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%I #32 Jan 05 2025 19:51:36
%S 12,20,110,510,131052,12751220,10000095,2162049150,124324220,1,
%T 920067411130599,43494229746440272890,
%U 12100324200007455010742303399999999999999999990,4201420328711160916072939999999999999999999999999999999999999996
%N 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).
%C Cooper and Kennedy (1993) proved that this sequence contains 20 terms. - _Sergio Pimentel_, Sep 18 2008
%C a(16) = 50757686696033684694106416498959861492*10^280 - 9 and a(17) = 14107593985876801556467795907102490773681*10^280 - 10. - _Max Alekseyev_, Apr 07 2013
%C 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
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 110, p. 39, Ellipses, Paris 2008.
%H C. N. Cooper and R. E. Kennedy, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/31-2/cooper.pdf">On consecutive Niven numbers</a>, Fibonacci Quart, (1993) 21, 146-151.
%H H. G. Grundman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/32-2/grundman.pdf">Sequences of consecutive n-Niven numbers</a>, Fibonacci Quarterly, (1994), 32 (2): 174-175.
%H B. Wilson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/35-2/wilson.pdf">Construction of 2n Consecutive n-Niven Numbers</a>, Fibonacci Quarterly, (1997), 35, 122-128.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_129.htm">Puzzle 129. Earliest sets of K consecutive Harshad Numbers</a>, The Prime Puzzles and Problems Connection.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Harshad_number">Harshad number</a>
%e 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]
%Y Cf. A005349.
%K fini,hard,nonn,base
%O 1,1
%A _Carlos Rivera_, Mar 12 2001
%E a(8) is found by _Jud McCranie_, Nov 13 2001
%E a(11)-a(13) are found by _Giovanni Resta_, Feb 21 2008
%E a(14), a(16)-a(17) from _Max Alekseyev_, Apr 07 2013