This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A153013 #51 Jul 20 2025 15:01:42 %S A153013 0,1,2,3,4,5,6,9,10,15,16,11,12,25,50,147,220,6125,1968750, %T A153013 89142864525,84252896510182189218, %U A153013 34892570216750728458698250328871491829901861750593684043 %N A153013 Starting with input 0, find the binary value of the input. Then interpret resulting string of 1's and 0's as prime-based numbers, as follows: 0's are separators, uninterrupted strings of 1's are interpreted from right to left as exponents of the prime numbers. Output is returned as input for the next number in sequence. %C A153013 From _Antti Karttunen_, Oct 15 2016: (Start) %C A153013 Iterates of map f : n -> A005940(1+n), (Doudna-sequence, but with starting offset zero) starting from the initial value 0. Conversely, the unique infinite sequence such that a(n) = A156552(a(n+1)) and a(0) = 0. %C A153013 Note that map f can also form cycles, like 7 <-> 8 (A005940(1+7) = 8, A005940(1+8) = 7). %C A153013 On the other hand, this sequence cannot ever fall into a loop because 0 is not in the range of map f, for n=0.., while f is injective on [1..]. Thus the values obtained by this sequence are not bounded, although there might be more nonmonotonic positions like for example there is from a(10) = 16 to a(11) = 11. %C A153013 The formula A008966(a(n+1)) = A085357(a(n)) tells that the squarefreeness of the next term a(n+1) is determined by whether the previous term a(n) is a fibbinary number (A003714) or not. Numerous other such correspondences hold, and they hold also for any other trajectories outside of this sequence. %C A153013 Even and odd terms alternate. No two squares can occur in succession because A106737 obtains even values for all squares > 1 and A000005 is odd for all squares. More directly this is seen from the fact that the rightmost 1-bit in the binary expansion of any square is always alone. %C A153013 (End) %H A153013 Yang Haoran, <a href="/A153013/b153013.txt">Table of n, a(n) for n = 0..23</a> %F A153013 From _Antti Karttunen_, Oct 15 2016: (Start) %F A153013 a(0) = 0; for n >= 1, a(n) = A005940(1+a(n-1)). %F A153013 A008966(a(n+1)) = A085357(a(n)). [See the comment.] %F A153013 A181819(a(1+n)) = A246029(a(n)). %F A153013 A000005(a(n+1)) = A106737(a(n)). %F A153013 (End) %e A153013 101 is interpreted as 3^1 * 2^1 = 6. 1110011 is interpreted as 5^3 * 2^2 = 500. %t A153013 NestList[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #] &@ Flatten@ MapIndexed[If[Total@ #1 == 0, ConstantArray[0, Boole[First@ #2 == 1] + Length@ #1 - 1], Length@ #1] &, Reverse@ Split@ IntegerDigits[#, 2]] &, 0, 21] (* _Michael De Vlieger_, Oct 17 2016 *) %o A153013 (PARI) step(n)=my(t=1,v); forprime(p=2,, v=valuation(n+1,2); t*=p^v; n>>=v+1; if(!n, return(t))) %o A153013 t=0; concat(0,vector(20,n, t=step(t))) \\ _Charles R Greathouse IV_, Sep 01 2015 %o A153013 (Scheme) %o A153013 ;; With memoization-macro definec. %o A153013 (definec (A153013 n) (if (zero? n) n (A005940 (+ 1 (A153013 (- n 1)))))) %o A153013 ;; _Antti Karttunen_, Oct 15 2016 %Y A153013 Cf. A000005, A003714, A005940, A008966, A085357, A106737, A156552, A181819, A246029. %Y A153013 Cf. also A109162, A328316 for similar iteration sequences. %K A153013 nonn %O A153013 0,3 %A A153013 Mark Zegarelli (mtzmtz(AT)gmail.com), Dec 16 2008 %E A153013 a(20)-a(22) from _Yang Haoran_, Aug 31 2015