A373177 Integers k such that 2k + 1 and 4k + 3 are anagrams of k.

15632, 126530, 130265, 150632, 152630, 156329, 162530, 163025, 1265030, 1265300, 1265309, 1300265, 1302650, 1302659, 1500632, 1502630, 1506329, 1526300, 1526309, 1563299, 1566332, 1625030, 1625300, 1625309, 1630025, 1630250, 1630259, 1656332, 12650030

Offset: 1

Author

Gonzalo Martínez, May 26 2024

Keywords

• nonn
• base

Comments

The terms of this sequence begin with decimal digits 1 or 2, otherwise 4*k + 3 has more digits than k and cannot be an anagram. The first term whose first digit is 2 is a(3931) = 2055114278.

This sequence has infinitely many terms, since 1500*10^m + 632 is a term for all positive integers m.

All terms == 8 (mod 9). - Hugo Pfoertner, May 27 2024

Examples

			15632 is a term, since 2*15632 + 1 = 31265 and 4*15632 + 3 = 62531 are both permutations of the digits of 15632.
		

Crossrefs

Cf. A323711, A371623.

Programs

Maple

filter:= proc(n) local L;
  L:= sort(convert(n,base,10));
  sort(convert(2*n+1,base,10))=L
  and sort(convert(4*n+3,base,10))=L
end proc:
R:= NULL: count:= 0:
for d from 1 while count < 100 do
 for x from 10^(d-1) + 7 by 9 to (10^d-3)/4 while count < 100 do
   if filter(x) then R:= R,x; count:= count+1 fi
od od:
R; # Robert Israel, May 27 2024

Mathematica

sid[n_] := Sort[IntegerDigits[n]]; Select[Range[13000000], sid[#] == sid[2*# + 1] == sid[4*# + 3] &] (* Amiram Eldar, May 27 2024 *)

PARI

isok(k) = my(d=vecsort(digits(k))); (d == vecsort(digits(2*k+1))) && (d == vecsort(digits(4*k+3))); \\ Michel Marcus, May 28 2024

Python

from itertools import count, islice
def agen(): # generator of terms
    for e in count(1):
        for k in range(10**(e-1), 10**e//4):
            if sorted(str(k)) == sorted(str(2*k+1)) == sorted(str(4*k+3)):
                yield k
print(list(islice(agen(), 30))) # Michael S. Branicky, May 26 2024