- #1

Raphie

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**Prime Indices & the Divisors of p'_n - 1 such that Divisors & Indices are equivalent**

for...

**p'_n an element of {N} | d'(n) < or = to 2**

This set of integers is equivalent to 1 Union the Prime Numbers aka "Primes at the beginning of the 20th Century"

where...

**d'_n denotes the number of divisors of n < or = to n**

Thus, by this definition d'(0) = 0 since 0 cannot divide 0 and all other N are greater than 0

Proposition:

**n an element of {N} | d'(p'_n - 1) = n**

n = 0,1,2,3,4,6

(and no other?)

The truth of this statement seems to me apparent, because the increase in the prime number indices far outpaces the growth of the divisors associated with them. For example, 5040 (7!) is the first integer with 60 divisors, and 5041 is the 675th prime. By comparison, 1081079 is the 84357th prime and is one less than the first integer with 256 divisors. By the time you get up to 11! = 39916800 (with a "mere" 540 divisors), you're at the 2428957th prime - 1. It seems logical to me that the number of divisors is just never going to "catch up" to the index number of the nth prime after n = 6.

So... what piece of key information am I missing here to be able to prove (or disprove) that which appears obvious?

TIA,

RaphieA Generalized Form of the Above:

**n an element of {N} | d'(p'_n - 1) = n/(k!/|k-1|!)**

e.g.

d'(p'_00 - 1) = d'(001 - 1) = d'(000) = 00/1 = 00; k = 0, n = 00

d'(p'_01 - 1) = d'(002 - 1) = d'(001) = 01/1 = 01; k = 1, n = 01

d'(p'_02 - 1) = d'(003 - 1) = d'(002) = 02/1 = 02; k = 1, n = 02

d'(p'_03 - 1) = d'(005 - 1) = d'(004) = 03/1 = 03; k = 1, n = 03

d'(p'_04 - 1) = d'(007 - 1) = d'(006) = 04/1 = 04; k = 1, n = 04

d'(p'_06 - 1) = d'(013 - 1) = d'(012) = 06/1 = 06; k = 1, n = 06

d'(p'_24 - 1) = d'(089 - 1) = d'(088) = 24/2 = 12; k = 2, n = 24 --> d'(n) is Monotonic (strictly increasing) to here

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d'(p'_28 - 1) = d'(107 - 1) = d'(106) = 28/7 = 04; k = 7, n = 28

d'(p'_30 - 1) = d'(113 - 1) = d'(112) = 30/3 = 10; k = 3, n = 30

d'(p'_32 - 1) = d'(131 - 1) = d'(130) = 32/4 = 08; k = 4, n = 32

d'(p'_36 - 1) = d'(151 - 1) = d'(150) = 36/3 = 12; k = 3, n = 36

etc.

Then for k = 0 or 1...

n = 0, 1, 2, 3, 4, 6

d'(p'_n - 1) = 0, 1, 2, 3, 4, 6

p'_n = 1, 2, 3, 5, 7, 13

And a few observations re: 1,2,3,4,6 (special case of k = 1)

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A) 1,2,3,4,6 is the complete set of integers > 0 with a totient of 1 or 2

B) 1,2,3,4,6 is the complete set of the proper divisors of 12

C) Let a_n = 1,2,3,4,6, LK_n denote "Laminated Lattice Kissing Number" for n = (1-->4) Then...

((a_n * a_(n+1)) + 0) * (a_n^0 + 0)

= {02, 06, 012, 024}

= K_(a_n + 0)

((a_n * a_(n+1)) + 6) * (a_n^1 + 4)

= {40, 72, 126, 240}

= K_(a_n + 4)

This is the complete set of Laminated Lattice Kissing Numbers up to Dimension 8, and also the highest known Non-Lattice Kissing Numbers up to Dimension 8

TWO RELATED SEQUENCES

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**n(n+ceiling(2^n/12))**

0, 2, 6, 12, 24, 40, 72, 126, 240, 468, 960, 2002...

Sloane's A029929 http://www.research.att.com/~njas/sequences/A029929

**Maximal kissing number of n-dimensional laminated lattice**

0, 2, 6, 12, 24, 40, 72, 126, 240, 272, 336, 438, 648, 906...

Sloane's A002336 http://www.research.att.com/~njas/sequences/A002336

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