Search: a096115 -id:a096115
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1, 2, 2, 3, 3, 6, 6, 4, 4, 8, 8, 12, 12, 24, 24, 5, 5, 10, 10, 15, 15, 30, 30, 20, 20, 40, 40, 60, 60, 120, 120, 6, 6, 12, 12, 18, 18, 36, 36, 24, 24, 48, 48, 72, 72, 144, 144, 30, 30, 60, 60, 90, 90, 180, 180, 120, 120, 240, 240, 360, 360, 720, 720, 7, 7, 14, 14, 21, 21
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OFFSET
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0,2
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COMMENTS
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Each n > 1 occurs 2*A045778(n) times in the sequence.
If the binary indices of n (row n of A048793) are the positions 1's in its reversed binary expansion, then a(n) is the product of all binary indices of n + 1. The number of binary indices of n is A000120(n), their sum is A029931(n), and their average is A326699(n)/A326700(n). - Gus Wiseman, Jul 27 2019
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LINKS
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FORMULA
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G.f.: ( prod(k>=1, 1+k*x^(2^(k-1)) )- 1 ) / x. - Vladeta Jovovic, Nov 08 2005
a(n) is the product of the exponents in the binary expansion of 2*n + 2. - Peter Kagey, Apr 24 2016
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MAPLE
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f:= proc(n) local L;
L:= convert(2*n+2, base, 2);
convert(subs(0=NULL, zip(`*`, L, [$0..nops(L)-1])), `*`);
end proc:
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MATHEMATICA
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CoefficientList[(Product[1 + k x^(2^(k - 1)), {k, 7}] - 1)/x, x] (* Michael De Vlieger, Apr 08 2016 *)
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1]; Table[Times@@bpe[n+1], {n, 0, 100}] (* Gus Wiseman, Jul 26 2019 *)
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PROG
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(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))
(PARI)
N=166; q='q+O('q^N);
gf= (prod(n=1, 1+ceil(log(N)/log(2)), 1+n*q^(2^(n-1)) ) - 1) / q;
Vec(gf)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A052330
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Let S_k denote the first 2^k terms of this sequence and let b_k be the smallest positive integer that is not in S_k; then the numbers b_k*S_k are the next 2^k terms.
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+10
63
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1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 7, 14, 21, 42, 28, 56, 84, 168, 35, 70, 105, 210, 140, 280, 420, 840, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 63, 126, 189, 378, 252, 504, 756, 1512, 315, 630, 945, 1890
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OFFSET
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0,2
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COMMENTS
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This sequence is not exactly a permutation because it has offset 0 but doesn't contain 0. A052331 is its exact inverse, which has offset 1 and contains 0. See also A064358.
Are there any other values of n besides 4 and 36 with a(n) = n? - Thomas Ordowski, Apr 01 2005
4 = 100 = 4^1 * 3^0 * 2^0, 36 = 100100 = 9^1 * 7^0 * 5^0 * 4^1 * 3^0 * 2^0. - Thomas Ordowski, May 26 2005
Ordering of positive integers by increasing "Fermi-Dirac representation", which is a representation of the "Fermi-Dirac factorization", term implying that each prime power with a power of two as exponent may appear at most once in the "Fermi-Dirac factorization" of n. (Cf. comment in A050376; see also the OEIS Wiki page.) - Daniel Forgues, Feb 11 2011
The subsequence consisting of the squarefree terms is A019565. - Peter Munn, Mar 28 2018
Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH-number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k). A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. Then a(n) is the number whose binary indices are the parts of the strict integer partition with FDH-number n. - Gus Wiseman, Aug 19 2019
The set of indices of odd-valued terms has asymptotic density 0. In this sense (using the order they appear in this permutation) 100% of numbers are even. - Peter Munn, Aug 26 2019
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LINKS
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Michael De Vlieger, Fan style binary tree showing a(n), n = 0..16383, with a color function representing primes in red, perfect powers of primes in gold, squarefree composites in greens, and numbers neither squarefree nor prime powers in blue or purple. Purple represents powerful numbers that are not prime powers. This is a 15 level version of Karttunen's diagram in the example.
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FORMULA
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The binary representation of n, n = Sum_{i=0..1+floor(log_2(n))} n_i * 2^i, n_i in {0,1}, is taken as the "Fermi-Dirac representation" (A182979) of a(n), a(n) = Product_{i=0..1+floor(log_2(n))} (b_i)^(n_i) where b_i is A050376(i), i.e., the i-th "Fermi-Dirac prime" (prime power with exponent being a power of 2). - Daniel Forgues, Feb 12 2011
(End)
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EXAMPLE
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Terms following 5 are 10, 15, 30, 20, 40, 60, 120; this is followed by 7 as 6 has already occurred. - Philippe Deléham, Jun 03 2015
This sequence can be regarded also as an irregular triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., that is, it can be represented as a binary tree, where each left hand child contains A300841(k), and each right hand child contains 2*A300841(k), when their parent contains k:
1
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...................2...................
3 6
4......../ \........8 12......../ \........24
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
5 10 15 30 20 40 60 120
7 14 21 42 28 56 84 168 35 70 105 210 140 280 420 840
etc.
(End)
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MATHEMATICA
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a = {1}; Do[a = Join[a, a*Min[Complement[Range[Max[a] + 1], a]]], {n, 1, 6}]; a (* Ivan Neretin, May 09 2015 *)
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PROG
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(PARI)
up_to_e = 13; \\ Good for computing up to n = (2^13)-1
v050376 = vector(up_to_e);
ispow2(n) = (n && !bitand(n, n-1));
i = 0; for(n=1, oo, if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e, break));
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CROSSREFS
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Cf. A050030, A052331 (inverse), A096111, A096113, A096114, A096115, A096116, A096118, A096119, A182979, A207901, A300841, A302023, A302783.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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1, 2, 3, 5, 4, 9, 7, 6, 5, 11, 12, 14, 9, 10, 8, 7, 6, 13, 14, 16, 15, 20, 18, 17, 11, 12, 13, 15, 10, 11, 9, 8, 7, 15, 16, 18, 17, 22, 20, 19, 18, 24, 25, 27, 22, 23, 21, 20, 13, 14, 15, 17, 16, 21, 19, 18, 12, 13, 14, 16, 11, 12, 10, 9, 8, 17, 18, 20, 19, 24, 22, 21, 20, 26
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OFFSET
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1,2
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COMMENTS
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Each n > 1 occurs A025147(n) times in the sequence.
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LINKS
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MATHEMATICA
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a = {1}; Do[AppendTo[a, If[BitAnd[n - 1, n - 2] == 0, Log2[n - 1] + 2, 2 + Floor[Log2[n - 1]] + a[[2 + BitXor[n - 1, 2^Ceiling[Log2[n]] - 1]]]]], {n, 2, 74}]; a (* Ivan Neretin, Jun 24 2016 *)
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PROG
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(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A122198
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Permutation of natural numbers: a recursed variant of A122155.
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+10
4
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0, 1, 2, 3, 4, 7, 6, 5, 8, 15, 14, 13, 12, 9, 10, 11, 16, 31, 30, 29, 28, 25, 26, 27, 24, 17, 18, 19, 20, 23, 22, 21, 32, 63, 62, 61, 60, 57, 58, 59, 56, 49, 50, 51, 52, 55, 54, 53, 48, 33, 34, 35, 36, 39, 38, 37, 40, 47, 46, 45, 44, 41, 42, 43, 64, 127, 126, 125, 124, 121
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A122199
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Permutation of natural numbers: a recursed variant of A122155.
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+10
4
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0, 1, 2, 3, 4, 7, 6, 5, 8, 13, 14, 15, 12, 11, 10, 9, 16, 25, 26, 27, 28, 31, 30, 29, 24, 21, 22, 23, 20, 19, 18, 17, 32, 49, 50, 51, 52, 55, 54, 53, 56, 61, 62, 63, 60, 59, 58, 57, 48, 41, 42, 43, 44, 47, 46, 45, 40, 37, 38, 39, 36, 35, 34, 33, 64, 97, 98, 99, 100, 103, 102
(list;
graph;
refs;
listen;
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text;
internal format)
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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