weak primes

A prime is said to be weak if it smaller than the average of the two surrounding primes

For example, 13 is a weak prime since it is less than the average of the two surrounding primes 11 and 17.

Primes which are neither balanced nor weak are called strong primes.

Erdös conjectured that there are infinitely many consecutive pairs of weak primes (that he called early primes, and offered $100 for a proof and $25000 for a disproof.

The first run of 1,2,..., 13 consecutive weak primes start at 3, 19, 349, 2909, 15377, 128983, 1319411, 17797519, 94097539, 6927837559, 48486712787, 968068681519, and 1472840004019, respectively.

The first weak primes are 3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199 more terms

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

Useful links Wikipedia, Weak prime
OEIS, Sequence A051635

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

a-pointer 13 103 181 293 + 99998611 aban 13 19 23 31 + 99000991 alt.fact. 19 619 35899 3301819 alternating 23 43 47 61 + 89898983 amenable 13 61 73 89 + 99999941 apocalyptic 443 647 823 859 + 29989 arithmetic 13 19 23 31 + 9999991 Bell 27644437 bemirp 1601 106861 168601 198901 + 19986091 c.decagonal 31 61 151 661 + 99926851 c.heptagonal 43 463 547 953 + 99673813 c.pentagonal 31 181 601 1051 + 99713851 c.square 13 61 113 181 + 99531941 c.triangular 19 31 109 199 + 99401611 canyon 103 109 313 317 + 98765419 Carol 47 3967 1046527 16769023 Chen 13 19 23 31 + 99999971 congruent 13 23 31 47 + 9999991 Cunningham 31 401 577 677 + 99720197 Curzon 89 113 233 293 + 99999353 cyclic 13 19 23 31 + 9999991 d-powerful 43 89 283 463 + 9979643 de Polignac 337 509 997 1259 + 99999787 deficient 13 19 23 31 + 9999991 dig.balanced 19 139 601 647 + 67076353 economical 13 19 23 31 + 19999927 emirp 13 31 73 113 + 99999827 equidigital 13 19 23 31 + 19999927 esthetic 23 43 89 4567 + 89876789 Eulerian 1013 16369 67108837 evil 23 43 83 89 + 99999971 fibodiv 19 47 61 199 + 67645819 Fibonacci 13 89 233 514229 Friedman 12101 12109 15629 16381 + 995347 Gilda 683 997 2207 happy 13 19 23 31 + 9999991 hex 19 61 271 547 + 98756719 Hogben 13 31 43 73 + 99990001 Honaker 131 1039 1433 1571 + 99973061 house 271 hungry 2003 iban 23 43 47 73 + 777743 iccanobiF 13 4139 idoneal 13 inconsummate 383 443 491 761 + 999931 Jacobsthal 43 683 2731 174763 junction 103 109 113 313 + 99999259 katadrome 31 43 61 73 + 98764321 Kynea 23 66047 lonely 23 2179 3967 24281 + 206699 Lucas 47 199 2207 9349 3010349 lucky 13 31 43 73 + 9999049 m-pointer 23 61 1231 1321 + 61114211 magnanimous 23 43 47 61 + 48804809 metadrome 13 19 23 47 + 23456789 modest 13 19 23 89 + 99702439 Motzkin 15511 mountain 131 151 181 193 + 89876431 nialpdrome 31 43 61 73 + 99999971 oban 13 19 23 73 + 997 odious 13 19 31 47 + 99999941 Ormiston 1913 18379 19013 34613 + 99999131 palindromic 131 151 181 313 + 9989899 palprime 131 151 181 313 + 9989899 pancake 1129 1327 1831 2017 + 99863779 panconsummate 23 31 43 61 + 1291 pandigital 19 pernicious 13 19 31 47 + 9999991 Perrin 43721 Pierpont 13 19 73 109 + 63700993 plaindrome 13 19 23 47 + 89999999 prime 13 19 23 31 + 99999971 primeval 13 113 1013 1237 + 10123579 Proth 13 113 193 241 + 99598337 repfigit 19 47 61 1084051 repunit 13 31 43 73 + 99990001 self 31 233 389 547 + 99999827 self-describing 10233221 10311533 10322321 12313319 + 33311519 Sophie Germain 23 83 89 113 + 99999563 star 13 73 181 337 + 99853921 straight-line 4567 76543 23456789 strobogrammatic 181 619 16091 61819 + 68969689 super-d 19 31 131 181 + 9999931 tetranacci 401 773 tribonacci 13 trimorphic 31249 49999 74218751 truncatable prime 13 23 31 43 + 99979337 twin 13 19 31 43 + 99999589 uban 13 19 23 31 + 99000079 Ulam 13 47 131 241 + 9999481 undulating 131 151 181 313 + 1212121 upside-down 19 73 1559 7193 + 99955111 weakly prime 294001 971767 1282529 1524181 + 98750609 Wieferich 1093 Woodall 23 383 4373 5119 + 3124999 zygodrome 11177 22111 44777 55333 + 99955111