magic square constants

In its most basic form an $n\times n$ magic square is an arrangement of the numbers from $1$

on a square grid in such a way the entries on the rows, columns and two main diagonals have the same sum.

Magic squares exists for every positive $n\neq2$ and the common sum, which we can call magic constant is $M(n)=(n^3+n)/2$ .

For example, $3\times3$ and $4\times4$ magic squares like these

$\begin{array}{|c|c|c|}\hline 6 & 7 & 2\\\hline 1 & 5 & 9\\\hline 8 & 3 & 4\\\hline \end{array}\quad\quad\quad\begin{array}{|c|c|c|c|}\hline 13 & 2 & 3 & 16 \\\hline 8 & 11 & 10 & 5\\\hline 12 & 7 & 6 & 9 \\\hline 1 & 14 & 15 & 4 \\\hline \end{array}$

have magic constant $M(3)=15$

and

The first magic constants are 1, 5, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439 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 magic constants are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Magic Constant
Wikipedia, Magic square
OEIS, Sequence A006003

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

aban 15 34 65 + 928166000645 abundant 260 870 6924 + 48668230 admirable 87836 alternating 34 65 369 + 727210385 amenable 65 260 369 + 995433385 apocalyptic 671 1105 1695 + 29679 arithmetic 15 65 111 + 9951391 astonishing 15 Bell 15 binomial 15 2925 brilliant 15 671 c.pentagonal 2287231 c.square 1105 cake 15 canyon 505 2056 83215 9732689 Carmichael 1105 2465 congruent 15 34 65 + 9951391 constructible 15 34 2056 8388736 Cullen 65 Cunningham 15 65 73599240 Curzon 65 369 870 + 193710609 cyclic 15 65 671 + 9951391 D-number 15 111 d-powerful 175 2465 66351 + 5848655 de Polignac 2465 78759 102719 + 80939585 decagonal 175 1105 deceptive 14911 1826209 deficient 15 34 65 + 9951391 dig.balanced 15 260 8801 + 199344496 double fact. 15 Duffinian 65 111 175 + 9951391 eban 34 2056 32000002000 economical 15 111 175 + 18797855 emirpimes 15 1379 12209 + 93084991 equidigital 15 111 175 + 18797855 eRAP 4969188575 esthetic 34 65 evil 15 34 65 + 997809119 fibodiv 66351 Fibonacci 34 Friedman 125055 256040 265761 + 976625 frugal 6826079 161414771 193710609 609093750 gapful 260 671 1695 + 99230675185 happy 671 5335 8801 + 9095855 Harshad 111 870 2465 + 9885306184 heptagonal 34 19669 hex 14911 hexagonal 15 hoax 27455 143781 578865 + 93574910 Hogben 111 iban 111 4010 idoneal 15 inconsummate 65 4641 21455 + 976625 insolite 111 interprime 15 34 111 + 91625500 junction 111 505 2925 + 72766051 katadrome 65 870 Lehmer 15 1105 1695 + 910382081071 Leyland 4294968320 lucky 15 111 1105 + 9951391 Lynch-Bell 15 175 magnanimous 34 65 2465 8801 metadrome 15 34 369 1379 modest 111 21621951 Moran 111 3439 23346 mountain 175 260 671 + 48986321 nialpdrome 65 111 870 + 665555 nonagonal 111 nude 15 111 175 + 336111126 O'Halloran 260 oban 15 65 369 + 870 octagonal 65 2465 odious 369 505 671 + 995433385 palindromic 111 505 5335 pancake 1379 panconsummate 15 34 pandigital 15 210975 partition 15 pernicious 34 65 260 + 9841635 persistent 51646873290 plaindrome 15 34 111 + 23346 Poulet 1105 2465 6998881 139101047324161 practical 260 870 16400 + 8001630 prim.abundant 87836 primeval 1379 pronic 870 Proth 65 25345 364545 pseudoperfect 260 870 6924 + 864060 rare 65 repdigit 111 repunit 15 111 Ruth-Aaron 15 369 44657535 150381835 self 1379 4641 6924 + 981258130 semiprime 15 34 65 + 93084991 sliding 65 Smith 143781 578865 629910 + 96550565 sphenic 1105 1695 2465 + 98061761 straight-line 111 369 strobogrammatic 111 super Niven 500050 4000100 500000500 4000001000 super-d 369 7825 19669 + 9410681 tau 2056 6924 2048080 + 702464560 tetrahedral 2925 tetranacci 15 triangular 15 uban 15 65 Ulam 175 260 2056 + 8889921 undulating 505 unprimeable 6924 16400 21455 + 9841635 untouchable 16400 32020 296394 + 629910 upside-down 1379 wasteful 34 65 260 + 9951391 weird 10990 Zuckerman 15 111 175 Zumkeller 260 870 6924 + 87836 zygodrome 111 665555 665500005500 665500000055000