Original entry on oeis.org
540, 2100, 5460, 11340, 20460, 33540, 51300, 74460, 103740, 139860, 183540, 235500, 296460, 367140, 448260, 540540, 644700, 761460, 891540, 1035660, 1194540, 1368900, 1559460, 1766940, 1992060, 2235540, 2498100, 2780460, 3083340
Offset: 0
Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Mar 26 2002
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Differences[Table[(n + 1)^6 - n^6, {n, 0, 30}], 2] (* Harvey P. Dale, Dec 27 2011 *)
Offset changed from 1 to 0 and added a(0)=540 by
Bruno Berselli, Feb 25 2015
A069476
First differences of A069475, successive differences of (n+1)^6-n^6.
Original entry on oeis.org
1800, 2520, 3240, 3960, 4680, 5400, 6120, 6840, 7560, 8280, 9000, 9720, 10440, 11160, 11880, 12600, 13320, 14040, 14760, 15480, 16200, 16920, 17640, 18360, 19080, 19800, 20520, 21240, 21960, 22680, 23400, 24120, 24840, 25560, 26280, 27000
Offset: 0
Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Mar 26 2002
Offset changed from 1 to 0 and added a(0)=1800 by
Bruno Berselli, Feb 25 2015
A069475
First differences of A069474, successive differences of (n+1)^6-n^6.
Original entry on oeis.org
1560, 3360, 5880, 9120, 13080, 17760, 23160, 29280, 36120, 43680, 51960, 60960, 70680, 81120, 92280, 104160, 116760, 130080, 144120, 158880, 174360, 190560, 207480, 225120, 243480, 262560, 282360, 302880, 324120, 346080, 368760, 392160, 416280
Offset: 0
Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Mar 26 2002
Offset changed from 1 to 0 and added a(0)=1560 by
Bruno Berselli, Feb 25 2015
A353890
a(n) is the period of the binary sequence {b(m)} defined by b(m) = 1 if (m+1)^n - m^n and (m+2)^n - 2*(m+1)^n + m^n are coprime, 0 otherwise.
Original entry on oeis.org
1, 1, 5, 11, 91, 1247, 3485, 263017, 852841, 1241058127, 74966255, 243641132605417, 181556731572385303, 718802057694183783881, 6582662048285, 943422576750791493013356207217, 487331778345355477261, 607088607861933740557075591887834842297
Offset: 2
For n=2 and n=3, the first and second differences are coprime for all m. Each of their sequences {b(m)} consist only of 1's, which can be described trivially as [1] with a period of 1, so a(2) = a(3) = 1.
For n > 3, the first and second differences are coprime for some m values, but not for all. Each repeating periodic sequence {b(m)} begins at m=1, and can be used to predict what b(m) will be at any higher m value for that power n.
n=4 has the 5-term repeating sequence, beginning at m=1:
[0 0 1 1 1], so a(4) = 5.
The sequence is repeating, so for example, f(41)..f(45) is also [0 0 1 1 1].
n=5 has the 11-term repeating sequence
[1 1 0 1 1 0 1 1 1 1 1]
so a(5) = 11.
n=6 has the 91-term repeating sequence
[0 0 0 0 0 0 1 0 0 0 0 1 1
1 0 0 0 0 0 1 1 0 0 0 0 1
1 1 0 0 0 0 1 1 1 0 0 0 0
1 1 1 0 0 0 0 1 1 1 0 0 0
0 1 1 1 0 0 0 0 1 1 1 0 0
0 0 1 1 0 0 0 0 0 1 1 1 0
0 0 0 1 0 0 0 0 0 0 1 1 1]
so a(6) = 91.
The period for higher n values has yet to be found. If they exist, it seems they would be quite large given the large expansion from 5, 11, to 91.
Example: the 233rd term in the sequence of values for n=6 is calculated by using m=233 and n=6. Define the first difference for the 233rd term as 234^6 - 233^6 = 4164782373647. The second difference for the 233rd term is 235^6 - 2*234^6 + 233^6 = 89948228762. The terms 4164782373647 and 89948228762 share a common factor, so the 233rd term of the sequence for 6th powered terms is denoted 0 (not coprime). Because the 6th powered terms repeat their tendency of being coprime or not every 91 terms, we could instead look at 233 mod 91 = 51, and from the table for n=6 above, the 51st term is 0.
Cf.
A005408,
A007395,
A003215,
A008588,
A005917,
A005914,
A022521,
A068236,
A022522,
A069473,
A069925,
A001045,
A002587.
Showing 1-4 of 4 results.
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