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 A087207 A binary representation of the primes that divide a number, shown in decimal. 37
 0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc. For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017 From Antti Karttunen, Jun 18 & 20 2017: (Start) A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n). For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself. Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565. If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612. (End) LINKS N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from T. D. Noe] FORMULA Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003 a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003 A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003 G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009 From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start) a(n) = A048675(A007947(n)). a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)). A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.] A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.] a(A293214(n)) = A218403(n). a(A293442(n)) = A267116(n). A069010(a(n)) = A287170(n). A007088(a(n)) = A276379(n). A038374(a(n)) = A300820(n) for n >= 1. (End) From Peter Munn, Jan 08 2020: (Start) a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)). a(A003961(n)) = 2*a(n). a(n^2) = a(n). a(n) = A267116(A225546(n)). a(A225546(n)) = A267116(n). (End) EXAMPLE a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001). a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13. MATHEMATICA a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *) PROG (Haskell) a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row -- Reinhard Zumkeller, Jul 16 2013 (PARI) a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[, 1]), v = concat(v, vecsearch(f[, 1], p)!=0); ); fromdigits(Vecrev(v), 2)); } \\ Michel Marcus, Jun 05 2017 (PARI) A087207(n)=vecsum(apply(p->1< n), and also A286611, A286612. A003986, A003961, A059896 are used to express relationship between terms of this sequence. Related to A267116 via A225546. Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17). Sequence in context: A058354 A085930 A258863 * A179206 A331740 A074987 Adjacent sequences:  A087204 A087205 A087206 * A087208 A087209 A087210 KEYWORD nonn,base,nice AUTHOR Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003 EXTENSIONS More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003 Name clarified by Antti Karttunen, Jun 18 2017 STATUS approved

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Last modified October 22 07:38 EDT 2020. Contains 337950 sequences. (Running on oeis4.)