A273227 -id:A273227 - cp's OEIS frontend

A273777 Consider all ways of writing the n-th composite number as the product of two divisors d1d2 = d3d4 = ... where each divisor is larger than 1; a(n) is the maximum of the sums {d1 + d2, d3 + d4, ...}.

4, 5, 6, 6, 7, 8, 9, 8, 10, 11, 12, 10, 13, 14, 10, 15, 12, 16, 17, 18, 14, 19, 12, 20, 21, 16, 22, 23, 24, 18, 25, 26, 14, 27, 20, 28, 29, 16, 30, 22, 31, 32, 33, 24, 34, 18, 35, 36, 26, 37, 38, 39, 28, 40, 18, 41, 42, 30, 43, 44, 22, 45, 32, 46, 47, 20, 48

Offset: 1

Michel Lagneau, May 30 2016

nonn

The divisors must be > 1 and < n.

For the minimum sums see A273227.

			a(14) = 14 because A002808(14) = 24 = 2*12 = 3*8 = 4*6 and 2+12 = 14 is the maximum sum.

Cf. A002808, A020639, A046343, A063655, A273227.

with(numtheory):nn:=100:lst:={}:
for n from 1 to nn do:
it:=0:lst:={}:
d:=divisors(n):n0:=nops(d):
  if n0>2 then
  for i from 2 to n0-1 do:
   p:=d[i]:
    for j from i to n0-1 do:
      q:=d[j]:
       if p*q=n then
        lst:=lst union {p+q}:
        else
       fi:
     od:
    od:
    n0:=nops(lst):printf(`%d, `, lst[n0]):
   fi:
   od:

Function[n, Max@ Map[Plus[#, n/#] &, Rest@ Take[#, Ceiling[Length[#]/2]]] &@ Divisors@ n] /@ Select[Range@ 120, CompositeQ] (* Michael De Vlieger, May 30 2016 *)

lista(nn) = {forcomposite(n=2, nn, m = 0; fordiv(n, d, if ((d != 1) && (d != n), m = max(m, d+n/d));); print1(m, ", "););} \\ Michel Marcus, Sep 13 2017

Let m = A002808(n). Then a(n) = A020639(m) + m / A020639(m).

Name edited by Jon E. Schoenfield, Sep 12 2017

A287273 Consider all ways of writing the composite Fibonacci number A090206(n+3) as product of two divisors d1d2 = d3d4 = ... The sequence a(n) gives the minimum sums of {d1+d2, d3+d4,...}.

Original entry on oeis.org

6, 10, 19, 16, 24, 42, 71, 68, 106, 150, 178, 447, 288, 431, 3026, 754, 1119, 1220, 1902, 2974, 3194, 19979, 5168, 142026, 7728, 13098, 13530, 136187, 20276, 62158, 34130, 52994, 119811, 92736, 138673, 6169970, 241310, 6379215, 392836, 55946694, 587943, 768686

Offset: 1

Michel Lagneau, May 22 2017

nonn

This sequence is included in A273227.

The primes of the sequence are 19, 71, 431, 19979,...

			a(5) = 24 because A090206(8) = 144 = 1*144 = 2*72 = 3*48 = 4*36 = 6*24 = 8*18 = 9*16 = 12*12 and the minimum sum of {145, 74, 51, 40, 30, 26, 25, 24} is 24.

Cf. A000045, A002808, A090206, A273227.

Function[n, Min@ Map[# + n/# &, Take[#, Ceiling[Length[#]/2]]] &@ Divisors@ n] /@ Select[Fibonacci@ Range@ 55, CompositeQ] (* Michael De Vlieger, May 24 2017 *)

for(n=1, 100, d=divisors(fibonacci(n)); t=#d; k=if(t%2, 2*d[t\2+1], d[t\2]+d[t\2+1]); print1(k", ")).

cp's OEIS Frontend

A273777 Consider all ways of writing the n-th composite number as the product of two divisors d1d2 = d3d4 = ... where each divisor is larger than 1; a(n) is the maximum of the sums {d1 + d2, d3 + d4, ...}.

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A287273 Consider all ways of writing the composite Fibonacci number A090206(n+3) as product of two divisors d1d2 = d3d4 = ... The sequence a(n) gives the minimum sums of {d1+d2, d3+d4,...}.

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cp's OEIS Frontend

A273777 Consider all ways of writing the n-th composite number as the product of two divisors d1*d2 = d3*d4 = ... where each divisor is larger than 1; a(n) is the maximum of the sums {d1 + d2, d3 + d4, ...}.

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Maple

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A287273 Consider all ways of writing the composite Fibonacci number A090206(n+3) as product of two divisors d1*d2 = d3*d4 = ... The sequence a(n) gives the minimum sums of {d1+d2, d3+d4,...}.

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Programs

Mathematica

PARI

A273777 Consider all ways of writing the n-th composite number as the product of two divisors d1d2 = d3d4 = ... where each divisor is larger than 1; a(n) is the maximum of the sums {d1 + d2, d3 + d4, ...}.

A287273 Consider all ways of writing the composite Fibonacci number A090206(n+3) as product of two divisors d1d2 = d3d4 = ... The sequence a(n) gives the minimum sums of {d1+d2, d3+d4,...}.