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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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explanation: This document is an excerpt from the On-Line Encyclopedia of Integer Sequences (OEIS) and describes a specific integer sequence, A192293. The sequence lists numbers that are (2,3)-anti-perfect, meaning that applying the sum of anti-divisors twice to these numbers results in three times the original number. The document provides examples, a Maple program, and a Python program to generate the sequence. It also includes cross-references to related sequences and mentions the status of the sequence.
  
  The document does not contain a specific math question or a question and answer. It is primarily a reference entry for the 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: 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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distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: This document is an excerpt from the On-Line Encyclopedia of Integer Sequences (OEIS) and describes a specific integer sequence, A192293. The sequence lists numbers that are (2,3)-anti-perfect, meaning that applying the sum of anti-divisors twice to these numbers results in three times the original number. The document provides examples, a Maple program, and a Python program to generate the sequence. It also includes cross-references to related sequences and mentions the status of the sequence.
  
  The document does not contain a specific math question or a question and answer. It is primarily a reference entry for the sequence.
  
  Label: no-question