centered pentagonal numbers

Centered pentagonal numbers are figurate numbers (see picture aside) defined as $P'_n=(5n^2-5n+2)/2$

Two formulas involving centered pentagonal numbers:

$\sum_{n=1}^{\infty}\frac{1}{P'_n}= \frac{2 \pi \tanh \left(\frac{1}{2} \sqrt{\frac{3}{5}} \pi \right)}{\sqrt{15}}\quad\mbox{and}\quad\sum_{n=1}^{\infty}\frac{P'_n}{2^n}=11\,.$

The first centered pentagonal numbers are 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051 more terms

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

remainders of centered pentagonal numbers

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

divisibility of centered pentagonal numbers

Useful links Mathworld, Centered pentagonal Number
Wikipedia, Centered pentagonal number
OEIS, Sequence A005891

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

a-pointer 181 19141 126001 + 8604688891 aban 16 31 51 + 998846000476 abundant 276 456 1266 + 49851726 admirable 1266 1626 6126 + 97390806 alternating 16 76 141 + 989676781 amenable 16 76 141 + 999950001 apocalyptic 226 1501 1626 + 29976 arithmetic 31 51 141 + 9995001 automorphic 76 balanced p. 11731 17431 17851 + 9482550391 bemirp 119010099181 binomial 276 1891 88831 + 2965643812176 brilliant 391 1501 1891 + 996852481 c.decagonal 31 1051 35701 + 54865086577351 c.heptagonal 106 15016 2132131 + 866724458635141 c.nonagonal 1891 88831 196049701 + 954928097357851 c.octagonal 22801 1666681 47411143081 3465632747641 c.square 181 58141 18721081 + 625008553923781 c.triangular 31 1891 117181 + 107213302675081 Carmichael 399001 1746692641 Chen 31 181 1381 + 99335281 congruent 31 141 181 + 9985006 constructible 16 51 2176 Cunningham 31 226 324901 + 974193625805626 Curzon 141 1626 6126 + 198269826 cyclic 31 51 141 + 9965031 D-number 51 141 681 + 7018251 d-powerful 226 3706 4951 + 9394456 de Polignac 331 195301 345031 + 99745431 decagonal 109726 2164576 228160269751 4500937275451 deceptive 399001 196049701 295039081 + 9210162781 deficient 16 31 51 + 9995001 dig.balanced 141 226 601 + 199741956 Duffinian 16 391 1156 + 9955051 eban 40006 50056 40030006 + 40000030000006 economical 16 31 106 + 19930381 emirp 31 1381 11731 + 196980631 emirpimes 51 226 766 + 99619141 enlightened 2176 equidigital 16 31 106 + 19930381 esthetic 76 456 evil 51 106 141 + 999850006 Friedman 273076 328516 387106 746656 frugal 22801 1254931 1363456 + 775060141 gapful 11056 36301 55131 + 99906521856 good prime 331 9151 143401 + 146669851 happy 31 226 331 + 9875391 Harshad 2176 2976 9456 + 9996821151 hex 331 159391 76825981 + 17848365484951 hexagonal 276 1891 88831 + 954928097357851 hoax 391 456 526 + 98486131 Hogben 31 601 43891 + 131602780302451 Honaker 11731 139831 256801 + 990472801 hyperperfect 3901 163201 iban 141 4101 7701 + 741201 idoneal 16 impolite 16 inconsummate 276 1266 1626 + 968766 interprime 76 1756 4731 + 99903406 Jordan-Polya 16 junction 3516 15016 17016 + 99713851 katadrome 31 51 76 + 987531 Lehmer 51 1891 23281 + 981056721331 Lucas 76 lucky 31 51 141 + 9726891 magic 2287231 magnanimous 16 76 601 + 177556 metadrome 16 456 modest 1266 40006 136306 + 1808823256 Moran 29431 189751 409051 + 92461606 Motzkin 51 nialpdrome 31 51 76 + 99966631 nonagonal 856 20026 17259391 + 8140708519516 nude 1266 1626 6126 + 211163226 oban 16 76 526 + 856 octagonal 7701 219781 1789106881 + 415648888795141 odious 16 31 76 + 999950001 Ormiston 139831 2644531 11273131 + 1939265131 palindromic 141 181 10401 + 620873909378026 palprime 181 pancake 16 106 4951 + 364750076378431 panconsummate 31 331 pandigital 141 69129556 91279516 + 380103076 pentagonal 51 3151 195301 + 178688837791801 pernicious 31 76 181 + 9995001 Perrin 51 persistent 11835492076 12519743806 15478913206 + 98426737051 plaindrome 16 226 456 + 444444455555556 Poulet 219781 399001 2537641 + 18823846860001 power 16 1156 22801 + 3465632747641 powerful 16 1156 22801 + 68366835443776 practical 16 276 456 + 9815856 prim.abundant 1266 1626 6126 + 97390806 prime 31 181 331 + 999977762881 pronic 8556 12337656 17790891306 25654452925506 pseudoperfect 276 456 1266 + 978126 repunit 31 601 43891 + 131602780302451 Ruth-Aaron 16 5406 3865731 + 97500319351 self 31 681 951 + 998550526 semiprime 51 106 141 + 99619141 Smith 391 526 1626 + 98486131 sphenic 1266 1626 2806 + 99871801 square 16 1156 22801 + 68366835443776 star 181 3901 695641 + 221235542824081 straight-line 456 951 strobogrammatic 181 strong prime 331 4951 9151 + 99998251 super-d 31 106 181 + 9965031 tau 276 856 2176 + 999750016 triangular 276 1891 88831 + 954928097357851 trimorphic 51 76 truncatable prime 31 twin 31 181 601 + 995654731 uban 16 31 51 + 90000075000016 Ulam 16 106 456 + 9935106 undulating 141 181 unprimeable 1266 2326 4306 + 9985006 untouchable 276 766 1266 + 981256 upside-down 456 951 95551 43482676 wasteful 51 76 226 + 9995001 weak prime 31 181 601 + 99713851 Zumkeller 276 456 1266 + 94576