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 A289271 A bijective binary representation of the prime factorization of a number, shown in decimal (see Comments for precise definition). 9
 0, 1, 2, 4, 8, 3, 16, 32, 64, 5, 128, 6, 256, 9, 10, 512, 1024, 17, 2048, 12, 18, 33, 4096, 34, 8192, 65, 16384, 20, 32768, 7, 65536, 131072, 66, 129, 24, 36, 262144, 257, 130, 40, 524288, 11, 1048576, 68, 72, 513, 2097152, 258, 4194304, 1025, 514, 132 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For n > 0, with prime factorization Product_{i=1..k} p_i ^ e_i (all p_i distinct and all e_i > 0): - let S_n = A000961 \ { p_i ^ (e_i + j) with i=1..k and j > 0 }, - a(n) = Sum_{i=1..k} 2^#{ s in S_n with 1 < s < p_i ^ e_i }. In an informal way, we encode the prime powers > 1 that are unitary divisors of n as 1's in binary, while discarding the 0's corresponding to their "proper" multiples. a(A002110(n)) = 2^n-1 for any n >= 0. a(A000961(n+1)) = 2^(n-1) for any n > 0. A000120(a(n)) = A001221(n) for any n > 0 (each prime divisor p of n (alongside the p-adic valuation of n) is encoded as a single 1 bit in the base-2 representation of a(n)). A000961(2+A007814(a(n))) = A034684(n) for any n > 1 (the least significant bit of a(n) encodes the smallest unitary divisor of n that is larger than 1). This sequence establishes a bijection between the positive numbers and the nonnegative numbers; see A289272 for the inverse of this sequence. The numbers 4, 36, 40 and 532 equal their image; are there other such numbers? This sequence has connections with A034729 (which encodes the divisors of a number, and is not surjective) and A087207 (which encodes the prime divisors of a number, and is not injective). LINKS Rémy Sigrist, Table of n, a(n) for n = 1..10000 Rémy Sigrist, Illustration of the first terms Rémy Sigrist, PARI program for A289271 EXAMPLE For n = 204 = 2^2 * 3 * 17: - S_204 = A000961 \ { 2^3, 2^4, ..., 3^2, ... }         = { 1, 2, 3, 4, 5, 7, 11, 13, 17, ... }, - a(204) = 2^#{ 2, 3 } + 2^#{ 2 } + 2^#{ 2, 3, 4, 5, 7, 11, 13 }          = 2^2 + 2^1 + 2^7          = 134. See also the illustration of the first terms in Links section. PROG (PARI) See Links section. (PARI) A289271(n) = { my(f = factor(n), pps = vecsort(vector(#f~, i, f[i, 1]^f[i, 2])), s=0, x=1, pp=1, k=-1); for(i=1, #f~, while(pp < pps[i], pp++; while(!isprimepower(pp)||(gcd(pp, x)>1), pp++); k++); s += 2^k; x *= pp); (s); }; \\ Antti Karttunen, Jan 01 2019 CROSSREFS Cf. A000120, A000961, A001221, A002110, A007814, A034684, A034729, A087207, A289272 (inverse), A322988, A322990, A322991, A322992, A322995. Cf. also A156552, A052331 for similar constructions. Sequence in context: A317503 A243505 A243065 * A223699 A231610 A225124 Adjacent sequences:  A289268 A289269 A289270 * A289272 A289273 A289274 KEYWORD nonn,base AUTHOR Rémy Sigrist, Jun 30 2017 STATUS approved

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Last modified October 14 05:08 EDT 2019. Contains 327995 sequences. (Running on oeis4.)