Editing Harshad number (section)

== Consecutive harshad numbers ==

=== Maximal runs of consecutive harshad numbers ===
Cooper and Kennedy [[mathematical proof|proved]] in 1993 that no 21 consecutive integers are all harshad numbers in base 10.<ref>{{citation | zbl=0776.11003 | last1=Cooper | first1=Curtis | last2=Kennedy | first2=Robert E. | title=On consecutive Niven numbers | journal=[[Fibonacci Quarterly]] | volume=31 | number=2 | pages=146–151 | year=1993 | doi=10.1080/00150517.1993.12429304 | issn=0015-0517 |url=http://www.fq.math.ca/Scanned/31-2/cooper.pdf}}</ref><ref name=HBII382>{{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | url=https://archive.org/details/handbooknumberth00sand_741 | url-access=limited | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | zbl=1079.11001|page=[https://archive.org/details/handbooknumberth00sand_741/page/n381 382]}}</ref> They also constructed infinitely many 20-tuples of consecutive integers that are all 10-harshad numbers, the smallest of which exceeds 10<sup>44363342786</sup>.

{{harvs|authorlink=Helen G. Grundman|first=H. G.|last=Grundman|year=1994|txt}} extended the Cooper and Kennedy result to show that there are 2''b'' but not 2''b'' + 1 consecutive ''b''-harshad numbers for any base ''b''.<ref name=HBII382/><ref>{{citation | last = Grundman | first = H. G. | author-link=Helen G. Grundman | title = Sequences of consecutive ''n''-Niven numbers | journal = [[Fibonacci Quarterly]] | volume = 32 | issue = 2 | year=1994 | pages = 174–175 | doi = 10.1080/00150517.1994.12429245 | zbl=0796.11002 | issn=0015-0517 |url=http://www.fq.math.ca/Scanned/32-2/grundman.pdf}}</ref> 
This result was strengthened to show that there are infinitely many runs of 2''b'' consecutive ''b''-harshad numbers for ''b'' = 2 or 3 by {{harvs|authorlink=T. Tony Cai|first=T.|last=Cai|year=1996|txt}}<ref name=HBII382/> and for arbitrary ''b'' by [[Brad Wilson (mathematician)|Brad Wilson]] in 1997.<ref>{{citation | last1=Wilson | first1=Brad | title=Construction of 2''n'' consecutive ''n''-Niven numbers | journal=[[Fibonacci Quarterly]] | volume=35 | pages=122–128 | year=1997 | issue=2 | doi=10.1080/00150517.1997.12429006 | issn=0015-0517 |url=http://www.fq.math.ca/Scanned/35-2/wilson.pdf}}</ref>

In [[binary numeral system|binary]], there are thus infinitely many runs of four consecutive harshad numbers and in [[ternary numeral system|ternary]] infinitely many runs of six.

In general, such maximal sequences run from ''N''·''b<sup>k</sup>'' − ''b'' to ''N''·''b<sup>k</sup>'' + (''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:
* Inserting zeroes into ''N'' will not change the sequence of digital sums (just as 21, 201 and 2001 are all 10-harshad numbers).
* If we insert ''n'' zeroes after the first digit, ''α'' (worth ''αb<sup>i</sup>''), we increase the value of ''N'' by <math>\alpha b^i \left (b^n - 1 \right )</math>.
* If we can ensure that ''b<sup>n</sup>'' − 1 is divisible by all digit sums in the sequence, then the divisibility by those sums is maintained.
* If our initial sequence is chosen so that the digit sums are [[coprime]] to ''b'', we can solve ''b<sup>n</sup>'' = 1 [[modular arithmetic|modulo]] all those sums.
* If that is not so, but the part of each digit sum not coprime to ''b'' divides ''αb<sup>i</sup>'', then divisibility is still maintained.
* ''(Unproven)'' The initial sequence is so chosen.
Thus <!-- any solution implies --> our initial sequence yields an infinite set of solutions.

=== First runs of exactly {{mvar|n}} consecutive 10-harshad numbers ===
The smallest naturals starting runs of ''exactly'' {{mvar|n}} consecutive 10-harshad numbers (i.e., the smallest {{mvar|x}} such that <math>x, x+1, \cdots, x+n-1</math> are harshad numbers but <math>x-1</math> and <math>x+n</math> are not) are as follows {{OEIS|id=A060159}}:

<div style="overflow:auto">
{{alternating rows table|class=wikitable|style=text-align: right}}
|-
| {{mvar|n}} || 1 || 2 || 3 || 4 || 5
|-
| {{mvar|x}} || 12 || 20 || 110 || 510 || {{val|131052}}
|-
| {{mvar|n}} || 6 || 7 || 8 || 9 || 10
|-
| {{mvar|x}} || {{val|12751220}}  || {{val|10000095}}  || {{val|2162049150}}  || {{val|124324220}}  || 1
|-
| {{mvar|n}} || 11 || 12 || 13 || 14 || 15
|-
| {{mvar|x}} || {{val|920067411130599}}  || {{val|43494229746440272890}}  || <small>{{val|121003242000074550107423034e20|s=&nbsp;− 10}}</small>  || <small>{{val|420142032871116091607294e40|s=&nbsp;− 4}}</small>  || {{unknown}}
|-
| {{mvar|n}} || 16 || 17 || 18 || 19 || 20
|-
| {{mvar|x}} || <small>{{val|50757686696033684694106416498959861492e280|s=&nbsp;− 9}}</small>  || <small>{{val|14107593985876801556467795907102490773681e280|s=&nbsp;− 10}}</small>  || {{unknown}} || {{unknown}} || {{unknown}}
|-
|}</div>

By the previous section, no such {{mvar|x}} exists for <math>n > 20.</math>