148757 + 19369541451 Sastry 183 self 31 211 703 + 999160491 semiprime 21 57 91 + 99910021 sliding 133 Smith 4557 44733 190533 + 99351057 sphenic 273 651 1407 + 99950007 star 13 73 1261 + 64514252901853 straight-line 111 strobogrammatic 111 10101 1001001 + 100000010000001 strong prime 307 757 3907 + 99211561 super-d 31 381 601 + 9976123 taxicab 2436429101743 12166921836433 triangular 21 91 703 + 160829915470003 tribonacci 13 truncatable prime 13 31 43 + 5113 twin 13 31 43 + 998907631 uban 13 21 31 + 96040049000007 Ulam 13 57 241 + 9894171 undulating 343 757 10101 upside-down 73 91 98392919181721 wasteful 57 91 273 + 9995083 weak prime 13 31 43 + 99990001 weakly prime 23402421463 Wieferich 22953 Zuckerman 111 zygodrome 111 33777110011 77333770011 + 2255660055111