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A329893
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Product_{k=0..floor(log_2(n))} (1 + A004718(floor(n/(2^k)))), where A004718 is Per Nørgård's "infinity sequence".
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2
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1, 2, 0, 6, 0, 0, -6, 24, 0, 0, 0, 0, -18, 0, -48, 120, 0, 0, 0, 0, 0, 0, 0, 0, 18, -72, 0, 0, -192, 48, -360, 720, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 144, -360, 0, 0, 0, 0, 384, -960, 144, 0, -1800, 720, -2880, 5040, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -54, 216
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OFFSET
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0,2
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COMMENTS
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The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.
From Mikhail Kurkov, May 22 2019 (comments originally given in A325803): (Start)
Number of positive terms on the interval 2^m <= n < 2^(m+1) for m > 0 equals f(m-1,2,1) (and f(m-1,4,3) for negative) with f(m,g,h) = binomial(m, floor(m/2) + floor((m+g)/4) - floor((m+h)/4)), so total number of nonzero terms equals binomial(m, floor(m/2)) = A001405(m).
Sum_{n=0..2^m-1} a(n) = 3^m, m >= 0.
More generally, if we define a(n,k) = (-1)^(n+1)*a(floor(n/k),k) + n mod k, a(0,k) = 0, so Sum_{n=0..k^m-1} Product_{i=0..floor(log_k(n))} (1 + a(floor(n/(k^i)),k)) = binomial(k+1, 2)^m for any k = 2p, p > 0.
(End)
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Will Sawin, Voyage into the golden screen (sequence defined by recurrence relation), Answer to question 333031 on Mathoverflow, June 1, 2019.
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FORMULA
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a(0) = 1; for n > 1, a(n) = (1+A004718(n)) * a(floor(n/2)).
a(n) = Product_{k=0..floor(log_2(n))} (1 + A004718(floor(n/(2^k)))).
a(A325804(n)) = A325803(n).
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PROG
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(PARI)
up_to = 65537;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ After code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n, n, v004718[n]);
A329893(n) = { my(m=1); while(n, m *= 1+A004718(n); n >>= 1); (m); };
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CROSSREFS
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Cf. A001405, A004718, A325803 (nonzero terms), A325804 (their positions).
Cf. also A284005, A293233.
Sequence in context: A153059 A151336 A180491 * A047918 A321981 A322481
Adjacent sequences: A329890 A329891 A329892 * A329894 A329895 A329896
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KEYWORD
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sign
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AUTHOR
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Antti Karttunen (after Mikhail Kurkov's A325803), Dec 10 2019
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STATUS
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approved
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