A305671 Most common value of sigma (A000203) among all composites (A073255) up to composite(n) = A002808(n) inclusive, or 0 if there is a tie.

7, 0, 0, 0, 0, 0, 0, 24, 24, 24, 24, 24, 24, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Felix Fröhlich, Jun 08 2018

Keywords

• nonn
• look

Examples

			In the following table, column A lists the n-th composite and column B lists sigma(A(n)).
   n |  A |   B | a(n)
  ---------------------
   1 |  4 |   7 |  7
   2 |  6 |  12 |  0
   3 |  8 |  15 |  0
   4 |  9 |  13 |  0
   5 | 10 |  18 |  0
   6 | 12 |  28 |  0
   7 | 14 |  24 |  0
   8 | 15 |  24 | 24 <--- first time a value of sigma occurs twice
   9 | 16 |  31 | 24
  10 | 18 |  39 | 24
  11 | 20 |  42 | 24
  12 | 21 |  32 | 24
  13 | 22 |  36 | 24
  14 | 24 |  60 | 24
  15 | 25 |  31 |  0 <--- second time a value of sigma occurs twice
  16 | 26 |  42 |  0
  17 | 27 |  40 |  0
  18 | 28 |  56 |  0
  19 | 30 |  72 |  0
  20 | 32 |  63 |  0
  21 | 33 |  48 |  0
  22 | 34 |  54 |  0
  23 | 35 |  48 |  0
  24 | 36 |  91 |  0
  25 | 38 |  60 |  0
  26 | 39 |  56 |  0
  27 | 40 |  90 |  0
  28 | 42 |  96 |  0
  29 | 44 |  84 |  0
  30 | 45 |  78 |  0
  31 | 46 |  72 |  0
  32 | 48 | 124 |  0
  33 | 49 |  57 |  0
  34 | 50 |  93 |  0
  35 | 51 |  72 | 72 <--- first time a value of sigma occurs three times
  36 | 52 |  98 | 72
  37 | 54 | 120 | 72
  38 | 55 |  72 | 72 <--- fourth occurrence of the value 72
  39 | 56 | 120 | 72
  40 | 57 |  80 | 72
  41 | 58 |  90 | 72
  42 | 60 | 168 | 72
  43 | 62 |  96 | 72
  44 | 63 | 104 | 72
  45 | 64 | 127 | 72
  46 | 65 |  84 | 72
  47 | 66 | 144 | 72
  48 | 68 | 126 | 72
  49 | 69 |  96 | 72
  50 | 70 | 144 | 72
  51 | 72 | 195 | 72
  52 | 74 | 114 | 72
  53 | 75 | 124 | 72
  54 | 76 | 140 | 72
  55 | 77 |  96 |  0 <--- another value apart from 72 occurs four times
  56 | 78 | 168 |  0
		

Crossrefs

Cf. A000203, A002808, A073255, A305672, A305673, A305674, A305675, A305676.

Programs

Maple

N:= 100: # to get a(1)..a(N)
cmax:= 3*N: Counts:= Vector(cmax):
i:= 0:
for n from 4 do
  if isprime(n) then next fi;
  i:= i+1;
  if i > N then break fi;
  s:= numtheory:-sigma(n);
  if s > cmax then cmax:= s; Counts(s):= 1;
  else Counts[s]:= Counts[s]+1;
  fi;
  vmax:= max[index](Counts):
  if max(Counts[1..vmax-1]) = Counts[vmax] or max(Counts[vmax+1..-1])=Counts[vmax] then A[i]:= 0 else A[i]:= vmax fi
od:
seq(A[i],i=1..N); # Robert Israel, Jun 12 2018

Mathematica

Block[{c = Select[Range@ 120, CompositeQ], s}, s = DivisorSigma[1, c]; Array[If[Length@ # == 1, #[[1, 1]], 0] &@ Last@ SplitBy[SortBy[Tally@ Take[s, #], Last], Last] &, Length@ s]] (* Michael De Vlieger, Jun 14 2018 *)

PARI

add_sigma(vec, val) = if(val > #vec, vec=concat(vec, vector(val-#vec))); vec[val]++; vec
max_pos(vec) = if(#setintersect(vecsort(vec), vector(#vec, t, vecmax(vec))) > 1, return(0), for(k=1, #vec, if(vec[k]==vecmax(vec), return(k))))
terms(n) = my(sig=[], i=0); forcomposite(c=1, , sig=add_sigma(sig, sigma(c)); print1(max_pos(sig), ", "); i++; if(i==n, break))
terms(100) \\ Print initial 100 terms of the sequence