Honaker primes

A prime

is a Honaker prime if its index $n$

and

itself have the same sum of digits.

For example, $p_{32}=131$ is a Honaker prime because $3+2=1+3+1$ .

The two smallest Honaker primes $p_n$ for which the sum of the digits is less than the number of digits of $n$ are $p_{1300010120}=30000101111$ and $p_{1000122031021}=30000011223001$ .

$(p_{88}=457, p_{457}=3229)$ is the earliest chain of length 2. Chains of length 3 and 4 start with $p_{248}$ and $p_{496657}$ .

The smallest prime which is Honaker in all the bases from 2 to 10 is $p_{277308991}= 5949670231$ .

The first Honaker primes are 131, 263, 457, 1039, 1049, 1091, 1301, 1361, 1433, 1571, 1913, 1933, 2141, 2221, 2273, 2441 more terms and their indexes are, 32, 56, 88, 175, 176, 182, 212, 218, 227, 248, 293, 295, 323, 331, 338, 362, respectively.

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

Useful links OEIS, Sequence A033548

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

a-pointer 1933 + 998369213 998552251 998963701 aban 131 + 998000411 999000953 999000991 alternating 2141 + 989050303 989250121 989252101 amenable 457 + 999821261 999822521 999843421 apocalyptic 1301 + 29131 29243 29423 arithmetic 131 + 9983203 9985231 9987001 balanced p. 263 + 999212503 999313141 999821261 bemirp 1091 + 118880911 161080811 169806181 c.decagonal 1361 + 902563301 907541281 995531051 c.heptagonal 1933 + 938205647 940843471 980855191 c.pentagonal 11731 + 765581251 822240901 990472801 c.square 1301 + 936362813 942257461 959176201 c.triangular 10711 + 856492381 934839391 955548541 canyon 1039 + 984301567 986501357 986532103 Carol 1046527 Chen 131 + 99970391 99972317 99973061 congruent 263 + 9973021 9973631 9985231 Cunningham 3137 + 921365317 951722501 960752017 Curzon 1049 + 199612001 199803041 199820213 cyclic 131 + 9983203 9985231 9987001 d-powerful 2803 + 9832351 9921473 9972233 de Polignac 3433 + 99962381 99970391 99972317 deficient 131 + 9983203 9985231 9987001 dig.balanced 2141 + 199541201 199805003 199805033 economical 131 + 19984301 19984501 19984511 emirp 1091 + 199710041 199803013 199901021 equidigital 131 + 19984301 19984501 19984511 esthetic 34543 3212123 343456543 565454543 Eulerian 67108837 evil 263 + 999802109 999810311 999831103 fibodiv 1301 67645819 123047543 Friedman 58921 + 723169 902507 902563 good prime 3433 + 198463801 198464323 199022107 happy 263 + 9939301 9940703 9982211 hex 126691 + 960317317 961068907 972162007 Hogben 5701 + 946085323 954779101 975906361 iban 1301 + 772703 773147 773273 inconsummate 1049 + 990287 991619 996011 junction 2719 + 99933047 99954541 99954551 lucky 1039 + 9941257 9971341 9985231 magnanimous 2221 + 600881 2666021 6600227 metadrome 457 12347 13457 modest 1433 + 993011111 997001221 998111111 mountain 131 + 345796543 467965321 567896321 nialpdrome 2221 + 998441111 998773211 998831111 odious 131 + 999822203 999822521 999843421 Ormiston 1913 + 998511109 998779031 999003913 palindromic 131 + 971141179 981151189 982323289 palprime 131 + 971141179 981151189 982323289 pancake 13367 + 928266329 950981467 960424879 pernicious 131 + 9973021 9983203 9985231 plaindrome 457 + 333355559 333444679 345555569 prime 131 + 999822521 999831103 999843421 primeval 100279 Proth 3137 + 944570369 949420033 955383809 repunit 5701 + 946085323 954779101 975906361 self 457 + 999414313 999632021 999831103 self-describing 18331031 19143133 19143331 21183223 Sophie Germain 131 + 999363251 999412313 999821261 star 11353 + 942230953 955359253 981120937 strobogrammatic 111689111 + 609080609 661000199 666101999 strong prime 457 + 99970391 99972221 99972703 super-d 131 + 9973021 9973631 9985231 truncatable prime 3137 + 613564937 627543853 915613967 twin 1049 + 999600421 999611309 999815041 uban 17000077 + 60000049 63000089 77000081 Ulam 131 + 9925193 9962321 9973021 undulating 131 upside-down 7283 + 96537541 96637441 98746321 weak prime 131 + 99970861 99972317 99973061 weakly prime 6173731 + 941179201 944375701 945808223 zygodrome 22277 + 992223311 994400333 999112211