Frugal number

Let ${\displaystyle b>1}$  be a number base, and let ${\displaystyle K_{b}(n)=\lfloor \log _{b}{n}\rfloor +1}$  be the number of digits in a natural number ${\displaystyle n}$  for base ${\displaystyle b}$ . A natural number ${\displaystyle n}$  has the prime factorisation

${\displaystyle n=\prod _{\stackrel {p\,\mid \,n}{p{\text{ prime}}}}p^{v_{p}(n)}}$ 

where ${\displaystyle v_{p}(n)}$  is the p-adic valuation of ${\displaystyle n}$ , and ${\displaystyle n}$  is an frugal number in base ${\displaystyle b}$  if

${\displaystyle K_{b}(n)>\sum _{\stackrel {p\,\mid \,n}{p{\text{ prime}}}}K_{b}(p)+\sum _{\stackrel {p^{2}\,\mid \,n}{p{\text{ prime}}}}K_{b}(v_{p}(n)).}$ 

• Equidigital number
• Extravagant number

• R.G.E. Pinch (1998), Economical Numbers