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Proth numbers
If  $k$  is an odd number,  $n0$">  and  $2^nk$">, then  $k\cdot2^n+1$  is a Proth number.

For example,  $448$  is a Proth number because it can be written as  $7\cdot2^6+1$, where  $2^67$">.

Cullen numbers are a subset of Proth numbers.

Proth numbers have been studied in relation to their primality. In particular, according to Proth's theorem, a Proth number  $n$  is prime if and only if there exists an integer  $k$  such that

\[ k^{(n-1)/2} \equiv {-1 \pmod n}\,. \]

The first Proth numbers are 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 289, 321, 353, 385, 417, 449, 481, 513, 545, 577, 609, 641, 673, 705, 737, 769, 801 more terms

The first Proth primes are 3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113

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

A graph displaying how many Proth numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of Proth numbers
Useful links Mathworld, Proth Number
Wikipedia, Proth number
OEIS, Sequence A080075 Proth numbers
OEIS, Sequence A080076 Proth primes
Proth numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 13 1153 1409 + 9767878657 ABA 81 aban 13 17 25 + 999424000001 abundant 7425 49665 318465 + 47734785 admirable 7425 alternating 25 41 49 + 989036545 amenable 13 17 25 + 999981057 apocalyptic 865 929 1217 + 29953 arithmetic 13 17 33 + 9998337 automorphic 25 90625 balanced p. 257 4993 9473 + 9963962369 bemirp 1601 16001 brilliant 25 49 209 + 999653377 c.octagonal 25 49 81 + 274878955521 c.square 13 25 41 + 549756862465 Carmichael 1729 8355841 40280065 53282340865 Chen 13 17 41 + 99893249 congruent 13 41 65 + 9879553 constructible 17 257 65537 Cullen 25 65 161 + 584115552257 Cunningham 17 33 65 + 998848331777 Curzon 33 41 65 + 199720961 cyclic 13 17 33 + 9994241 D-number 33 57 129 + 6987777 d-powerful 209 2241 2753 + 9875457 de Polignac 1985 5761 7169 + 99893249 deceptive 481 1729 19201 + 385351681 deficient 13 17 25 + 9998337 dig.balanced 41 49 177 + 67092481 Duffinian 25 49 57 + 9998337 economical 13 17 25 + 19996673 emirp 13 17 97 + 199229441 emirpimes 49 129 177 + 99844097 equidigital 13 17 25 + 19996673 esthetic 65 321 545 + 56565432321 Eulerian 57 evil 17 33 57 + 999948289 fibodiv 79361 Fibonacci 13 Friedman 25 289 6145 + 983041 frugal 10625 14337 16129 + 999817217 gapful 225 385 1665 + 99978051585 Gilda 49 good prime 17 41 97 + 190857217 happy 13 49 97 + 9994241 Harshad 81 209 225 + 9999613953 heptagonal 81 25857 6705153 + 439803183105 hoax 1665 1921 2625 + 99581953 Hogben 13 57 241 + 274877382657 Honaker 3137 57089 64513 + 955383809 hungry 17 hyperperfect 112803841 463743221761 iban 17 41 177 + 777217 iccanobiF 13 idoneal 13 25 33 + 385 inconsummate 65 161 993 + 994305 interprime 81 129 225 + 98828289 junction 113 513 1217 + 99778561 katadrome 41 65 81 + 8321 Lehmer 481 1729 4033 + 888547573761 Leyland 17 57 145 177 lucky 13 25 33 + 9990145 magic 65 25345 364545 magnanimous 25 41 49 + 26881 metadrome 13 17 25 + 13569 modest 13 49 209 + 1484718081 Moran 209 481 801 + 99368961 nialpdrome 33 41 65 + 98877441 nonagonal 1089 74241 4788225 + 19633799169 nude 33 3393 319717377 oban 13 17 25 + 993 octagonal 65 225 481 + 824635817985 odious 13 25 41 + 999981057 Ormiston 464897 925697 986113 + 1938358273 palindromic 33 161 353 + 946783387649 palprime 353 929 96769 panconsummate 57 81 257 + 481 pandigital 177 225 11265 + 8736210945 partition 385 pentagonal 145 2625 43265 + 733006004225 pernicious 13 17 25 + 9994241 Perrin 17 209 persistent 4296015873 6803947521 10277486593 + 98761703425 Pierpont 13 17 97 + 880602513409 plaindrome 13 17 25 + 1124466689 Poulet 1729 4033 8321 + 137439477761 power 25 49 81 + 274878955521 powerful 25 49 81 + 274878955521 prim.abundant 7425 49665 2392065 32743425 prime 13 17 41 + 999977648129 primeval 13 113 pseudoperfect 7425 49665 318465 + 963585 rare 65 repdigit 33 repfigit 1537 repunit 13 57 241 + 274877382657 Ruth-Aaron 25 49 224257 + 983483547649 self 97 209 929 + 999915521 semiprime 25 33 49 + 99991553 sliding 25 65 641 Smith 1921 3649 7809 + 99581953 Sophie Germain 41 113 641 + 9996206081 sphenic 385 609 705 + 99942401 square 25 49 81 + 274878955521 star 13 straight-line 321 strobogrammatic 609 strong prime 17 41 97 + 99893249 super-d 81 481 769 + 9965569 tau 225 1089 16641 + 67125249 taxicab 1729 405274625 tribonacci 13 81 trimorphic 25 49 90625 truncatable prime 13 17 97 + 986113 twin 13 17 41 + 995622913 uban 13 17 25 + 57031000065 Ulam 13 57 97 + 9936897 undulating 161 353 545 + 929 unprimeable 1345 2945 3265 + 9990145 upside-down 7553 14573569 162658254849 795855552513 vampire 180225 wasteful 33 57 65 + 9998337 weak prime 13 113 193 + 99598337 weakly prime 43447484417 62688854017 Wieferich 14209 Woodall 17 Zeisel 1729 Zumkeller 7425 49665 zygodrome 33 8833 11777 + 222554488833