Editing Harshad number (section)

== 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 [[zero of a function|zeros]] of certain [[Dedekind zeta function]]s, Sanna proved that there exists a positive integer <math>k</math> such that every natural number is the sum of at most <math>k</math> harshad numbers, that is, the set of harshad numbers is an [[additive basis]].<ref>{{citation|first1=Carlo|last1=Sanna|title=Additive bases and Niven numbers|journal=[[Bulletin of the Australian Mathematical Society]]|volume=104|issue=3|date=March 2021|pages=373–380|doi=10.1017/S0004972721000186|s2cid=231639019 |doi-access=free|arxiv=2101.07593}}.</ref>

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, ... {{OEIS|id=A337853}}.

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, ... {{OEIS|id=A337854}}.