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de Polignac numbers
A de Polignac number is an odd number  $n$  that cannot be expressed as  $n=2^k+p$, for  $p$  prime.

The name comes from the fact that de Polignac erroneously conjectured that every odd number can be expressed in that way.

Erdös proved that there are infinite such numbers, for example all the numbers of the form 1260327937 + 2863311360k.

The smallest composite number in this class is 905, while the first square is 40401.

Roger Crocker proved that there are infinite odd numbers not of the form  $p+2^a+2^b$, with  $p$  prime and  $a,b>0$. The first numbers of this kind are 1, 3, 5, 6495105, 848629545, 1117175145, 2544265305,...

The smallest 3 × 3 magic square whose entries are de Polignac numbers is

4589109491649
278957298669
98095096869

The first de Polignac numbers are 1, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 905, 907, 959, 977, 997 more terms

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

a-pointer 701 2213 3643 + 99951043 ABA 1856465 aban 127 149 251 + 99000971 abundant 1561875 2012985 2410065 + 49722435 Achilles 704099 757687 1119371 + 97749971 alternating 127 149 509 + 89898941 amenable 149 337 373 + 99999937 apocalyptic 251 977 1019 + 29983 arithmetic 127 149 251 + 9999997 balanced p. 373 977 3637 + 99999617 Bell 877 bemirp 10061 16061681 16880681 + 18999191 binomial 46971 79003 93961 + 99934453 brilliant 1207 1271 1541 + 99400891 c.decagonal 5951 12251 15401 + 99837461 c.heptagonal 2843 3697 4663 + 99823291 c.nonagonal 79003 93961 166753 + 99934453 c.octagonal 40401 62001 96721 + 99620361 c.pentagonal 331 195301 345031 + 99745431 c.square 1985 6161 7813 + 99899113 c.triangular 50509 111385 176989 + 99866161 cake 7807 29317 166751 + 91895351 Carmichael 2465 63973 126217 + 92625121 Carol 959 3967 65023 Chen 127 149 251 + 99999839 congruent 127 149 373 + 9999997 constructible 16843009 cube 205379 226981 389017 + 91733851 Cullen 4718593 20971521 44040193 92274689 Cunningham 127 3845 10001 + 99800099 Curzon 509 809 905 + 99999041 cyclic 127 149 251 + 9999997 D-number 7431 8031 11541 + 7042281 d-powerful 373 2203 2263 + 9994657 decagonal 7267 9457 31417 + 99865045 deceptive 10001 63973 79003 + 98661277 deficient 127 149 251 + 9999997 dig.balanced 149 809 905 + 67078145 Duffinian 905 959 1207 + 9999997 economical 127 149 251 + 19999963 emirp 149 337 701 + 99999343 emirpimes 1211 1243 1807 + 99999937 equidigital 127 149 251 + 19999963 eRAP 182939 247475 1220549 + 99740309 esthetic 4543 4567 8789 + 89878987 Eulerian 65519 evil 149 337 373 + 99999937 fibodiv 149 24719 40331 + 30324247 Fibonacci 1597 121393 514229 Friedman 127 2503 12595 + 979769 frugal 14375 26875 27097 + 99977369 gapful 1207 6161 10021 + 99999755 Gilda 997 18633637 30571351 good prime 127 149 251 + 99904457 happy 331 907 1211 + 9999983 Harshad 2465 4311 6021 + 99996401 heptagonal 9517 96727 101707 + 99026649 hex 127 331 1657 + 99930637 hexagonal 79003 93961 166753 + 99934453 hoax 3505 4855 8023 + 99993307 Hogben 757 1807 17557 + 99930013 Honaker 3433 4153 13217 + 99972317 house 12131 165425 197107 + 81473093 hungry 161449 hyperperfect 1232053 1570153 1618597 + 61599553 iban 127 373 701 + 777701 iccanobiF 792517 inconsummate 1271 1541 1649 + 999931 interprime 1207 1243 3353 + 99999419 junction 509 1207 1211 + 99999857 Kaprekar 94520547 katadrome 6521 7431 8621 + 98654321 Kynea 1087 263167 Lehmer 2465 10963 14611 + 99815821 Leyland 1649 32993 lonely 3967 24281 38501 + 47326801 Lucas 15127 54018521 lucky 127 331 997 + 9999997 m-pointer 15121 61211 1111211 + 61114211 magic 2465 78759 102719 + 80939585 magnanimous 809 2429 2465 + 28862465 metadrome 127 149 1259 + 23456789 modest 509 599 809 + 99901517 Moran 4311 10963 14383 + 99987085 Motzkin 127 nialpdrome 331 877 977 + 99999883 nonagonal 99541 125875 478225 + 96271975 nude 39393 71575 155575 + 97799373 oban 337 373 509 + 997 octagonal 1541 2465 14981 + 97903681 odious 127 251 331 + 99999897 Ormiston 34631 35897 40031 + 99999113 palindromic 373 757 959 + 99944999 palprime 373 757 11411 + 9965699 pancake 1541 1597 2279 + 99962731 panconsummate 127 331 337 pandigital 16405 17735 23935 + 16433683 partition 17977 21637 23338469 pentagonal 17767 20827 31901 + 99825367 pernicious 127 251 331 + 9999997 Pierpont 629857 746497 1492993 86093443 plaindrome 127 149 337 + 88889999 Poulet 2465 31417 41665 + 97255801 power 40401 62001 96721 + 99620361 powerful 40401 62001 96721 + 99620361 prim.abundant 2012985 13777785 16819275 + 96527475 prime 127 149 251 + 99999839 primeval 10079 10237 10379 + 10123579 Proth 1985 5761 7169 + 99893249 repdigit 11111111 repunit 127 757 1807 + 99930013 Rhonda 259333 1583197 2156517 + 98655551 Ruth-Aaron 5405 12727 24433 + 99704375 self 905 1087 1199 + 99999895 self-describing 10143133 10153331 10183133 + 33193117 semiprime 905 959 1199 + 99999937 sliding 10001 48830173 Smith 3505 4855 8023 + 99993307 sphenic 2465 4503 4543 + 99999897 square 40401 62001 96721 + 99620361 star 337 23437 31537 + 99267337 straight-line 4567 98765 11111111 23456789 strobogrammatic 6119 10001 18881 + 69911669 strong prime 127 149 251 + 99999839 super-d 127 331 337 + 9999869 tau 40401 62001 660969 + 87665769 taxicab 3242197 5772403 10765603 + 89576767 tetrahedral 302621 366145 657359 + 56677949 tetranacci 14564533 triangular 46971 79003 93961 + 99934453 tribonacci 149 trimorphic 251 499999 7890625 87109375 truncatable prime 337 373 599 + 99336373 twin 149 599 809 + 99996131 uban 1000061 1000099 2000039 + 99000077 Ulam 905 1985 2789 + 10000109 undulating 373 757 959 + 73737373 unprimeable 3505 3665 4195 + 9999775 upside-down 1199 1649 1829 + 99973111 vampire 124483 371893 489955 + 83119297 wasteful 905 959 1199 + 9999997 weak prime 337 509 997 + 99999787 weakly prime 584141 604171 4393139 + 98750609 Wieferich 3837523 Woodall 38879 49151 52487 + 46137343 Zeisel 336611 982513 2263811 + 96931639 Zuckerman 1111117 11111111 zygodrome 1199 7799 44111 + 99996677