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Please do not rely on any information it contains.
69 is an integer.
Membership in core sequences
Odd numbers
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..., 63, 65, 67, 69, 71, 73, 79, ...
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A005843
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Squarefree numbers
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..., 65, 66, 67, 69, 70, 71, 73, ...
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A005117
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In Pascal's triangle, 69 occurs twice.
Sequences pertaining to 69
Multiples of 69
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69, 138, 207, 276, 345, 414, 483, 552, 621, 690, 759, 828, ...
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69-gonal numbers
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1, 69, 204, 406, 675, 1011, 1414, 1884, 2421, 3025, 3696, ...
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Partitions of 69
There are 3554345 partitions of 69.
Roots and powers of 69
In the table below, irrational numbers are given truncated to eight decimal places.
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8.30662386
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A010521
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69 2
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4761
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4.10156592
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A010639
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69 3
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328509
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2.88212141
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A011061
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69 4
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22667121
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2.33222162
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A011154
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69 5
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1564031349
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2.02523231
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69 6
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107918163081
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1.83101847
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69 7
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7446353252589
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1.69768118
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69 8
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513798374428641
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1.60072440
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69 9
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35452087835576229
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1.52716129
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69 10
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2446194060654759801
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Logarithms and 69th powers
In the OEIS specifically and mathematics in general, refers to the natural logarithm of , whereas all other bases are specified with a subscript.
PLACEHOLDER
Values for number theoretic functions with 69 as an argument
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1
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–1
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19
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96
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4
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44
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2
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2
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22
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This is the Carmichael lambda function.
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1
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This is the Liouville lambda function.
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69!
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1.71122452... × 10 98
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2.48003554... × 10 96
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Factorization of some small integers in a quadratic integer ring adjoining ,
The commutative quadratic integer ring with unity , with units of the form (), is a unique factorization domain. But it is not norm-Euclidean. If we picked numbers out of a hat to put through the Euclidean algorithm using the absolute value of the norm, we'd probably believe that the algorithm works. That is, unless the numbers we picked happened to be certain multiples of and . Let's call the former number and see if we can express the latter number as such that , and —or we could also say .
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1
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25
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−803
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853
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2
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−44
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−2
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−44
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Part of what makes this all so frustrating is that we're talking about numbers that don't seem to be that far apart on the number line. is approximately 23.959935794377 while is approximately 34.61324772. And yet the norms of the remainders give such an impression of insurmountable distance.
Is it usually this difficult to find a suitable remainder? Let's for a moment consider a superficially similar situation: in , compute . We're looking to solve so that . If we try , we get and . Easy.
Returning now to the vexed question of finding a suitable remainder in , what if we simply made another adjustment to the norm function? If is prime in , then And if is a product of primes, then . Essentially, if is coprime to 23, then .
The following table gives the factorizations of the integers from 2 to 22 in .
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2
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Prime
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3
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4
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2 2
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5
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6
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7
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Prime
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8
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2 3
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9
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10
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11
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12
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13
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14
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2 × 7
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15
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16
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2 4
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17
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18
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19
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Prime
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20
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21
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22
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And so for 23, the primes that make it up each have a norm of −23, but with the specially adjusted norm function, that becomes 26. gives us , which with its norm of 25 is now a suitable remainder. We can then move on to solving then the of the [FINISH WRITING]
Now, if we have to know the prime factorizations of numbers before we can apply the adjusted norm function, doesn't that mean the Euclidean algorithm is hopelessly inefficient in this domain? Perhaps. But with the relatively recently proven fact that the adjusted norm function described above is a valid Euclidean function for this domain,[1] at least we know that this is a Euclidean domain after all.
In regards to Z[-69], the class number the of the [FINISH WRITING]
Factorization of 69 in some quadratic integer rings
PLACEHOLDER
TABLE GOES HERE
Representation of 69 in various bases
Base
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12
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13
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14
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15
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16
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17
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18
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19
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20
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Representation
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1000101
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2220
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1011
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234
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153
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126
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105
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76
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69
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63
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59
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54
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4D
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49
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45
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41
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3F
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3C
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39
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REMARKS GO HERE
See also
References
- ↑ David A. Clark, "A quadratic field which is Euclidean but not norm-Euclidean" Manuscripta Math. 83, 327-330 (1994)