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 A192293 Let sigma*_m (n) be the result of applying the sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; this sequence gives the (2,3)-anti-perfect numbers. 6
 32, 98, 2524, 199282, 1336968 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Like A019281 but using anti-divisors. a(6) > 2*10^7. - Chai Wah Wu, Dec 02 2014 LINKS EXAMPLE sigma*(32)= 3+5+7+9+13+21=58; sigma*(58)= 3+4+5+9+13+23+39=96 and 3*32=96. sigma*(98)= 3+4+5+13+15+28+39+65=172; sigma*(172)= 3+5+7+8+15+23+49+69+115=294 and 3*98=294. sigma*(2524)= 3+7+8+9+11+17+27+33+49+51+99+103+153+187+297+459+561+721+1683=4478; sigma*(4478)= 3+4+5+9+13+15+45+53+169+199+597+689+995+1791+2985=7572 and 3*2524=7572. MAPLE with(numtheory): P:= proc(n) local i, j, k, s, s1; for i from 3 to n do k:=0; j:=i; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; s:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2; k:=0; j:=s; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; s1:=sigma(2*s+1)+sigma(2*s-1)+sigma(s/2^k)*2^(k+1)-6*s-2; if s1/i=3 then print(i); fi; od; end: P(10^9); PROG (Python) from sympy import divisors def antidivisors(n): ....return [2*d for d in divisors(n) if n > 2*d and n % (2*d)] + \ ...........[d for d in divisors(2*n-1) if n > d >=2 and n % d] + \ ...........[d for d in divisors(2*n+1) if n > d >=2 and n % d] A192293_list = [] for n in range(1, 10**4): ....if 3*n == sum(antidivisors(sum(antidivisors(n)))): ........A192293_list.append(n) # Chai Wah Wu, Dec 02 2014 CROSSREFS Cf. A019281, A066272, A192290, A192291, A192292. Sequence in context: A197904 A273554 A218901 * A188862 A228686 A172517 Adjacent sequences:  A192290 A192291 A192292 * A192294 A192295 A192296 KEYWORD nonn,more,less AUTHOR Paolo P. Lava, Jun 29 2011 EXTENSIONS a(4)-a(5) from Chai Wah Wu, Dec 01 2014 STATUS approved

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The text passage is a detailed explanation of a specific integer sequence from the Online Encyclopedia of Integer Sequences (OEIS). It defines the sequence, provides examples, and includes code in Maple and Python to generate the sequence.
no_question

the passage does not contain any explicit mathematical questions or problems to solve. it is purely descriptive and provides information about the sequence, its properties, and how to generate 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
This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A192293 Let sigma*_m (n) be the result of applying the sum of anti-divisors m times to n; call n (m,k)-anti-perfect if sigma*_m (n) = k*n; this sequence gives the (2,3)-anti-perfect numbers. 6
 32, 98, 2524, 199282, 1336968 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Like A019281 but using anti-divisors. a(6) > 2*10^7. - Chai Wah Wu, Dec 02 2014 LINKS EXAMPLE sigma*(32)= 3+5+7+9+13+21=58; sigma*(58)= 3+4+5+9+13+23+39=96 and 3*32=96. sigma*(98)= 3+4+5+13+15+28+39+65=172; sigma*(172)= 3+5+7+8+15+23+49+69+115=294 and 3*98=294. sigma*(2524)= 3+7+8+9+11+17+27+33+49+51+99+103+153+187+297+459+561+721+1683=4478; sigma*(4478)= 3+4+5+9+13+15+45+53+169+199+597+689+995+1791+2985=7572 and 3*2524=7572. MAPLE with(numtheory): P:= proc(n) local i, j, k, s, s1; for i from 3 to n do k:=0; j:=i; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; s:=sigma(2*i+1)+sigma(2*i-1)+sigma(i/2^k)*2^(k+1)-6*i-2; k:=0; j:=s; while j mod 2 <> 1 do k:=k+1; j:=j/2; od; s1:=sigma(2*s+1)+sigma(2*s-1)+sigma(s/2^k)*2^(k+1)-6*s-2; if s1/i=3 then print(i); fi; od; end: P(10^9); PROG (Python) from sympy import divisors def antidivisors(n): ....return [2*d for d in divisors(n) if n > 2*d and n % (2*d)] + \ ...........[d for d in divisors(2*n-1) if n > d >=2 and n % d] + \ ...........[d for d in divisors(2*n+1) if n > d >=2 and n % d] A192293_list = [] for n in range(1, 10**4): ....if 3*n == sum(antidivisors(sum(antidivisors(n)))): ........A192293_list.append(n) # Chai Wah Wu, Dec 02 2014 CROSSREFS Cf. A019281, A066272, A192290, A192291, A192292. Sequence in context: A197904 A273554 A218901 * A188862 A228686 A172517 Adjacent sequences:  A192290 A192291 A192292 * A192294 A192295 A192296 KEYWORD nonn,more,less AUTHOR Paolo P. Lava, Jun 29 2011 EXTENSIONS a(4)-a(5) from Chai Wah Wu, Dec 01 2014 STATUS approved

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user
The text passage is a detailed explanation of a specific integer sequence from the Online Encyclopedia of Integer Sequences (OEIS). It defines the sequence, provides examples, and includes code in Maple and Python to generate the sequence.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems to solve. It is purely descriptive and provides information about the sequence, its properties, and how to generate it.