Editing Harshad number (section)

== Nivenmorphic numbers ==

A '''Nivenmorphic number''' or '''harshadmorphic number''' for a given number base is an integer {{mvar|t}} such that there exists some harshad number {{mvar|N}} whose [[digit sum]] is {{mvar|t}}, and {{mvar|t}}, written in that base, terminates {{mvar|N}} written in the same base.

For example, 18 is a Nivenmorphic number for base 10:

  16218 is a harshad number
  16218 has 18 as digit sum
     18 terminates 16218

Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except [[11 (number)|11]].<ref>{{citation|first=Sandro|last=Boscaro|title=Nivenmorphic integers|journal=[[Journal of Recreational Mathematics]]|volume=28|issue=3|year=1996–1997|pages=201–205}}.</ref> In fact, for an [[parity (mathematics)|even]] integer ''n'' > 1, all positive integers except ''n''+1 are Nivenmorphic numbers for base ''n'', and for an [[parity (mathematics)|odd]] integer ''n'' > 1, all positive integers are Nivenmorphic numbers for base ''n''. e.g. the Nivenmorphic numbers in [[duodecimal|base 12]] are {{oeis|A011760}} (all positive integers except 13).

The smallest number with base 10 digit sum ''n'' and terminates ''n'' written in base 10 are: (0 if no such number exists)

: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, 78329, 99930, 585931, 388832, 1098933, 198934, 289835, 99936, 99937, 478838, 198939, 1999840, 2988941, 2979942, 2979943, 999944, 999945, 4698946, 4779947, 2998848, 2998849, 9999950, ... {{OEIS|id=A187924}}