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 + 99986539 99998611 aban 13 19 + 99000977 99000991 alt.fact. 19 619 35899 3301819 alternating 23 43 + 89898929 89898983 amenable 13 61 + 99999721 99999941 apocalyptic 443 647 + 29959 29989 arithmetic 13 19 + 9999973 9999991 Bell 27644437 bemirp 1601 106861 + 19880981 19986091 c.decagonal 31 61 + 98856811 99926851 c.heptagonal 43 463 + 99487123 99673813 c.pentagonal 31 181 + 99335281 99713851 c.square 13 61 + 99475513 99531941 c.triangular 19 31 + 99181939 99401611 Carol 47 3967 1046527 16769023 Chen 13 19 + 99999847 99999971 congruent 13 23 + 9999973 9999991 Cunningham 31 401 + 99400901 99720197 Curzon 89 113 + 99998933 99999353 cyclic 13 19 + 9999973 9999991 d-powerful 43 89 + 9979337 9979643 de Polignac 337 509 + 99999373 99999787 deficient 13 19 + 9999973 9999991 dig.balanced 19 139 + 67059841 67076353 economical 13 19 + 19999909 19999927 emirp 13 31 + 99999643 99999827 equidigital 13 19 + 19999909 19999927 esthetic 23 43 + 87676789 89876789 Eulerian 1013 16369 67108837 evil 23 43 + 99999847 99999971 fibodiv 19 47 + 60999623 67645819 Fibonacci 13 89 233 514229 Friedman 12101 12109 + 995341 995347 Gilda 683 997 2207 happy 13 19 + 9999533 9999991 hex 19 61 + 98515891 98756719 Hogben 13 31 + 99930013 99990001 Honaker 131 1039 + 99972317 99973061 house 271 hungry 2003 iban 23 43 + 777421 777743 iccanobiF 13 4139 idoneal 13 inconsummate 383 443 + 999773 999931 Jacobsthal 43 683 2731 174763 junction 103 109 + 99998557 99999259 katadrome 31 43 + 97654321 98764321 Kynea 23 66047 lonely 23 2179 + 203713 206699 Lucas 47 199 + 9349 3010349 lucky 13 31 + 9998971 9999049 m-pointer 23 61 + 19122211 61114211 magnanimous 23 43 + 8608081 48804809 metadrome 13 19 + 1234789 23456789 modest 13 19 + 99619999 99702439 Motzkin 15511 nialpdrome 31 43 + 99999941 99999971 oban 13 19 + 983 997 odious 13 19 + 99999827 99999941 Ormiston 1913 18379 + 99999113 99999131 palindromic 131 151 + 9981899 9989899 palprime 131 151 + 9981899 9989899 pancake 1129 1327 + 99468461 99863779 panconsummate 23 31 + 1093 1291 pandigital 19 pernicious 13 19 + 9999973 9999991 Perrin 43721 Pierpont 13 19 + 57395629 63700993 plaindrome 13 19 + 88888999 89999999 prime 13 19 + 99999941 99999971 primeval 13 113 + 10123457 10123579 Proth 13 113 + 99532801 99598337 repfigit 19 47 61 1084051 repunit 13 31 + 99930013 99990001 self 31 233 + 99999401 99999827 self-describing 10233221 10311533 + 33161831 33311519 Sophie Germain 23 83 + 99999551 99999563 star 13 73 + 99462673 99853921 straight-line 4567 76543 23456789 strobogrammatic 181 619 + 68811889 68969689 super-d 19 31 + 9999667 9999931 tetranacci 401 773 tribonacci 13 trimorphic 31249 49999 74218751 truncatable prime 13 23 + 99962683 99979337 twin 13 19 + 99999541 99999589 uban 13 19 + 99000059 99000079 Ulam 13 47 + 9998753 9999481 undulating 131 151 + 95959 1212121 upside-down 19 73 + 99864211 99955111 weakly prime 294001 971767 + 98643439 98750609 Wieferich 1093 Woodall 23 383 + 524287 3124999 zygodrome 11177 22111 + 99944111 99955111