[seqfan] Re: Iterative transform of integer sequence

[M. F. Hasler]

On Thu, Jun 1, 2017 at 4:40 AM, zak seidov via SeqFan <seqfan at list.seqfan.eu
> wrote:

>  Iterative transform of A00040
>  {{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
> 71, 73, 79, 83,...},
>   {4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69,
> 72, 76, 81, 86, 93,...},
>   {5, 28, 32, 53, 58, 62, 74, 96, 136, 141, 147, 152, 157, 173, 178, 183,
> 189, 211, 221,...},
>   {30, 60, 68, 85, 116, 144, 165, 186, 200, 216, 260, 270, 359, 399, 420,
> 426, 445,...},
>   {...}}
>
> 1st column: {2,4,5,30,45,97,160,359,536,2866,4983,...}
> 2nd column: {3,6,28,60,64,223,558,713,1553,7100,15369,...}
> Old hat? Interesting? Worth submitting?


I don't think this has been considered earlier, because it seems a bit
"artificial" to take half sums when you have terms of mixed parity, but
only want integer terms and therefore simply drop the non-integer terms you
have produced.
I think it is much more natural not to divide by two upon addition, so
your're sure to get always integers. The result would be:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43...]=A000040
[5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100,...]=A001043
<https://oeis.org/A001043>
[13, 20, 30, 42, 54, 66, 78, 94, 112, 128, 146, 162, 175,...]=A096277
[33, 50, 72, 96, 120, 144, 172, 206, 240, 274, 308, 336,....]=A096278
<https://oeis.org/A096278>
[83, 122, 168, 216, 264, 316, 378, 446, 514, 582, 644,] = A096279
<https://oeis.org/A096279>
...
1st column = A007443 = Binomial transform of primes.
The list of indices of primes in that sequence (A096280) was not yet in the
OEIS, I suggest to add it.

- Maximilian

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