# [seqfan] Re: Boris Stechkin and A055004

Zak Seidov zakseidov at mail.ru
Fri Jun 3 06:32:24 CEST 2016

``` BTW The long-standing Q of mine:
Is it any way to make some classical texts on NT
*as, e.g., R. K. Guy, Unsolved Problems in Number Theory
available (free! online!) at least to registered users of OEIS.

>Пятница,  3 июня 2016, 4:33 +03:00 от israel at math.ubc.ca:
>
>A055004 is titled "Boris Stechkin's function". It has a reference to R. K.
>Guy, Unsolved Problems Number Theory, A17. No formula is given, but
>according to the Maple code (which does match the Data) it would be
>
>a(n) = Sum_{m=2..n} (m-1)*floor(n*(m-1)/m).
>
>But this definition doesn't seem to match anything stated in A17 of
>"Unsolved Problems in Number Theory", which instead quotes three theorems
>of Boris Stechkin referring to the function
>
>S(n) = # {m: 2 <= m <= n, (m-1) | floor(n(m-1)/m)}
>
>For example, the first is that n-1 is prime iff S(n) = d(n), the number of
>divisors of n.
>
>This function S would produce (for 0 <= n <= 100) 0, 0, 1, 2, 3, 3, 4, 4,
>4, 5, 5, 4, 6, 6, 4, 6, 7, 5, 6, 6, 6, 8, 6, 4, 8, 9, 5, 6, 8, 6, 8, 8, 6,
>8, 6, 6, 11, 9, 4, 6, 10, 8, 8, 8, 6, 10, 8, 4, 10, 11, 7, 8, 8, 6, 8, 10,
>10, 10, 6, 4, 12, 12, 4, 8, 11, 9, 10, 8, 6, 8, 10, 8, 12, 12, 4, 8, 10, 8,
>10, 8, 10, 13, 7, 4, 12, 14, 6, 6, 10, 8, 12, 14, 8, 8, 6, 6, 14, 12, 6,
>10, 13
>
>which does not seem to be in OEIS. I think it should be. But what to do
>about A055004? It's an old sequence, so we shouldn't change the Data. Is
>there a reference to this as a function of Stechkin? Does it have any
>relationship to be primes?
>
>Cheers,
>Robert
>
>--
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```