cp's OEIS Frontend

A356274 a(n) is the number whose base-(n+1) expansion equals the binary expansion of n.

%I A356274 #72 Aug 23 2022 09:42:38
%S A356274 1,3,5,25,37,56,73,729,1001,1342,1741,2366,2941,3615,4369,83521,
%T A356274 104977,130340,160021,194922,234741,280393,332377,406250,474553,
%U A356274 551151,636637,732511,837901,954304,1082401,39135393,45435425,52521910,60466213,69345326,79236613
%N A356274 a(n) is the number whose base-(n+1) expansion equals the binary expansion of n.
%C A356274 If the binary expansion of n is n = bit0*2^0 + bit1*2^1 + bit2*2^2 + ... then a(n) = bit0*(n+1)^0 + bit1*(n+1)^1 + bit2*(n+1)^2 + ... . In other words: Interpret the binary expansion of n as digits in base n+1.
%F A356274 a(2^n) = (2^n + 1)^n = A136516(n).
%F A356274 a(2^n - 1) = (2^(n^2) - 1)/(2^n - 1) = A128889(n).
%F A356274 a(2^n + 1) = 1 + (2^n + 2)^n.
%F A356274 a(n) = A104257(n+1, n).
%F A356274 a(n) = (1/n)*Sum_{j>=1} floor((n + 2^(j-1))/2^j) * ((n-1)*(n+1)^(j-1) + 1).
%F A356274 a(n) = (1/n)*Sum_{j=1..n} ((n-1)*(n+1)^A007814(j) + 1).
%F A356274 a(2*n) = A104258(2*n+1) - 1.
%F A356274 (2*m+1)^n divides a((2*m+1)^n-1) for positive m and n > 0.
%e A356274 n=4 in binary is 100 and interpreting those digits as base n+1 = 5 is a(4) = 25.
%t A356274 a[n_] := FromDigits[IntegerDigits[n, 2], n + 1]; Array[a, 40] (* _Amiram Eldar_, Aug 19 2022 *)
%o A356274 (PARI) a(n) = fromdigits(digits(n, 2), n+1)
%o A356274 (Python)
%o A356274 def a(n): return sum((n+1)**i*int(bi) for i, bi in enumerate(bin(n)[2:][::-1]))
%o A356274 print([a(n) for n in range(1, 39)]) # _Michael S. Branicky_, Aug 02 2022
%Y A356274 Cf. A000523, A007814, A104257, A104258, A128889, A136516.
%K A356274 nonn,base
%O A356274 1,2
%A A356274 _Thomas Scheuerle_, Aug 02 2022