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 A347870 a(n) = A003415(sigma(n)) mod 2, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n. 19
 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1 COMMENTS If a(k) = 0 for all terms k of A342923, then there cannot be any odd perfect numbers, as k + 3*A003415(k) is odd for any k of the form 4u+2. See comments in A005820 and A235991, also in A347887. LINKS Antti Karttunen, Table of n, a(n) for n = 1..65537 Index entries for characteristic functions Index entries for sequences related to sigma(n) FORMULA a(n) = A000035(A342925(n)) = A165560(A000203(n)). a(n) = A000035(n) XOR A347871(n). MATHEMATICA ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); a[n_] := Mod[ad[DivisorSigma[1, n]], 2]; Array[a, 105] (* Amiram Eldar, Sep 18 2021 *) PROG (PARI) A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1])); A342925(n) = A003415(sigma(n)); A347870(n) = (A342925(n)%2); CROSSREFS Cf. A000035, A000203, A003415, A005820, A165560, A235991, A342923, A342925, A347871, A347887, A349909 (partial sums). Characteristic function of A347877, while its complement A347878 gives the positions of zeros. Sequence in context: A288741 A341684 A327183 * A188967 A090171 A316832 Adjacent sequences: A347867 A347868 A347869 * A347871 A347872 A347873 KEYWORD nonn AUTHOR Antti Karttunen, Sep 17 2021 STATUS approved

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Last modified July 23 23:47 EDT 2024. Contains 374575 sequences. (Running on oeis4.)