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 A192274 Numbers which are both Zumkeller numbers and anti-Zumkeller numbers. 3
 42, 70, 78, 88, 126, 160, 176, 228, 234, 258, 270, 280, 308, 342, 350, 368, 378, 380, 390, 396, 402, 438, 448, 462, 468, 490, 500, 522, 532, 540, 552, 558, 560, 572, 580, 588, 608, 618, 620, 630, 644, 650, 690, 702, 732, 756, 770, 780, 798, 812, 822, 852, 858 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Numbers n whose sets of divisors and anti-divisors can each be partitioned into two disjoint sets whose sums are sigma(n)/2 for the sets in the divisors partition and sigma*(n)/2 for the anti-divisors partition, where sigma*(n) is the sum of the anti-divisors of n. LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10000 EXAMPLE 270-> divisors: 1,2,3,5,6,9,10,15,18,27,30,45,54,90,135,270; sigma(270)/2=360; 1+2+3+5+6+9+10+15+18+27+30+45+54+135=90+270=360. 270-> anti-divisors: 4,7,11,12,20,36,49,60,77,108,180; sigma*(270)/2=282; 4+7+11+20+60+180=12+36+49+77+108=282. MAPLE with(combstruct); with(numtheory); P:=proc(i) local S, R, Stop, Comb, a, b, c, d, k, m, n, s; for n from 3 to i do   a:={};   for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a:=a union {k}; fi; od;   b:=nops(a); c:=op(a); s:=0;    if b>1 then for k from 1 to b do s:=s+c[k]; od;    else s:=c;   fi;   if (modp(s, 2)=0 and 2*n<=s) then      S:=1/2*s-n; R:=select(m->m<=S, [c]); Stop:=false; Comb:=iterstructs(Combination(R));      while not (finished(Comb) or Stop) do Stop:=add(d, d=nextstruct(Comb))=S; od;      if Stop then         s:=sigma(n);         if (modp(s, 2)=0 and 2*n<=s) then           S:=1/2*s-n; R:=select(m->m<=S, divisors(n)); Stop:=false;       Comb:=iterstructs(Combination(R));           while not (finished(Comb) or Stop) do Stop:=add(d, d=nextstruct(Comb))=S; od;           if Stop then print(n); fi;         fi;      fi;   fi; od; end: P(10000); PROG (Python3) from sympy import divisors from sympy.combinatorics.subsets import Subset 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] for n in range(1, 10**3): ....d = divisors(n) ....s = sum(d) ....if not s % 2 and max(d) <= s/2: ........for x in range(1, 2**len(d)): ............if sum(Subset.unrank_binary(x, d).subset) == s/2: ................d = antidivisors(n) ................s = sum(d) ................if not s % 2 and max(d) <= s/2: ....................for x in range(1, 2**len(d)): ........................if sum(Subset.unrank_binary(x, d).subset) == s/2: ............................print(n, end=', ') ............................break ................break # Chai Wah Wu, Aug 14 2014 (Python) from sympy import divisors import numpy as np A192274 = [] for n in range(3, 10**3): ....d = divisors(n) ....s = sum(d) ....if not s % 2 and 2*n <= s: ........d.remove(n) ........s2, ld = int(s/2-n), len(d) ........z = np.zeros((ld+1, s2+1), dtype=int) ........for i in range(1, ld+1): ............y = min(d[i-1], s2+1) ............z[i, range(y)] = z[i-1, range(y)] ............z[i, range(y, s2+1)] = np.maximum(z[i-1, range(y, s2+1)], z[i-1, range(0, s2+1-y)]+y) ............if z[i, s2] == s2: ................d2 = [2*x for x in d if n > 2*x and n % (2*x)] + \ ................[x for x in divisors(2*n-1) if n > x >=2 and n % x] + \ ................[x for x in divisors(2*n+1) if n > x >=2 and n % x] ................s, dmax = sum(d2), max(d2) ................if not s % 2 and 2*dmax <= s: ....................d2.remove(dmax) ....................s2, ld = int(s/2-dmax), len(d2) ....................z = np.zeros((ld+1, s2+1), dtype=int) ....................for i in range(1, ld+1): ........................y = min(d2[i-1], s2+1) ........................z[i, range(y)] = z[i-1, range(y)] ........................z[i, range(y, s2+1)] = np.maximum(z[i-1, range(y, s2+1)], z[i-1, range(0, s2+1-y)]+y) ........................if z[i, s2] == s2: ............................A192274.append(n) ........................break ................break # Chai Wah Wu, Aug 19 2014 CROSSREFS Cf. A083207, A066272, A192273. Sequence in context: A118074 A115957 A255989 * A291319 A226168 A248430 Adjacent sequences:  A192271 A192272 A192273 * A192275 A192276 A192277 KEYWORD nonn AUTHOR Paolo P. Lava, Jun 28 2011 EXTENSIONS Corrected entries and comment by Chai Wah Wu, Aug 13 2014 STATUS approved

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Last modified May 26 00:08 EDT 2020. Contains 334613 sequences. (Running on oeis4.)