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 A325563 a(1) = 1; for n > 1, a(n) is the largest proper divisor d of n such that A048720(d,k) = n for some k. 7
 1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 7, 11, 1, 12, 1, 13, 9, 14, 1, 15, 1, 16, 3, 17, 7, 18, 1, 19, 3, 20, 1, 21, 1, 22, 15, 23, 1, 24, 7, 25, 17, 26, 1, 27, 1, 28, 3, 29, 1, 30, 1, 31, 21, 32, 5, 33, 1, 34, 1, 35, 1, 36, 1, 37, 15, 38, 1, 39, 1, 40, 1, 41, 1, 42, 17, 43, 1, 44, 1, 45, 1, 46, 31, 47, 19, 48, 1, 49, 33, 50 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS For n > 1, a(n) is the largest proper divisor d of n for which it holds that when the binary expansion of d is converted to a (0,1)-polynomial (e.g., 13=1101 encodes X^3 + X^2 + 1), then that polynomial is a divisor of (0,1)-polynomial similarly converted from n, when the polynomial division is done over field GF(2). See the example. LINKS Antti Karttunen, Table of n, a(n) for n = 1..16384 Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537 Index entries for sequences related to polynomials in ring GF(2)[X] FORMULA For all n, a(n) <= A032742(n). EXAMPLE For n = 39 = 3*13, A032742(39) = 13, but 13 is not the answer because X^3 + X^2 + 1 does not divide X^5 + X^2 + X + 1 (39 is "100111" in binary) over GF(2). However, the next smaller divisor 3 works because X^5 + X^2 + X + 1 = (X^1 + 1)(X^4 + X^3 + X^2 + 1) when multiplication is done over GF(2). Note that 39 = A048720(3,29), where 29 is "11101" in binary. Thus a(39) = 3. PROG (PARI) A325563(n) = if(1==n, n, my(p = Pol(binary(n))*Mod(1, 2)); fordiv(n, d, if((d>1), my(q = Pol(binary(n/d))*Mod(1, 2)); if(0==(p%q), return(n/d))))); (PARI) A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2); A325563(n) = if(1==n, n, fordiv(n, d, if((d>1), for(t=1, n, if(A048720(n/d, t)==n, return(n/d)))))); \\ (Slow) CROSSREFS Cf. A032742, A048720, A325560, A325564, A325567, A325643. Cf. A325559 (positions of ones, after the initial 1). Sequence in context: A353271 A326139 A325641 * A159353 A032742 A060654 Adjacent sequences: A325560 A325561 A325562 * A325564 A325565 A325566 KEYWORD nonn AUTHOR Antti Karttunen, May 11 2019 STATUS approved

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Last modified September 29 23:48 EDT 2023. Contains 365781 sequences. (Running on oeis4.)