A060159 Initial term of a series of exactly n consecutive Harshad or Niven numbers (a Harshad number is such that is divided by the sum of its digits).

12, 20, 110, 510, 131052, 12751220, 10000095, 2162049150, 124324220, 1, 920067411130599, 43494229746440272890, 12100324200007455010742303399999999999999999990, 4201420328711160916072939999999999999999999999999999999999999996

Offset: 1

Author

Carlos Rivera, Mar 12 2001

Keywords

• fini
• hard
• nonn
• base

Comments

Cooper and Kennedy (1993) proved that this sequence contains 20 terms. - Sergio Pimentel, Sep 18 2008

a(16) = 50757686696033684694106416498959861492*10^280 - 9 and a(17) = 14107593985876801556467795907102490773681*10^280 - 10. - Max Alekseyev, Apr 07 2013

H. G. Grundman (1994) extended the Cooper and Kennedy result to show that there are 2b but not 2b + 1 consecutive Harshad numbers in any base b. - Jianing Song, Dec 16 2024

Examples

			a(3) = 110 since (110, 111, 112) is the earliest run of 3 consecutive Harshad numbers: 110 is divisible by 1+1+0=2, 111 is divisible by 1+1+1=3, 112 is divisible by 1+1+2=4, but 109 is not divisible by 1+0+9=10, 113 is not divisible by 1+1+3=5, and there are no earlier runs of 3 consecutive numbers with this property. [Clarified by _Jianing Song_, Dec 16 2024]
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 110, p. 39, Ellipses, Paris 2008.

Links

• C. N. Cooper and R. E. Kennedy, On consecutive Niven numbers, Fibonacci Quart, (1993) 21, 146-151.
• H. G. Grundman, Sequences of consecutive n-Niven numbers, Fibonacci Quarterly, (1994), 32 (2): 174-175.
• B. Wilson, Construction of 2n Consecutive n-Niven Numbers, Fibonacci Quarterly, (1997), 35, 122-128.
• Carlos Rivera, Puzzle 129. Earliest sets of K consecutive Harshad Numbers, The Prime Puzzles and Problems Connection.
• Wikipedia, Harshad number

Crossrefs

Cf. A005349.

Extensions

a(8) is found by Jud McCranie, Nov 13 2001

a(11)-a(13) are found by Giovanni Resta, Feb 21 2008

a(14), a(16)-a(17) from Max Alekseyev, Apr 07 2013