heptagonal numbers

Heptagonal numbers are figurate numbers of the form $n(5n-3)/2$

(see picture aside).

If $h$ is a heptagonal number then $5h+1$ is a triangular number.

The sum of the reciprocals of heptagonal numbers has a quite complicated closed form:

$\frac{1}{6} \left(2 \sqrt{1-\frac{2}{\sqrt{5}}} \pi +5 \log (5)-2 \sqrt{5} \coth ^{-1}\left(\sqrt{5}\right)\right)$

The first heptagonal numbers are 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970 more terms

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

A graph displaying how many heptagonal numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Heptagonal Number
Wikipedia, Heptagonal number
OEIS, Sequence A000566

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

ABA 18 81 1877922 194706720450 aban 18 34 55 + 996926000998 abundant 18 112 342 + 49967896 Achilles 28252567 admirable 7756 98704 2458672 13510575 alternating 18 34 81 + 989298783 amenable 81 112 148 + 999970000 apocalyptic 286 540 970 + 29539 arithmetic 55 189 235 + 9977013 binomial 55 286 27405 + 593128762435 brilliant 403 697 2059 + 999670027 c.decagonal 1901737531 c.heptagonal 148 21022 2984983 + 8545594544488 c.nonagonal 55 121771 5720653 + 593128762435 c.octagonal 81 5929 2307361 + 350709705290025 c.square 4141 85285 429332905 + 916808615026525 c.triangular 235 2839 902101 + 160992923703385 cake 697 Carmichael 670033 173085121 9836283601 congruent 34 55 112 + 9987004 constructible 34 Cunningham 342 783 143640 + 1324147312657 Curzon 18 81 189 + 199393506 cyclic 235 403 469 + 9967027 D-number 21483 d-powerful 2205 2356 3940 + 9591264 de Polignac 9517 96727 101707 + 99026649 decagonal 540 2887450 1123674201 6004054625647 deceptive 3367 4141 401401 + 98288089981 deficient 34 55 81 + 9987004 dig.balanced 970 2205 2673 + 199840291 Duffinian 55 81 189 + 9967027 eban 34 46036 economical 81 112 189 + 19792269 emirpimes 1177 1651 2059 + 94025023 equidigital 81 112 189 + 19792269 eRAP 978751 658913710 129621298968 esthetic 34 evil 18 34 189 + 999870004 Fibonacci 34 55 Friedman 13950 45360 59059 + 589761 frugal 16281 478953 589761 + 870740928 gapful 286 2059 3367 + 99994700070 happy 469 874 970 + 9987004 Harshad 18 81 112 + 9998149140 hex 469 3367 226051 + 181728138938347 hexagonal 121771 12625478965 highly composite 45360 hoax 1404 2356 2839 + 99814924 Hogben 18361 191407 1342123 + 29093062258603 hyperperfect 697 iban 112 342 403 + 747202 idoneal 18 112 inconsummate 3186 3744 6426 + 991305 interprime 18 34 81 + 99593892 junction 616 1918 5221 + 99562336 Kaprekar 55 356643 670033 katadrome 81 540 874 970 Lehmer 3367 4141 9211 + 999573669121 Lucas 18 lucky 189 235 2059 + 9967027 Lynch-Bell 281736 magic 34 19669 magnanimous 34 112 403 + 1353136 metadrome 18 34 148 + 1345789 modest 5688 8037 40132 + 1612099999 Moran 18 1651 nialpdrome 55 81 540 + 999999970000000 nonagonal 26884 542041975 10928650279834 nude 55 112 1288 + 494469144 oban 18 55 616 + 970 octagonal 297045 69010153345 odious 55 81 112 + 999970000 palindromic 55 616 3553 + 717685292586717 pancake 1177 14707 55279 + 594243385996267 panconsummate 18 34 81 pandigital 970 785961 2207590 + 9386472150 partition 3010 pentagonal 4347 16701685 64167869935 246532939589097 pernicious 18 34 55 + 9967027 persistent 1423809765 1640538297 10839469752 + 98630421751 plaindrome 18 34 55 + 444444468888889 Poulet 23377 164737 401401 + 999953670536617 power 81 5929 2307361 + 4797839017609 powerful 81 5929 2307361 + 350709705290025 practical 18 112 342 + 9997000 prim.abundant 18 2205 7756 + 69292665 pronic 342 711635652 Proth 81 25857 6705153 + 439803183105 pseudoperfect 18 112 342 + 997612 repdigit 55 repunit 6175 18361 27931 + 29093062258603 Rhonda 12251169 Ruth-Aaron 890127 5210674 6066073 + 913906407883 Saint-Exupery 944640 2075717340 self 1177 7756 10144 + 999270133 semiprime 34 55 235 + 99751747 Smith 2839 10791 24354 + 99878121 sphenic 286 874 970 + 99909727 square 81 5929 2307361 + 350709705290025 star 5221 20060473 77072333437 296111885005873 straight-line 13579 super Niven 540 351000 super-d 81 469 783 + 9957046 tau 18 3744 5688 + 996772608 tetrahedral 286 triangular 55 121771 5720653 + 593128762435 tribonacci 81 uban 18 55 81 + 90000081000018 Ulam 18 148 189 + 9927133 undulating 616 4141 60606 unprimeable 2512 3744 4558 + 9917172 untouchable 342 540 1288 + 981882 upside-down 55 13579 74563 71628493 vampire 260338 1025024629 1643485540 + 7349168574 wasteful 18 34 55 + 9997000 Zuckerman 112 116316 24191136 212322816 Zumkeller 112 342 540 + 98704 zygodrome 55 1177 11122 + 881118800088111