A066505 f-amicable numbers where f(n) = n+1.
36, 62, 168, 326, 9936, 14056, 16198, 19862, 45304, 51910, 82662, 90152, 337688, 388102, 472902, 479672, 1970586, 2353756, 2969288, 3769942, 6319544, 8454886, 12276056, 13125574, 16783976, 17948854, 18818780, 20825882, 21738114, 22479040, 25960468, 31470614
Offset: 1
Keywords
A238765
Numbers k such that if x = Sum_{j|k, j
198, 608, 11322, 15450, 17874, 20826, 33894, 41022, 56608, 1259910, 1764414, 3055150, 565344850, 579667086, 907521650
Offset: 1
Comments
A066218 is a subsequence. It lists the fixed points of the transform n -> Sum_{j|n, j
Examples
Aliquot divisors of 15450 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 103, 150, 206, 309, 515, 618, 1030, 1545, 2575, 3090, 5150, 7725. Their respective sigma(k)-k are 0, 1, 1, 1, 6, 8, 9, 6, 42, 43, 49, 1, 222, 106, 107, 109, 630, 842, 951, 649, 4398, 4522, 5171 and their sum is equal to 17874. Aliquot divisors of 17874 are 1, 2, 3, 6, 9, 18, 27, 54, 331, 662, 993, 1986, 2979, 5958, 8937. Their respective sigma(k)-k are 0, 1, 1, 6, 4, 21, 13, 66, 1, 334, 335, 1998, 1337, 6990, 4343 and their sum is equal to 15450.
Programs
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Maple
with(numtheory); P:=proc(q) local a,b,c,i,n; for n from 1 to q do a:=sort([op(divisors(n))]); b:=0; for i from 1 to nops(a)-1 do b:=b+sigma(a[i])-a[i]; od; a:=sort([op(divisors(b))]); c:=0; for i from 1 to nops(a)-1 do c:=c+sigma(a[i])-a[i]; od; if n=c then print(n); fi; od; end: P(10^6);
Extensions
a(13)-a(15) from Michel Marcus, Mar 07 2014
A259536 Numbers n such that sigma(n) - n = Sum_{k|n, k < n} sigma*(k), where sigma(n) is the sum of the divisors of n and sigma*(k) is the sum of the anti-divisors of k.
14, 20, 42, 45, 52, 55, 121, 377, 451, 4331, 9211, 18042, 52093, 58743, 112853, 143498, 170322, 182293, 226118, 289811, 298489, 394067, 400831, 456745, 507678, 541362, 769667, 929582, 1132069, 1150595, 1218361, 1230403, 1262507, 1431109, 1516082, 1875155
Offset: 1
Keywords
Examples
Aliquot parts of 14 are 1, 2, 7. Anti-divisors of 7 are [2, 3, 5] while 1 and 2 have no anti-divisors. Finally 1 + 2 + 7 = 2 + 3 + 5 = 10. Aliquot parts of 20 are 1, 2, 4, 5, 10. Anti-divisors: for 4 is [3], for 5 are [2, 3], for 10 are [3, 4, 7] while 1 and 2 have no anti-divisors. Finally 1 + 2 + 4 + 5 + 10 = 3 + 2 + 3 + 3 + 4 + 7 = 22.
Programs
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Maple
with(numtheory): P:=proc(q) local a,b,i,j,k,n; for n from 3 to q do a:=sort([op(divisors(n))]); b:=0; for k from 2 to nops(a)-1 do i:=0; j:=a[k]; while j mod 2 <> 1 do i:=i+1; j:=j/2; od; b:=b+sigma(2*a[k]+1)+sigma(2*a[k]-1)+sigma(a[k]/2^i)*2^(i+1)-6*a[k]-2; od; if sigma(n)-n=b then print(n); fi; od; end: P(10^9);
Comments
Examples
Proper divisors of 36 = {1, 2, 3, 4, 6, 9, 12, 18}. f applied to these divisors = {2, 3, 4, 5, 7, 10, 13, 19}; their sum = 63. So D(36) = f(62). proper divisors of 62 = {1, 2, 31}. f applied to these divisors = {2, 3, 32}; their sum = 37. So D(62) = f(36). Therefore 36, 62 is an f-amicable pair.Links
Crossrefs
Programs
Mathematica
f[x_] := x + 1; d[x_] := Apply[ Plus, Map[ f, Divisors[ x] ] ] - f[ x]; m = Table[{x, y}, {x, 1, 1000}, {y, 1, 1000}]; Do[a = m[[i, j]]; If[ (a[[1]] < a[[2]]) && (f[a[[1]]] == d[a[[2]]]) && (f[a[[2]]] == d[a[[1]]]), Print[{i, j}]], {i, 1, 1000}, {j, 1, 1000}]Extensions