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 A234741 a(n) is the base-2 carryless product of the prime factors of n; Encoding of the product of the polynomials over GF(2) represented by the prime factors of n (with multiplicity). 32
 1, 2, 3, 4, 5, 6, 7, 8, 5, 10, 11, 12, 13, 14, 15, 16, 17, 10, 19, 20, 9, 22, 23, 24, 17, 26, 15, 28, 29, 30, 31, 32, 29, 34, 27, 20, 37, 38, 23, 40, 41, 18, 43, 44, 17, 46, 47, 48, 21, 34, 51, 52, 53, 30, 39, 56, 53, 58, 59, 60, 61, 62, 27, 64, 57, 58, 67 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS "Encoding" means the number whose binary representation is given by the coefficients of the polynomial, e.g., 13=1101[2] encodes X^3+X^2+1. The product is the usual multiplication of polynomials in GF(2)[X] (or binary multiplication without carry-bits, cf. A048720). a(n) <= n. [As all terms of the table A061858 are nonnegative] LINKS Antti Karttunen, Table of n, a(n) for n = 1..16384 Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537 FORMULA a(0)=0, a(1)=1, and for n > 1, a(n) = A048720(A020639(n),a(n/A020639(n))). [A048720 used as a bivariate function] Equally, for n with its unique prime factorization n = p_1 * ... * p_k, with the p_i not necessarily distinct primes, a(n) = p_1 x ... x p_k, where x stands for carryless multiplication defined in A048720, which is isomorphic to multiplication in GF(2)[X]. a(2n) = 2*a(n). More generally, if A061858(x,y) = 0, then a(x*y) = a(x)*a(y). a(A235034(n)) = A235034(n). A236378(n) = n - a(n). EXAMPLE a(9) = a(3*3) = 5, as when we multiply 3 ('11' in binary) with itself, and discard the carry-bits, using XOR (A003987) instead of normal addition, we get: 11 110 ----- 101 that is, 5, as '101' is its binary representation. In other words, a(9) = a(3*3) = A048720(3,3) = 5. Alternatively, 9 = 3*3, and 3=11[2] encodes the polynomial X+1, and (X+1)*(X+1) = X^2+1 in GF(2)[X], which is encoded as 101[2] = 5, therefore a(9) = 5. - M. F. Hasler, Feb 16 2014 PROG (Scheme, with Antti Karttunen's IntSeq-library) (definec (A234741 n) (if (< n 2) n (A048720bi (A020639 n) (A234741 (/ n (A020639 n)))))) ;; A048720bi is a bivariate-function for A048720. (PARI) A234741(n)={n=factor(n); n[, 1]=apply(t->Pol(binary(t)), n[, 1]); sum(i=1, #n=Vec(factorback(n))%2, n[i]<<(#n-i))} \\ M. F. Hasler, Feb 18 2014 CROSSREFS A235034 gives the k for which a(k)=k. A236833(n) gives the number of times n occurs in this sequence. A236841 gives the same sequence sorted and duplicates removed, A236834 gives the numbers that do not occur here, A236835 gives numbers that occur more than once. A325562(n) gives the number of iterations needed before one of the fixed points (terms of A235034) is reached. Cf. also A048720, A061858, A234742, A236378, A091202/A091203, A235041/A235042, A266195, A325561, A325562. Sequence in context: A307282 A245354 A097377 * A063917 A234344 A331298 Adjacent sequences: A234738 A234739 A234740 * A234742 A234743 A234744 KEYWORD nonn AUTHOR Antti Karttunen, Jan 22 2014 EXTENSIONS Term a(0) = 0 removed and a new primary definition added by Antti Karttunen, May 10 2019 STATUS approved

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Last modified December 7 04:29 EST 2022. Contains 358649 sequences. (Running on oeis4.)