A006336 - Unexpected Relation to Golden Ratio?

[T. D. Noe]

Dear seqfans,

I#ve been surprised not to find sequences of the following form in the OEIS:

a(n)=min(k in N: sigma(r,n)=sigma(r,k)) with sigma(r,n)=sum of the r-th
power of the divisors of n:

new[r_, n_] :=
  (If[Head[#1] === rep, #1 = n, #1] & )[rep[DivisorSigma[r, n]]]

for r=0 (number of divisors)

Clear[rep];
(new[0, #1] & ) /@ Range[0, 100]

{0, 1, 2, 2, 4, 2, 6, 2, 6, 4, 6, 2, 12, 2, 6, 6, 16, 2, 12, 2, 12,
  6, 6, 2, 24, 4, 6, 6, 12, 2, 24, 2, 12, 6, 6, 6, 36, 2, 6, 6, 24,
  2, 24, 2, 12, 12, 6, 2, 48, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2,
  60, 2, 6, 12, 64, 6, 24, 2, 12, 6, 24, 2, 60, 2, 6, 12, 12, 6,
  24, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2, 60, 6, 12, 6, 6, 6, 60,
  2, 12, 12, 36}

for r=1 (sum of divisors)

Clear[rep];
(new[1, #1] & ) /@ Range[0, 100]

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 12, 13, 14, 14, 16, 10, 18,
  19, 20, 21, 22, 14, 24, 16, 20, 27, 28, 29, 30, 21, 32, 33, 34,
  33, 36, 37, 24, 28, 40, 20, 42, 43, 44, 45, 30, 33, 48, 49, 50,
  30, 52, 34, 54, 30, 54, 57, 40, 24, 60, 61, 42, 63, 64, 44, 66,
  67, 68, 42, 66, 30, 72, 73, 74, 48, 76, 42, 60, 57, 80, 81, 68,
  44, 84, 85, 86, 54, 88, 40, 90, 91, 60, 93, 66, 54, 96, 52, 98,
  99, 100}

r=2 (sum of squares of divisors)

Clear[rep];
(new[2, #1] & ) /@ Range[0, 100]

{0, 1, 2, 3, 4, 5, 6, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
  19, 20, 21, 22, 23, 24, 25, 24, 27, 28, 29, 30, 31, 32, 33, 34,
  30, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 40, 48, 49, 50,
  51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,
  67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 66, 78, 79, 80, 81, 82,
  83, 84, 85, 86, 87, 88, 89, 90, 78, 92, 93, 94, 95, 96, 97, 98,
  99, 100}

and so on.

Are these of interest? And if so, up to which exponent r?

Peter



>What does this sequence count? It is similar to A00123 and A005704, which
>both have a recursion a(n)=a(n-1)+a([n/k]), where k is 2 and 3,
>respectively.  Those sequences count "number of partitions of k*n into
>powers of k". For sequence A006336, k=Phi.  Does A006336(n) count the
>number of partitions of n*Phi into powers of Phi?

Answering my own question.

If the recursion starts with a(0)=1, then I think we obtain "number of
partitions of n*Phi into powers of Phi".  That sequence is 1, 2, 4, 6, 10,
16, 22,..., which I just submitted as A131882.

We need negative powers of Phi also, letting p=Phi and q=1/Phi

n=0:	0*p = {}				for 1 partition

n=1:	1*p = p = 1+q				for 2 partitions

n=2:	2*p = p+p = 1+p+q = 1+1+q+q = p^2+q	for 4 partitions

etc. tedious!

So A006336(n), which starts with a(1)=1, counts 1/2 of the "number of
partitions of n*Phi into powers of Phi"

Tony