|
| |
|
|
A126176
|
|
Number of augmented infinitary amicable pairs (i,j) with i<j and i<=10^n.
|
|
5
|
|
|
|
0, 0, 0, 0, 0, 0, 8, 26, 48, 104, 227
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,7
|
|
|
COMMENTS
|
A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
|
|
|
LINKS
|
|
|
|
FORMULA
|
augmented infinitary amicable pairs (m,n) satisfy isigma(m)=isigma(n)=m+n-1, with m<n
|
|
|
EXAMPLE
|
a(9)=48 because there are 48 augmented infinitary amicable pairs (m,n) with m<n and m<=10^9
|
|
|
MATHEMATICA
|
ExponentList[n_Integer, factors_List] := {#, IntegerExponent[n, # ]} & /@ factors; InfinitaryDivisors[1] := {1}; InfinitaryDivisors[n_Integer?Positive] := Module[ { factors = First /@ FactorInteger[n], d = Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g] == g][ #, Last[ # ]]] & /@ Transpose[Last /@ ExponentList[ #, factors] & /@ d]], _?( And @@ # &), {1}]] ]] ] Null; properinfinitarydivisorsum[k_] := Plus @@ InfinitaryDivisors[k] - k; AugmentedInfinitaryAmicableNumberQ[n_] := If[properinfinitarydivisorsum[properinfinitarydivisorsum[ n] + 1] == n - 1 && ! properinfinitarydivisorsum[n] + 1 == n, True, False]; AugmentedInfinitaryAmicablePairList[k_] := (anlist = Select[Range[k], AugmentedInfinitaryAmicableNumberQ[ # ] &]; prlist = Table[ Sort[{anlist[[n]], properinfinitarydivisorsum[anlist[[n]]] + 1}], {n, 1, Length[anlist]}]; amprlist = Union[prlist, prlist]); data = AugmentedInfinitaryAmicablePairList[10^7]; Table[Length[Select[data, First[ # ] < 10^k &]], {k, 1, 7}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
hard,nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
