Harshad number

A Harshad number or Niven number is a natural number that  is divisible by its checksum , i.e. the sum of its digits (in the decimal system with base 10).

The term Harshad number was introduced by the Indian mathematician D. R. Kaprekar and is derived from the Sanskrit word harsha ("joy"), while the Niven number goes back to the mathematician Ivan M. Niven , who described these numbers at a congress in 1977.

Examples

777 is its cross sum is divisible and is thus a Harshad number: . ${\displaystyle 7+7+7=21}$ ${\displaystyle 777=21\cdot 37}$

The first Harshad numbers (in the decimal system) are:

${\displaystyle 1,2,3,4,5,6,7,8,9,10,12,18,20,21,24,27,30,36,40,42,45,48,50,54,60,63,70,72,80,81,84,90,100,\ldots }$(Follow A005349 in OEIS )

The smallest , making it a harshad number, are the following: ${\displaystyle k}$ ${\displaystyle k\cdot n}$

${\displaystyle 1,1,1,1,1,1,1,1,1,1,10,1,9,3,2,3,6,1,6,1,1,5,9,1,2,6,1,3,9,1,12,6,4,3,2,1,3,3,3,1,10,1,12,3,\ldots }$(Follow A144261 in OEIS )

d. h .: are Harshad numbers ${\displaystyle {\underline {1}}\cdot 1=1,{\underline {1}}\cdot 2=2,\ldots ,{\underline {1}}\cdot 10=10,{\underline {10}}\cdot 11=110,{\underline {1}}\cdot 12=12,{\underline {9}}\cdot 13=117,\ldots }$

The smallest , so that it is not a Harshad number, are the following: ${\displaystyle k}$ ${\displaystyle k\cdot n}$

${\displaystyle 11,7,5,4,3,11,2,2,11,13,1,8,1,1,1,1,1,161,1,8,5,1,1,4,1,1,7,1,1,13,1,1,1,1,1,83,1,1,1,4,\ldots }$(Follow A144262 in OEIS )

d. h .: No Harshad numbers ${\displaystyle {\underline {11}}\cdot 1=11,{\underline {7}}\cdot 2=14,{\underline {5}}\cdot 3=15,{\underline {4}}\cdot 4=16,{\underline {3}}\cdot 5=15,{\underline {11}}\cdot 6=66,\ldots }$

n-Harshad numbers

Harshad numbers are also called n-Harshad numbers (or n-Niven numbers ) if they are considered in the base n .

The first n-Harshad numbers in base 12 are (with the lack of further digits being used): ${\displaystyle A=10,B=11,C=12,\ldots }$

${\displaystyle 1,2,3,4,5,6,7,8,9,A,B,10,1A,20,29,30,38,40,47,50,56,60,65,70,74,80,83,90,92,A0,A1,B0,100,}$

${\displaystyle 10A,110,115,119,120,122,128,130,134,137,146,150,153,155,164,172,173,182,191,1A0,\ldots }$

Example:

${\displaystyle 172}$is not an n-Harshad number for base 10:

${\displaystyle N=172}$has the checksum , but it is not a factor of . ${\displaystyle 1+7+2=10}$ ${\displaystyle 10}$ ${\displaystyle 172}$

${\displaystyle 172_{12}}$ but is an n-Harshad number for base 12:

${\displaystyle N=172_{12}}$is the number in the decimal system . The checksum of is (in the decimal system that is ). It is actually a divisor of (in the decimal system ). ${\displaystyle {\underline {1}}\cdot 12^{2}+{\underline {7}}\cdot 12^{1}+{\underline {2}}\cdot 12^{0}=230}$ ${\displaystyle N=172_{12}}$ ${\displaystyle 1+7+2=A_{12}}$ ${\displaystyle 10}$ ${\displaystyle A_{12}}$ ${\displaystyle N=172_{12}=A_{12}\cdot 1B_{12}}$ ${\displaystyle 230=10\cdot 23}$

The smallest , so that a base 12 harshad number is n, are the following (written in the decimal system): ${\displaystyle k}$ ${\displaystyle k\cdot n}$

${\displaystyle 1,1,1,1,1,1,1,1,1,1,1,1,12,6,4,3,10,2,11,3,4,1,7,1,12,6,4,3,11,2,11,3,1,5,9,1,12,11,4,3,11,2,11,1,4,4,11,1,16\ldots }$

The smallest , so that there is no base 12 n-Harshad number, are the following (written in the decimal system): ${\displaystyle k}$ ${\displaystyle k\cdot n}$

${\displaystyle 13,7,5,4,3,3,2,2,2,2,13,16,1,1,1,1,1,1,1,1,1,157,1,8,1,1,1,1,1,1,1,1,13,1,1,6,1,1,1,1,1,1,1,157,1,1,1,4\ldots }$

properties

The example given above with the number 777 can be generalized to all 3-digit natural numbers of the same type:

• Every natural number of the form , where any digit from 1 to 9 can represent, is a Harshad number in the decimal system (so it can be divided by its checksum). ${\displaystyle nnn}$ ${\displaystyle n}$

The proof results from the following consideration:

${\displaystyle {\begin{aligned}nnn&=n\cdot 10^{2}+n\cdot 10^{1}+n\cdot 10^{0}\\&=n\cdot (100+10+1)\\&=n\cdot 111\\&=n\cdot (3\cdot 37)\\&=(n\cdot 3)\cdot 37\\\end{aligned}}}$

But now the sum of the digits is . ${\displaystyle nnn\colon ~n+n+n=n\cdot 3}$

So every natural number of the form is 37 times its checksum, i.e. a Harshad number. q. e. d. ${\displaystyle nnn}$

• All integers between 0 and base n are n -Harshad numbers.
• There are no 21 consecutive harshad numbers in the decimal system.
• In the decimal system there are an infinite number of 20 consecutive Harshad numbers. The smallest of these is larger than . ${\displaystyle 10^{44363342786}}$

• With base n there are no 2n + 1 consecutive n-Harshad numbers (generalization of the property above).
• With base n there are infinitely many 2n consecutive Harshad numbers (generalization of the property above).
• Let be the number of Harshad numbers and be . Then: ${\displaystyle N(x)}$ ${\displaystyle \leq x}$ ${\displaystyle \varepsilon >0}$

${\displaystyle x^{1-\varepsilon }\ll N(x)\ll {\frac {x\log \log x}{\log x}}}$

Example:

There are exactly 11872 Harshad numbers out of 100,000. So is and . And actually applies ${\displaystyle x=100000}$ ${\displaystyle N(x)=11872}$ ${\displaystyle x^{1-\varepsilon }=100000^{1-\varepsilon }\ll 100000^{1-0,185095}\approx N(x)=11872\ll 21223,7\approx {\frac {100000\cdot \log \log 100000}{\log 100000}}={\frac {x\log \log x}{\log x}}}$

Niven morph numbers

A niven morph number (or harshadmorph number ) for a base n is an integer t such that there exists a Harshad number N whose cross sum is t , and t , written in this base n , describes the number N in this base n .

Example 1:

${\displaystyle 18}$ is a niven morph number for base 10:

${\displaystyle N=16218}$is a Harshad number (based on n = 10). The checksum of is . It is actually a factor of . ${\displaystyle 16218}$ ${\displaystyle 1+6+2+1+8=18}$ ${\displaystyle 18}$ ${\displaystyle 16218=18\cdot 901}$

Example 2:

${\displaystyle 18_{12}}$ is a niven morph number for base 12:

${\displaystyle N=1A0_{12}}$is a Harshad number (based on n = 12) and is the number in the decimal system . The checksum of is (in the decimal system 11). It is actually a divisor of (in the decimal system ). ${\displaystyle {\underline {1}}\cdot 12^{2}+{\underline {10}}\cdot 12^{1}+{\underline {0}}\cdot 12^{0}=264}$ ${\displaystyle N=1A0_{12}}$ ${\displaystyle 1+A+0=B_{12}}$ ${\displaystyle B_{12}}$ ${\displaystyle N=1A0_{12}=B_{12}\cdot 20_{12}}$ ${\displaystyle 264=11\cdot 24}$

The next list gives the smallest number (in the decimal system) whose cross sum is n and which is divisible by n (if there is no such number, 0 is given):

${\displaystyle 1,2,3,4,5,6,7,8,9,910,0,912,11713,6314,915,3616,15317,918,17119,9920,18921,9922,82823,19824,9925,46826,18927,18928,}$

${\displaystyle 78329,99930,585931,388832,1098933,198934,289835,99936,99937,478838,198939,1999840,2988941,2979942,2979943,999944,999945,}$

${\displaystyle 4698946,4779947,2998848,2998849,9999950,\ldots }$(Follow A187924 in OEIS )

For example has the checksum and actually is a factor of . So is a niven morph number to base 10. ${\displaystyle 289835}$ ${\displaystyle 2+8+9+8+3+5=35}$ ${\displaystyle 35}$ ${\displaystyle 289835=35\cdot 8281}$ ${\displaystyle 35}$

Properties:

• All positive integers with a base of 10 are niven morph numbers, except for the number 11.
• All positive even integers with base n > 1 are niven morph numbers with base n , except for n + 1 .
• All positive odd integers with base n > 1 are niven morph numbers with base n .

Multiple Harshad numbers

A multiple harshad number is a harshad number which, divided by its checksum, results in another (different) harshad number.

Example 1 is a multiple Harshad number, because , , and also Harshad numbers are. This number is also known as MHN-4 , so you can make four (different) more Harshad numbers out of it. ${\displaystyle 6804}$ ${\displaystyle 6804/18=378}$ ${\displaystyle 378/18=21}$ ${\displaystyle 21/3=7}$ ${\displaystyle 7/7=1}$ ${\displaystyle 6804}$

Example 2: is an MHN-12 , so you can find 12 different other Harshad numbers by dividing with their respective checksums (the first checksum is ). ${\displaystyle 2016502858579884466176}$ ${\displaystyle 108}$

Example 3: is another, smaller MHN-12 . ${\displaystyle 10080000000000=1008\cdot 10^{10}}$

Example 4: is an MHN- (n + 2) . ${\displaystyle 1008\cdot 10^{n}}$

See also

• Happy number
• Lucky number
• Self-describing number

literature

• Curtis Cooper, Robert E. Kennedy: On consecutive Niven numbers . In: Fibonacci Quarterly , 31, 2, 1993, pp. 146-151
• Helen G. Grundmann: Sequences of Consecutive Niven Numbers . In: Fibonacci Quarterly , 32, 2, (1994), 174-175
• Brad Wilson: Construction of 2n consecutive n-Niven numbers . In: Fibonacci Quarterly , 35, 1997, pp. 122-128
• Jean-Marie DeKoninck, Nicolas Doyon: On the number of Niven numbers up to x . In: Fibonacci Quarterly , 41, 5, November 2003, pp. 431-440
• Jean-Marie DeKoninck, Nicolas Doyon, I. Katái: On the counting function for the Niven numbers . In: Acta Arithmetica , 106, 2003, pp. 265-275
• Sandro Boscaro: Nivenmorphic Integers . In: Journal of Recreational Mathematics , 28, 3, 1996-1997, pp. 201-205
• E. Bloem: Harshad numbers . In: Journal of Recreational Mathematics , 34, 2, 2005, p. 128

Web links

• Eric W. Weisstein : Harshad Number . In: MathWorld (English).
• József Sándor , Borislav Crstici: Handbook of Number Theory II. (PDF) Springer-Verlag, pp. 381–383 , accessed on May 27, 2018 (English). 
• Curtis Cooper, Robert E. Kennedy: On consecutive Niven numbers. (PDF) Fibonacci Quarterly , pp. 146–151 , accessed on May 28, 2018 (English). 
• Helen G. Grundman: Sequences of consecutive n -niven numbers. (PDF) Fibonacci Quarterly, pp. 174–175 , accessed on May 28, 2018 (English). 
• Brad Wilson: Construction of 2 n consecutive n -niven numbers. (PDF) Fibonacci Quarterly, pp. 122–128 , accessed on May 28, 2018 (English). 
• Jean-Marie DeKoninck, Nicolas Doyon: On the number of Niven numbers up to x. (PDF) Fibonacci Quarterly, pp. 431–440 , accessed on May 30, 2018 (English). 

Individual evidence

1. ↑ József Sándor , Borislav Crstici: Handbook of Number Theory II. (PDF) (No longer available online.) Springer-Verlag, pp. 381 and 451 , formerly in the original ; accessed on May 27, 2018 (English).  ( Page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.    
2. ^ Curtis Cooper, Robert E. Kennedy: On consecutive Niven numbers. (PDF) In: Fibonacci Quarterly . Pp. 146–151 , accessed on May 28, 2018 (English). 
3. ↑ ^{a b c} József Sándor , Borislav Crstici: Handbook of Number Theory II. (PDF) (No longer available online.) Springer-Verlag, p. 382 , formerly in the original ; accessed on May 28, 2018 (English).  ( Page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.    
4. ^ Curtis Cooper, Robert E. Kennedy: On consecutive Niven numbers. (PDF) In: Fibonacci Quarterly . P. 148 , accessed on May 28, 2018 (English). 
5. ↑ primepuzzles.net: Problems & Puzzles: Puzzle 129. Retrieved May 30, 2018 (English). 
6. ↑ ^{a b} Helen G. Grundman: Sequences of consecutive n -niven numbers. (PDF) In: Fibonacci Quarterly . Pp. 174–175 , accessed on May 28, 2018 (English). 
7. ^ Brad Wilson: Construction of 2 n consecutive n -niven numbers. (PDF) In: Fibonacci Quarterly . Pp. 122–128 , accessed on May 28, 2018 (English). 
8. ^ ^{A b} Jean-Marie DeKoninck, Nicolas Doyon: On the number of Niven numbers up to x. (PDF) In: Fibonacci Quarterly . Pp. 431–440 , accessed May 30, 2018 (English). 
9. ↑ Sandro Boscaro: Nivenmorphic integers . In: Journal of Recreational Mathematics . tape 28 , no. 3 , 1996, p. 201-205 .
10. ↑ E. Bloem: Harshad numbers . In: Journal of Recreational Mathematics . tape 34 , no. 2 , 2005, p. 128 .