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magnanimous numbers
A number (which we assume of at least 2 digits) such that the sum obtained inserting a "+" among its digit in any position gives a prime.

For example, 4001 is magnanimous because the numbers 4+001=5, 40+01=41 and 400+1=401 are all prime numbers.

Since all the prime numbers are odd, except for 2, all the magnanimous numbers, except for 11, are either a sequence of odd digits followed by an even digit, or a sequence of even digits followed by an odd digits.

It is conjectured that the magnanimous numbers are finite and that probably the largest one is 97393713331910, while the largest one which is also a prime number itself is probably 608844043.

The following table summarizes their frequency as a function of the number of digits, reporting the smallest, the largest and their quantitity.

d#m. minm. max
2 33 11 98
3 79 101 998
4 104 1001 9910
5 112 11116 99998
6 96 111112 999994
7 71 1115756 9959374
8 35 11771992 95559998
9 18 117711170 995955112
10 6 1777137770 9151995592
11 5 22226226625 84448000009
12 0 none
13 1 5391391551358
14 1 97393713331910
15 0 none

The first magnanimous numbers are 11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 110, 112, 116, 118, 130, 136, 152, 158, 170, 172, 203 more terms

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

a-pointer 11 101 ABA 32 50 98 + 512 aban 11 12 14 + 84448000009 abundant 12 20 30 + 5731136 admirable 12 20 30 + 975772 alt.fact. 101 alternating 12 14 16 + 101 amenable 12 16 20 + 995955112 apocalyptic 245 394 443 + 28429 arithmetic 11 14 20 + 9959374 automorphic 25 76 376 625 balanced p. 607 42209 Bell 52 203 binomial 20 21 56 + 6824665 brilliant 14 21 25 + 22600601 c.decagonal 11 61 101 + 661 c.heptagonal 43 316 736 + 1772 c.nonagonal 136 6824665 c.octagonal 25 49 625 + 22801 c.pentagonal 16 76 601 + 177556 c.square 25 41 61 + 88621 c.triangular 85 136 316 31756 cake 130 Carmichael 2465 6601 Carol 47 Catalan 14 Chen 11 23 29 + 48804809 congruent 14 20 21 + 9959374 constructible 12 16 20 + 512 cube 512 Cullen 25 65 Cunningham 50 65 101 + 40001 Curzon 14 21 29 + 62202281 cyclic 11 23 29 + 8844449 D-number 21 d-powerful 43 89 209 + 4488245 de Polignac 809 2429 2465 + 28862465 decagonal 52 85 370 6601 deceptive 6601 deficient 11 14 16 + 9959374 dig.balanced 11 12 21 + 62202281 Duffinian 16 21 25 + 8844449 eban 30 32 34 + 56 economical 11 14 16 + 8608081 emirpimes 49 58 85 + 26428645 equidigital 11 14 16 + 8608081 eRAP 20 98 170 esthetic 12 21 23 + 101 Eulerian 11 evil 12 20 23 + 882428665 fibodiv 14 47 61 Fibonacci 21 34 89 Friedman 25 625 736 frugal 512 625 2209 + 22801 gapful 110 130 170 + 22226226625 Gilda 29 49 110 + 683 Giuga 30 good prime 11 29 41 + 602081 happy 23 32 49 + 9777910 Harshad 12 20 21 + 62202281 heptagonal 34 112 403 + 1353136 hex 61 highly composite 12 hoax 58 85 94 + 66886483 Hogben 21 43 601 6007 Honaker 2221 8821 40427 + 6600227 house 32 hungry 74 hyperperfect 21 iban 11 12 14 + 777712 idoneal 12 16 21 + 130 impolite 16 32 512 inconsummate 65 443 536 + 931130 interprime 12 21 30 + 82486825 Jacobsthal 11 21 43 + 683 Jordan-Polya 12 16 32 512 junction 101 1001 1112 + 71553536 katadrome 20 21 30 + 9752 Kynea 23 Lehmer 85 2465 6601 88621 Leyland 32 512 lonely 23 Lucas 11 29 47 76 lucky 21 25 43 + 6640063 Lynch-Bell 12 m-pointer 23 61 magic 34 65 2465 8801 metadrome 12 14 16 + 2489 modest 23 29 49 + 80009 Moran 21 152 209 + 62202281 Motzkin 21 narcissistic 370 407 nialpdrome 11 20 21 + 999994 nude 11 12 112 + 111112 O'Halloran 12 20 116 oban 11 12 16 + 998 octagonal 21 65 736 2465 odious 11 14 16 + 995955112 palindromic 11 101 1001 palprime 11 101 pancake 11 16 29 + 821 panconsummate 11 12 14 + 661 pandigital 11 21 970 + 42265 partition 11 30 56 101 pentagonal 12 70 92 + 1001 pernicious 11 12 14 + 9959374 Perrin 12 29 158 + 1130 plaindrome 11 12 14 + 222245 Poulet 2465 6601 power 16 25 32 + 22801 powerful 16 25 32 + 22801 practical 12 16 20 + 115136 prim.abundant 12 20 30 + 5731136 prime 11 23 29 + 608844043 primorial 30 pronic 12 20 30 + 992 Proth 25 41 49 + 26881 pseudoperfect 12 20 30 + 975772 rare 65 repdigit 11 repfigit 14 47 61 repunit 21 43 85 + 6007 Rhonda 53176 Ruth-Aaron 16 25 49 + 370 self 20 110 209 + 22600601 semiprime 14 21 25 + 91959398 sliding 11 20 25 + 2225 Smith 58 85 94 + 66886483 sphenic 30 70 110 + 64622665 square 16 25 49 + 22801 star 661 2281 strobogrammatic 11 101 1001 strong prime 11 29 41 + 20266681 subfactorial 265 super Niven 12 20 30 + 110 super-d 281 310 334 + 9171170 superabundant 12 tau 12 56 136 + 995955112 tetrahedral 20 56 tetranacci 29 56 401 triangular 21 136 6824665 trimorphic 25 49 76 + 625 truncatable prime 23 29 43 + 4643 twin 11 29 41 + 228440489 uban 11 12 16 + 98 Ulam 11 16 38 + 7719370 undulating 101 unprimeable 512 518 536 + 9171170 untouchable 52 518 556 + 979592 wasteful 12 20 30 + 9959374 weak prime 23 43 47 + 48804809 weird 70 Woodall 23 Zuckerman 11 12 112 + 111112 Zumkeller 12 20 30 + 99712 zygodrome 11