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   A295827 a(n) = least odd k > 1 such that n and n*k have the same Hamming weight, or -1 if no such k exists. 2
   -1, -1, 3, -1, 13, 3, 3, -1, 57, 13, 35, 3, 21, 3, 3, -1, 241, 57, 7, 13, 13, 35, 39, 3, 169, 21, 5, 3, 21, 3, 3, -1, 993, 241, 11, 57, 7, 7, 5, 13, 3197, 13, 9, 35, 3, 39, 13, 3, 21, 169, 3, 21, 39, 5, 47, 3, 27, 21, 5, 3, 13, 3, 3, -1, 4033, 993, 491, 241 (list; graph; refs; listen; history; text; internal format)
   OFFSET 1,3 COMMENTS The Hamming weight of a number n is given by A000120(n). Apparently, a(n) = -1 iff n = 2^k for some k >= 0. Apparently, a(2^n + 1) = A020515(n) for any n > 1. a(2^n - 1) = 3 for any n > 1. a(n) = 3 iff n = A077459(k) for some k > 1. This sequence has similarities with A292849: here we want A000120(n*a(n)) = A000120(n), there we want A000120(n*a(n)) = A000120(a(n)). For any n > 0, if a(n) > 0 then A292849(a(n)) <= n. LINKS Rémy Sigrist, Table of n, a(n) for n = 1..8192 Rémy Sigrist, Logarithmic scatterplot of the sequence for n=1..2^17 and a(n) < 10^18 FORMULA a(2*n) = a(n) for any n > 0. EXAMPLE The first terms, alongside the binary representations of n and of n*a(n), are: n a(n) bin(n) bin(n*a(n)) -- ---- ------ ----------- 1 -1 1 -1 2 -1 10 -10 3 3 11 1001 4 -1 100 -100 5 13 101 1000001 6 3 110 10010 7 3 111 10101 8 -1 1000 -1000 9 57 1001 1000000001 10 13 1010 10000010 11 35 1011 110000001 12 3 1100 100100 13 21 1101 100010001 14 3 1110 101010 15 3 1111 101101 16 -1 10000 -10000 17 241 10001 1000000000001 18 57 10010 10000000010 19 7 10011 10000101 20 13 10100 100000100 MAPLE f:= proc(n) local k, w; if n = 2^padic:-ordp(n, 2) then return -1 fi; w:= convert(convert(n, base, 2), `+`); for k from 3 by 2 do if convert(convert(n*k, base, 2), `+`)=w then return k fi od end proc: map(f, [\$1..100]); # Robert Israel, Nov 28 2017 MATHEMATICA Table[SelectFirst[Range[3, 10^4 + 1, 2], SameQ @@ Map[DigitCount[#, 2, 1] &, {n, n #}] &] /. m_ /; MissingQ@ m -> -1, {n, 68}] (* Michael De Vlieger, Nov 28 2017 *) PROG (PARI) A057168(n)=n+bitxor(n, n+n=bitand(n, -n))\n\4+n \\ after M. F. Hasler at A057168 a(n) = n\=2^valuation(n, 2); if (n==1, -1, my(w=(n-1)/2); while(1, w=A057168(w); if((2*w+1)%n==0, return((2*w+1)/n)))) CROSSREFS Cf. A000120, A020515, A057168, A077459, A292849. Sequence in context: A248843 A170910 A134768 * A277197 A297898 A322384 Adjacent sequences: A295824 A295825 A295826 * A295828 A295829 A295830 KEYWORD sign,base AUTHOR Rémy Sigrist, Nov 28 2017 STATUS approved
  
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  Last modified April 19 19:02 EDT 2024. Contains 371798 sequences. (Running on oeis4.)
explanation: The provided text is an excerpt from the On-Line Encyclopedia of Integer Sequences (OEIS) describing the sequence A295827. This sequence is defined as the least odd \( k > 1 \) such that \( n \) and \( n \cdot k \) have the same Hamming weight, or -1 if no such \( k \) exists. The text includes the sequence definition, initial terms, comments, formulas, examples, and code implementations in Maple and Mathematica.
  
  The document does not contain a specific question or a question and answer. It is primarily a detailed description of a mathematical sequence.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: The OEIS is supported by the many generous donors to the OEIS Foundation.
  
   Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
   A295827 a(n) = least odd k > 1 such that n and n*k have the same Hamming weight, or -1 if no such k exists. 2
   -1, -1, 3, -1, 13, 3, 3, -1, 57, 13, 35, 3, 21, 3, 3, -1, 241, 57, 7, 13, 13, 35, 39, 3, 169, 21, 5, 3, 21, 3, 3, -1, 993, 241, 11, 57, 7, 7, 5, 13, 3197, 13, 9, 35, 3, 39, 13, 3, 21, 169, 3, 21, 39, 5, 47, 3, 27, 21, 5, 3, 13, 3, 3, -1, 4033, 993, 491, 241 (list; graph; refs; listen; history; text; internal format)
   OFFSET 1,3 COMMENTS The Hamming weight of a number n is given by A000120(n). Apparently, a(n) = -1 iff n = 2^k for some k >= 0. Apparently, a(2^n + 1) = A020515(n) for any n > 1. a(2^n - 1) = 3 for any n > 1. a(n) = 3 iff n = A077459(k) for some k > 1. This sequence has similarities with A292849: here we want A000120(n*a(n)) = A000120(n), there we want A000120(n*a(n)) = A000120(a(n)). For any n > 0, if a(n) > 0 then A292849(a(n)) <= n. LINKS Rémy Sigrist, Table of n, a(n) for n = 1..8192 Rémy Sigrist, Logarithmic scatterplot of the sequence for n=1..2^17 and a(n) < 10^18 FORMULA a(2*n) = a(n) for any n > 0. EXAMPLE The first terms, alongside the binary representations of n and of n*a(n), are: n a(n) bin(n) bin(n*a(n)) -- ---- ------ ----------- 1 -1 1 -1 2 -1 10 -10 3 3 11 1001 4 -1 100 -100 5 13 101 1000001 6 3 110 10010 7 3 111 10101 8 -1 1000 -1000 9 57 1001 1000000001 10 13 1010 10000010 11 35 1011 110000001 12 3 1100 100100 13 21 1101 100010001 14 3 1110 101010 15 3 1111 101101 16 -1 10000 -10000 17 241 10001 1000000000001 18 57 10010 10000000010 19 7 10011 10000101 20 13 10100 100000100 MAPLE f:= proc(n) local k, w; if n = 2^padic:-ordp(n, 2) then return -1 fi; w:= convert(convert(n, base, 2), `+`); for k from 3 by 2 do if convert(convert(n*k, base, 2), `+`)=w then return k fi od end proc: map(f, [\$1..100]); # Robert Israel, Nov 28 2017 MATHEMATICA Table[SelectFirst[Range[3, 10^4 + 1, 2], SameQ @@ Map[DigitCount[#, 2, 1] &, {n, n #}] &] /. m_ /; MissingQ@ m -> -1, {n, 68}] (* Michael De Vlieger, Nov 28 2017 *) PROG (PARI) A057168(n)=n+bitxor(n, n+n=bitand(n, -n))\n\4+n \\ after M. F. Hasler at A057168 a(n) = n\=2^valuation(n, 2); if (n==1, -1, my(w=(n-1)/2); while(1, w=A057168(w); if((2*w+1)%n==0, return((2*w+1)/n)))) CROSSREFS Cf. A000120, A020515, A057168, A077459, A292849. Sequence in context: A248843 A170910 A134768 * A277197 A297898 A322384 Adjacent sequences: A295824 A295825 A295826 * A295828 A295829 A295830 KEYWORD sign,base AUTHOR Rémy Sigrist, Nov 28 2017 STATUS approved
  
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  Last modified April 19 19:02 EDT 2024. Contains 371798 sequences. (Running on oeis4.)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an excerpt from the On-Line Encyclopedia of Integer Sequences (OEIS) describing the sequence A295827. This sequence is defined as the least odd \( k > 1 \) such that \( n \) and \( n \cdot k \) have the same Hamming weight, or -1 if no such \( k \) exists. The text includes the sequence definition, initial terms, comments, formulas, examples, and code implementations in Maple and Mathematica.
  
  The document does not contain a specific question or a question and answer. It is primarily a detailed description of a mathematical sequence.
  
  Label: no-question