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A262958
Numbers whose base-b expansions, for both b=3 and b=4, include no digits other than 1 and b-1.
4
1, 5, 7, 13, 23, 53, 125, 215, 373, 1367, 1373, 1375, 3551, 4093, 5471, 5495, 5503, 30581, 30589, 32765, 32767, 56821, 56831, 89557, 96119, 96215, 96223, 97655, 98135, 98141, 98143, 98167, 98293, 98303, 351743, 352093, 521599, 521693, 521717, 521719, 524119, 524149, 875893, 875903, 884725, 884735
OFFSET
1,2
COMMENTS
1, 7 and 32767 also share this property in base 2; their binary expansions consist only of a sequence of 1s.
EXAMPLE
53 is 1222 in base 3 and 311 in base 4; it only uses the digit 1 or the largest digit in the two bases and is therefore a term.
Similarly 215 is 21222 in base 3 and 3113 in base 4 so it is also a term.
MATHEMATICA
Select[Range@ 1000000, Last@ DigitCount[#, 3] == 0 && Total@ Rest@ Drop[DigitCount[#, 4], {3}] == 0 &] (* Michael De Vlieger, Oct 05 2015 *)
Join[{1, 5}, Flatten[Table[Select[FromDigits[#, 3]&/@Tuples[{1, 2}, n], Union[ IntegerDigits[ #, 4]] =={1, 3}&], {n, 20}]]] (* Harvey P. Dale, Jun 14 2016 *)
PROG
(PARI) is(n)=!setsearch(Set(digits(n, 3)), 0) && #setintersect(Set(digits(n, 4)), [0, 2])==0 \\ Charles R Greathouse IV, Oct 12 2015
(Python)
from gmpy2 import digits
def f1(n):
s = digits(n, 3)
m = len(s)
for i in range(m):
if s[i] == '0':
return(int(s[:i]+'1'*(m-i), 3))
return n
def f2(n):
s = digits(n, 4)
m = len(s)
for i in range(m):
if s[i] == '0':
return(int(s[:i]+'1'*(m-i), 4))
if s[i] == '2':
return(int(s[:i]+'3'+'1'*(m-i-1), 4))
return n
A262958_list = []
n = 1
for i in range(10**4):
m = f2(f1(n))
while m != n:
n, m = m, f2(f1(m))
A262958_list.append(m)
n += 1 # Chai Wah Wu, Oct 30 2015
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Robin Powell, Oct 05 2015
STATUS
approved