Re: Is this triangle number deficient or abundant?

Is the sum of the devisors of this
triangle number >(abundant) or <(deficient)

9.223372039002259456e+18

Just doing a little research of the Fermat prime
triangle numbers and the first 5 and possibly only
5 triangle numbers related to the Fermat primes. 3,10,136,32896,2147516416
are all deficient by 2
I believe all the rest of the even triangle numbers
except of course the even perfect numbers are abundant
or deficient but only the 5 above deficient by 2.

The even triangle number above is where 2^32 + 1
fails to be a 6th Fermat prime.
Its 3 factors are 2^31 * 641 * 6700417.

If the prime-power factorization of n is n = 2^31 * 641 >* 6700417,
then the sum of the divisors is given by
sigma(n) = (2^32 - 1) (642) (6700418),
whatever that is.

The 2 factors of each of the 5 Fermat triangle
numbers are --
2^0 * 3 = 3
2^1 * 5 = 10
2^3 * 17 = 136
2^7 * 257 = 32896
2^15 * 65537 = 2147516416
3,10,136,32896,2147516416 are all deficient by (2)
Are they the only even triangle # deficient by 2?

I don't know. Have a look at sequences A045768, >A088831, and A063785
at http://www.research.att.com/~njas/sequences/
and
MR0538520 (81b:10004) Makowski, Andrzej Some equations >involving the
sum of divisors. Elem. Math. 34 (1979), no. 4, 82.
and
MR0123509 (23 #A834) Makowski, A. Remarques sur les >fonctions $\theta
(n),\,\varphi (n)$ et $\sigma (n)$. (French) Mathesis >69 1960 302--303.

Interesting enough most terms in the sequence in A045768
are triangle numbers (-1). 20+1,104+1,464+1,1952+1,etc.
are triangle numbers.
Most of these terms deal with 2^(n-1) * (2^n-3)
or 2^(n-1) * (2^n-1).
Where my problem deals with triangle numbers of
the form --
2^0*2^1+1 = 3
2^1*2^2+1 = 10
2^3*2^4+1 = 136
2^7*2^8+1 = 32896
2^15*2^16+1 = 2147516416
For each triangle number it gives 5 Fermat primes
3,5,17,257,65537 as one of the factors.
I am still trying to comprehend the 3 sequences in OEIS.

Thanks for the info.
I have yet to lookup the other MR series info.

Is there an algorithm used in a calculator online
that gives the sum of divisors of larger integers?
I know there is just about everything else!

Dan

Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

Relevant Pages

  • Re: Why has no one confirmed or refuted this?
    ... even more rare than the even perfect ... numbers which are also triangle numbers ... Where an integer is deficient -- ... relationship with the 5 known Fermat primes. ...
    (sci.math)
  • Re: Why has no one confirmed or refuted this?
    ... even more rare than the even perfect ... numbers which are also triangle numbers ... Where an integer is deficient -- ... relationship with the 5 known Fermat primes. ...
    (sci.math)
  • Re: Is this triangle number deficient or abundant?
    ... triangle numbers and the first 5 and possibly only ... triangle numbers related to the Fermat primes. ... except of course the even perfect numbers are abundant ... or deficient but only the 5 above deficient by 2. ...
    (sci.math)
  • Re: Why has no one confirmed or refuted this?
    ... relationship to the Fermat primes. ... These deficient by 2 triangle numbers are ... are you the same danny who investigated Leroy sequences ?... ...
    (sci.math)
  • Re: Is this triangle number deficient or abundant?
    ... triangle numbers and the first 5 and possibly only ... or deficient but only the 5 above deficient by 2. ... and so it's easy to calculate the sum of all divisors. ... But to decide whether your number is deficient or abundant, ...
    (sci.math)
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