Harshad number explained

In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base.Harshad numbers in base are also known as -harshad (or -Niven) numbers.Harshad numbers were defined by D. R. Kaprekar, a mathematician from India.[1] The word "harshad" comes from the Sanskrit (joy) + (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.

Definition

Stated mathematically, let be a positive integer with digits when written in base, and let the digits be

ai

(

i=0,1,\ldots,m-1

). (It follows that

ai

must be either zero or a positive integer up to .) can be expressed as
m-1
X=\sum
i=0

aini.

is a harshad number in base if:

X\equiv0\bmod

m-1
{\sum
i=0

ai}.

A number which is a harshad number in every number base is called an all-harshad number, or an all-Niven number. There are only four all-harshad numbers: 1, 2, 4, and 6. The number 12 is a harshad number in all bases except octal.

Examples

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, ... .

Properties

Given the divisibility test for 9, one might be tempted to generalize that all numbers divisible by 9 are also harshad numbers. But for the purpose of determining the harshadness of, the digits of can only be added up once and must be divisible by that sum; otherwise, it is not a harshad number. For example, 99 is not a harshad number, since 9 + 9 = 18, and 99 is not divisible by 18.

The base number (and furthermore, its powers) will always be a harshad number in its own base, since it will be represented as "10" and 1 + 0 = 1.

All numbers whose base b digit sum divides b−1 are harshad numbers in base b.

For a prime number to also be a harshad number it must be less than or equal to the base number, otherwise the digits of the prime will add up to a number that is more than 1, but less than the prime, and will not be divisible. For example: 11 is not harshad in base 10 because the sum of its digits “11” is 1 + 1 = 2, and 11 is not divisible by 2; while in base 12 the number 11 may be represented as “”, the sum of whose digits is also . Since is divisible by itself, it is harshad in base 12.

Although the sequence of factorials starts with harshad numbers in base 10, not all factorials are harshad numbers. 432! is the first that is not. (432! has digit sum 3897 = 32 × 433 in base 10, thus not dividing 432!)

The smallest such that

kn

is a harshad number are

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 9, 3, 2, 3, 6, 1, 6, 1, 1, 5, 9, 1, 2, 6, 1, 3, 9, 1, 12, 6, 4, 3, 2, 1, 3, 3, 3, 1, 10, 1, 12, 3, 1, 5, 9, 1, 8, 1, 2, 3, 18, 1, 2, 2, 2, 9, 9, 1, 12, 6, 1, 3, 3, 2, 3, 3, 3, 1, 18, 1, 7, 3, 2, 2, 4, 2, 9, 1, ... .

The smallest such that

kn

is not a harshad number are

11, 7, 5, 4, 3, 11, 2, 2, 11, 13, 1, 8, 1, 1, 1, 1, 1, 161, 1, 8, 5, 1, 1, 4, 1, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 83, 1, 1, 1, 4, 1, 4, 1, 1, 11, 1, 1, 2, 1, 5, 1, 1, 1, 537, 1, 1, 1, 1, 1, 83, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 68, 1, 1, 1, 1, 1, 1, 1, 2, ... .

Other bases

The harshad numbers in base 12 are:

1, 2, 3, 4, 5, 6, 7, 8, 9,,, 10, 1, 20, 29, 30, 38, 40, 47, 50, 56, 60, 65, 70, 74, 80, 83, 90, 92, 0, 1, 0, 100, 10, 110, 115, 119, 120, 122, 128, 130, 134, 137, 146, 150, 153, 155, 164, 172, 173, 182, 191, 10, 10, 1, 200, ...where represents ten and represents eleven.

Smallest such that

kn

is a base-12 harshad number are (written in base 10):

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 6, 4, 3, 10, 2, 11, 3, 4, 1, 7, 1, 12, 6, 4, 3, 11, 2, 11, 3, 1, 5, 9, 1, 12, 11, 4, 3, 11, 2, 11, 1, 4, 4, 11, 1, 16, 6, 4, 3, 11, 2, 1, 3, 11, 11, 11, 1, 12, 11, 5, 7, 9, 1, 7, 3, 3, 9, 11, 1, ...

Smallest such that

kn

is not a base-12 harshad number are (written in base 10):

13, 7, 5, 4, 3, 3, 2, 2, 2, 2, 13, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 157, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 157, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1885, 1, 1, 1, 1, 1, 3, ...

Similar to base 10, not all factorials are harshad numbers in base 12. After 7! (= 5040 = 200 in base 12, with digit sum 13 in base 12, and 13 does not divide 7!), 1276! is the next that is not. (1276! has digit sum 14201 = 11 × 1291 in base 12, thus does not divide 1276!)

Consecutive harshad numbers

Maximal runs of consecutive harshad numbers

Cooper and Kennedy proved in 1993 that no 21 consecutive integers are all harshad numbers in base 10.[3] They also constructed infinitely many 20-tuples of consecutive integers that are all 10-harshad numbers, the smallest of which exceeds 1044363342786.

extended the Cooper and Kennedy result to show that there are 2b but not 2b + 1 consecutive b-harshad numbers for any base b.[3] This result was strengthened to show that there are infinitely many runs of 2b consecutive b-harshad numbers for b = 2 or 3 by [3] and for arbitrary b by Brad Wilson in 1997.

In binary, there are thus infinitely many runs of four consecutive harshad numbers and in ternary infinitely many runs of six.

In general, such maximal sequences run from N·bkb to N·bk + (b − 1), where b is the base, k is a relatively large power, and N is a constant.Given one such suitably chosen sequence, we can convert it to a larger one as follows:

Thus our initial sequence yields an infinite set of solutions.

First runs of exactly consecutive 10-harshad numbers

The smallest naturals starting runs of exactly consecutive 10-harshad numbers (i.e., the smallest such that

x,x+1,,x+n-1

are harshad numbers but

x-1

and

x+n

are not) are as follows :

|-| || 1 || 2 || 3 || 4 || 5|-| || 12 || 20 || 110 || 510 || |-| || 6 || 7 || 8 || 9 || 10|-| || || || || || 1|-| || 11 || 12 || 13 || 14 || 15|-| || || || || || unknown|-| || 16 || 17 || 18 || 19 || 20|-| || || || unknown || unknown || unknown|-|}

By the previous section, no such exists for

n>20.

Estimating the density of harshad numbers

If we let

N(x)

denote the number of harshad numbers

\lex

, then for any given

\varepsilon>0,

x1-\varepsilon\llN(x)\ll

xloglogx
logx

as shown by Jean-Marie De Koninck and Nicolas Doyon;[4] furthermore, De Koninck, Doyon and Kátai[5] proved that

N(x)=(c+o(1))x
logx

,

where

c=(14/27)log101.1939

and the

o(1)

term uses Big O notation.

Sums of harshad numbers

Every natural number not exceeding one billion is either a harshad number or the sum of two harshad numbers. Conditional to a technical hypothesis on the zeros of certain Dedekind zeta functions, Sanna proved that there exists a positive integer

k

such that every natural number is the sum of at most

k

harshad numbers, that is, the set of harshad numbers is an additive basis.[6]

The number of ways that each natural number 1, 2, 3, ... can be written as sum of two harshad numbers is:

0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 3, 2, 4, 3, 3, 4, 3, 3, 5, 3, 4, 5, 4, 4, 7, 4, 5, 6, 5, 3, 7, 4, 4, 6, 4, 2, 7, 3, 4, 5, 4, 3, 7, 3, 4, 5, 4, 3, 8, 3, 4, 6, 3, 3, 6, 2, 5, 6, 5, 3, 8, 4, 4, 6, ... .

The smallest number that can be written in exactly 1, 2, 3, ... ways as the sum of two harshad numbers is:

2, 4, 6, 8, 10, 51, 48, 72, 108, 126, 90, 138, 144, 120, 198, 162, 210, 216, 315, 240, 234, 306, 252, 372, 270, 546, 360, 342, 444, 414, 468, 420, 642, 450, 522, 540, 924, 612, 600, 666, 630, 888, 930, 756, 840, 882, 936, 972, 1098, 1215, 1026, 1212, 1080, ... .

Nivenmorphic numbers

A Nivenmorphic number or harshadmorphic number for a given number base is an integer such that there exists some harshad number whose digit sum is, and, written in that base, terminates written in the same base.

For example, 18 is a Nivenmorphic number for base 10:

16218 is a harshad number 16218 has 18 as digit sum 18 terminates 16218

Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except 11.[7] In fact, for an even integer n > 1, all positive integers except n+1 are Nivenmorphic numbers for base n, and for an odd integer n > 1, all positive integers are Nivenmorphic numbers for base n. e.g. the Nivenmorphic numbers in base 12 are (all positive integers except 13).

The smallest number with base 10 digit sum n and terminates n written in base 10 are: (0 if no such number exists)

1, 2, 3, 4, 5, 6, 7, 8, 9, 910, 0, 912, 11713, 6314, 915, 3616, 15317, 918, 17119, 9920, 18921, 9922, 82823, 19824, 9925, 46826, 18927, 18928, 78329, 99930, 585931, 388832, 1098933, 198934, 289835, 99936, 99937, 478838, 198939, 1999840, 2988941, 2979942, 2979943, 999944, 999945, 4698946, 4779947, 2998848, 2998849, 9999950, ...

Multiple harshad numbers

defines a multiple harshad number as a harshad number that, when divided by the sum of its digits, produces another harshad number.[8] He states that 6804 is "MHN-4" on the grounds that

\begin{align} 6804/18&=378\\ 378/18&=21\\ 21/3&=7\\ 7/7&=1 \end{align}

(it is not MHN-5 since

1/1=1

, but 1 is not "another" harshad number)

and went on to show that 2016502858579884466176 is MHN-12. The number 10080000000000 = 1008 × 1010, which is smaller, is also MHN-12. In general, 1008 × 10n is MHN-(n+2).

Notes and References

  1. D. R. Kaprekar, Multidigital Numbers, Scripta Mathematica 21 (1955), 27.
  2. Rosario D'Amico, A method to generate Harshad numbers, in Journal of Mathematical Economics and Finance, vol. 5, n. 1, giugno 2019, p. 19-26.
  3. Book: Sándor . Jozsef . Crstici . Borislav . Handbook of number theory II . limited . Dordrecht . Kluwer Academic . 2004 . 1-4020-2546-7 . 1079.11001. 382.
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