nialpdromes

A number is a nialpdrome in a given base $b$

(often 10 or 16) if its digits are in nonincreasing order in that base.

For example, 43210, 2222, 76652 and 9630 are all nialpdromes in base 10.

A nialpdrome in which the digits are strictly decreasing is called katadrome, while numbers whose digits are deincreasing and strictly decreasing are called plaindromes and metadromes.

The number $N^{(b)}_k$ of nialpdromes of 1$"> digits in base $b$ is equal to

$N^{(b)}_k = {{k+b-1}\choose {b-1}}-1\,,$

which, for $b=2$

simplifies to $k$

. In general $N^{(b)}_1=b$ , since we count also the 0 among the 1-digit nialpdromes.

The total number $T^{(b)}_n = N^{(b)}_1+N^{(b)}_2+\cdots+N^{(b)}_n$ of nialpdromes in base $b$ with at most $n$ digits is equal to

$T^{(b)}_n = {{n+b}\choose{n}}-n\,.$

Probably the largest nialpdrome primes with index respectively nialpdrome and plaindrome are $p_{664444}= 9997777$ and $p_{6441111}=112556669$ . See the plaindromes for the symmetric pairs.

The first nialpdromes (in base 10) are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 22, 30, 31, 32, 33, 40, 41, 42, 43, 44, 50 more terms

Below, the spiral pattern of nialpdromes up to 10000 . See the page on prime numbers for an explanation and links to similar pictures.

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

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

Useful links Mathworld, Nialpdrome
OEIS, Sequence A009996 (base 10)
OEIS, Sequence A023771 (base 16)

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

a-pointer 11 631 811 + 9999875321 ABA 32 50 64 + 8820000000000 aban 10 11 20 + 1000000000000 abundant 20 30 40 + 50000000 Achilles 72 200 432 + 50000000000000 admirable 20 30 40 + 99999844 alt.fact. 4421 alternating 10 21 30 + 987654321 amenable 20 21 32 + 1000000000 amicable 220 87633 88730 apocalyptic 220 222 411 + 30000 arithmetic 11 20 21 + 9999999 automorphic 76 balanced p. 53 211 653 + 9999997543 Bell 52 877 betrothed 75 binomial 10 20 21 + 222222111111 brilliant 10 21 221 + 999999863 c.decagonal 11 31 61 + 222222211111111 c.heptagonal 22 43 71 + 83333321 c.nonagonal 10 55 91 + 87777776521 c.octagonal 81 441 841 + 999887641 c.pentagonal 31 51 76 + 99966631 c.square 41 61 85 + 755221 c.triangular 10 31 64 + 886666652110 cake 42 64 93 + 988442 Carmichael 997633 Catalan 42 Chen 11 31 41 + 99999971 congruent 20 21 22 + 9999999 constructible 10 20 30 + 7710 cube 64 1000 8000 + 64000000000000 Cullen 65 Cunningham 10 31 33 + 99999999999999 Curzon 21 30 33 + 99999981 cyclic 11 31 33 + 9999997 D-number 21 33 51 + 6666531 d-powerful 43 63 332 + 9999865 de Polignac 331 877 977 + 99999883 decagonal 10 52 85 + 99999985000000 deceptive 91 6533 6541 + 11111111111 deficient 10 11 21 + 9999999 dig.balanced 10 11 21 + 88855500 droll 72 800 9984 640000 Duffinian 21 32 50 + 9999997 eban 30 32 40 + 66000000000000 economical 10 11 21 + 20000000 emirp 31 71 73 + 99999643 emirpimes 51 62 85 + 99999997 equidigital 10 11 21 + 9999991 eRAP 20 98 76640 + 976533100 esthetic 10 21 32 + 9876543210 Eulerian 11 66 evil 10 20 30 + 999999998 factorial 720 fibodiv 61 75 911 + 98876 Fibonacci 21 55 610 987 Friedman 54432 65531 65542 + 999964 frugal 10000 20000 32000 + 999887641 gapful 100 110 200 + 100000000000 Gilda 110 220 330 + 997 Giuga 30 good prime 11 41 53 + 99754411 happy 10 31 32 + 10000000 harmonic 6200 8872200 Harshad 10 20 21 + 10000000000 heptagonal 55 81 540 + 999999970000000 hex 61 91 331 + 87766666663321 hexagonal 66 91 630 + 87777776521 highly composite 60 720 840 + 554400 hoax 22 84 85 + 99999650 Hogben 21 31 43 + 998844421 Honaker 2221 6311 6553 + 998831111 house 32 652 933 hungry 74 hyperperfect 21 iban 10 11 20 + 777777 idoneal 10 21 22 + 840 impolite 32 64 inconsummate 62 63 65 + 999995 insolite 111 111111111 interprime 21 30 42 + 99999998 Jacobsthal 11 21 43 85 Jordan-Polya 32 64 72 + 86400 junction 111 210 311 + 99999970 Kaprekar 55 99 999 + 999999999999999 katadrome 10 20 21 + 9876543210 Lehmer 51 85 91 + 87777776521 Leyland 32 54 100 + 20000000000 lonely 53 211 Lucas 11 76 322 + 843 lucky 21 31 33 + 9999997 Lynch-Bell 432 864 6432 + 9864 m-pointer 61 2111 3221111 magic 65 111 870 + 665555 magnanimous 11 20 21 + 999994 modest 111 211 222 + 1111111111 Moran 21 42 63 + 99999965 Motzkin 21 51 nonagonal 75 111 651 + 88877542221 nude 11 22 33 + 444444444 O'Halloran 20 44 60 + 660 oban 10 11 20 + 999 octagonal 21 40 65 + 833333333333333 odious 11 21 22 + 1000000000 Ormiston 55331 66431 85331 + 999666431 palindromic 11 22 33 + 999999999999999 palprime 11 pancake 11 22 92 + 87777776522 panconsummate 10 11 20 + 721 pandigital 11 21 75 + 9876543210 partition 11 22 30 + 77 pentagonal 22 51 70 + 8888842110 pernicious 10 11 20 + 9999998 Perrin 10 22 51 + 853 persistent 9876543210 98765432100 98765432110 + 99876543210 Pierpont 73 97 433 plaindrome 11 22 33 + 999999999999999 Poulet 8321 83333 653333 + 87777776521 power 32 64 81 + 10000000000000 powerful 32 64 72 + 999887641000000 practical 20 30 32 + 10000000 prim.abundant 20 30 42 + 99999844 prime 11 31 41 + 999999999961 primorial 30 210 pronic 20 30 42 + 99999990000000 Proth 33 41 65 + 98877441 pseudoperfect 20 30 40 + 1000000 rare 65 repdigit 11 22 33 + 999999999999999 repfigit 61 75 742 repunit 21 31 40 + 111111111111111 Rhonda 885521 98655551 Ruth-Aaron 50 77 8855 + 66544333220 Saint-Exupery 60 4200 7500 + 999999999744000 self 20 31 42 + 999999994 self-describing 22 4444 444422 + 88888888666666 semiprime 10 21 22 + 99999997 sliding 11 20 52 + 700000000000000 Smith 22 85 94 + 99999920 Sophie Germain 11 41 53 + 9999988811 sphenic 30 42 66 + 99999994 square 64 81 100 + 999887641000000 star 73 433 541 + 777777661 straight-line 111 210 222 + 999999999999999 strobogrammatic 11 88 96 + 999999986666666 strong prime 11 41 71 + 99999931 subfactorial 44 super Niven 10 20 30 + 44444400000 super-d 31 81 310 + 9999981 superabundant 60 720 840 + 554400 tau 40 60 72 + 1000000000 taxicab 96666661000 96666661000000 tetrahedral 10 20 84 + 85333333320000 tetranacci 773 triangular 10 21 55 + 22222221111111 tribonacci 44 81 trimorphic 51 75 76 + 999999999999999 truncatable prime 31 43 53 + 9866653 twin 11 31 41 + 999998641 uban 10 11 20 + 99000000000000 Ulam 11 53 62 + 9999999 unprimeable 200 320 322 + 10000000 untouchable 52 88 96 + 999996 upside-down 55 64 73 + 999999951111111 wasteful 20 22 30 + 9999999 weak prime 31 43 61 + 99999971 weird 70 43330 54110 + 997430 Wieferich 7651 Woodall 63 80 99999999999 Zeisel 8444431 Zuckerman 11 111 432 + 9111111111 Zumkeller 20 30 40 + 100000 zygodrome 11 22 33 + 999999999999999