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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.)
The text passage is a detailed description and definition of a specific integer sequence (A295827) from the On-Line Encyclopedia of Integer Sequences (OEIS). It includes the sequence definition, initial terms, comments, formulas, examples, and code implementations in different programming languages.
no_question

the passage does not contain any explicit mathematical questions or problems to solve. it is primarily an informative entry about the sequence a295827, providing various details and properties related to it.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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.)
user
The text passage is a detailed description and definition of a specific integer sequence (A295827) from the On-Line Encyclopedia of Integer Sequences (OEIS). It includes the sequence definition, initial terms, comments, formulas, examples, and code implementations in different programming languages.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems to solve. It is primarily an informative entry about the sequence A295827, providing various details and properties related to it.