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A260193 Numbers k of the form abs(a - b + c - d) such that k^4 equals the concatenation of a//b//c//d and numbers k,b,c,d have the same number of digits.

%I A260193 #36 May 10 2025 09:03:58
%S A260193 198,220,221,287,352,364,484,562,627,638,672,715,716,780,793,858,901,
%T A260193 1095,1233,2328,8905,18183,39753,60248,85207,336734,2727274,5893504,
%U A260193 8620777,17769557,52818678,70710735,76470590,82230444,101318734,101636206,104263158,105262158,109891110,109942690,117883117,119722383,120826541
%N A260193 Numbers k of the form abs(a - b + c - d) such that k^4 equals the concatenation of a//b//c//d and numbers k,b,c,d have the same number of digits.
%C A260193 Leading zeros in b, c, and d are allowed.
%C A260193 Many numbers come in pairs, like: (220, 221), (715, 716), (140017877, 140017878).
%C A260193 Some numbers are also member of A259379, for example: 287, 715, 1095 and also the pair (140017877, 140017878).
%H A260193 Pieter Post, <a href="/A260193/b260193.txt">Table of n, a(n) for n = 1..239</a>
%e A260193 198^4 = 1536953616 and 198 = abs (1 - 536 + 953 - 616 ).
%e A260193 8905^4 = 6288335365950625 and 8905 = abs (6288 - 3353 + 6595 - 0625 ).
%t A260193 test[n_] := Block[{L=IntegerLength@ n, v}, v = IntegerDigits[ n^4, 10^L]; Length@ v == 4 && Abs@ Total[ {1, -1, 1, -1} v] == n]; Select[Range[10^5], test] (* _Giovanni Resta_, Aug 12 2015 *)
%o A260193 (Python)
%o A260193 def modb(n, m):
%o A260193     kk = 0
%o A260193     l = 1
%o A260193     while n > 0:
%o A260193         na = n % m
%o A260193         l += 1
%o A260193         kk += ((-1)**l) * na
%o A260193         n //= m
%o A260193     return abs(kk)
%o A260193 for n in range (100, 10**9):
%o A260193     ll = len(str(n))
%o A260193     if modb(n**4, 10**ll) == n and n**4 >= 10**(ll*3):
%o A260193          print (n, end=', ') # corrected by _David Radcliffe_, May 09 2025
%Y A260193 Cf. A056733, A101311, A118937, A118938, A228381, A257796, A258482, A259379.
%K A260193 nonn,base
%O A260193 1,1
%A A260193 _Pieter Post_, Jul 22 2015