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#242 by Michael De Vlieger at Mon Sep 04 16:37:57 EDT 2023
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#241 by Jon E. Schoenfield at Mon Sep 04 14:18:58 EDT 2023
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#240 by Jon E. Schoenfield at Mon Sep 04 14:18:53 EDT 2023
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| COMMENTS
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It is a result that goes back to Mirsky that the set of primes p for which p-1 is squarefree has density A, where A = A005596 denotes the Artin constant. More precisely, Sum_{p <= x} mu(p-1)^2 = AxA*x/log x + o(x/log x) as x tends to infinity. Conjecture: Sum_{p <= x, mu(p-1)=1} 1 = (A/2))*x/log x + o(x/log x) and Sum_{p <= x, mu(p-1)=-1} 1 = (A/2))*x/log x + o(x/log x). - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003
With the exception of the first two terms {2,5}, the continued fraction (1 + sqrt(p))/2 has period 3. - Artur Jasinski, Feb 03 2010
With the exception of the first term {2}, congruent to 1 (mod 4). - Artur Jasinski, Mar 22 2011
With the exception of the first two terms, congruent to 1 or 17 (mod 20). - Robert Israel, Oct 14 2014
These primes are the primitive terms which generate the sequence of integers with only one prime factor and whose Euler's totient is a square: A054755. So this sequence is a subsequence of A054755 and of A039770. Additionally, the terms of this sequence also have a square cototient, so this sequence is a subsequence of A063752 and A054754.
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| FORMULA
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There are O(sqrt(n)/log(n)) membersterms of this sequence up to n. But this is just an upper bound. See the Bateman-Horn or Wolf papers, for example, for the conjectured for what is believed to be the correct density.
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| STATUS
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approved
editing
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#239 by Alois P. Heinz at Wed Sep 07 21:43:35 EDT 2022
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#238 by Alois P. Heinz at Wed Sep 07 21:40:05 EDT 2022
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#237 by Kevin Ryde at Wed Sep 07 21:26:39 EDT 2022
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Discussion
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| Wed Sep 07
| 21:39
| Alois P. Heinz: agreed ...
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#236 by Kevin Ryde at Wed Sep 07 21:23:54 EDT 2022
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| COMMENTS
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For a(n)>17, the fractional part of square root of a(n) starts with digit 0 (see A034096). - Charles Kusniec, Sep 05 2022
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| STATUS
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proposed
editing
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Discussion
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| Wed Sep 07
| 21:26
| Kevin Ryde: Yes, no, let's not. (Don't think I'm out on a limb on that!)
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#235 by Michel Marcus at Wed Sep 07 16:51:20 EDT 2022
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#234 by Michel Marcus at Wed Sep 07 16:51:13 EDT 2022
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| COMMENTS
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For a(n)>17, the fractional part of square root of a(n) starts with digit 0. - _ (see A034096). - _Charles Kusniec_, Sep 05 2022
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| STATUS
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proposed
editing
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#233 by Charles Kusniec at Tue Sep 06 13:41:17 EDT 2022
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Discussion
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| Tue Sep 06
| 13:50
| Michel Marcus: did you see A034096 ?
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| 21:30
| Charles Kusniec: Michel, I just saw that you put references in the three sequences. Several things are different. This is because your references contemplate all integers that have the same digit. For example in this case, if you take all elements >17 and subtract 2, the digit will never be more 0 (2 elements with 8 and all others with 9). But in your reference you will find elements subtracted from 2 with digit 0. The same happens in the other sequences when we do the shift (+ or -).
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| Wed Sep 07
| 02:45
| Joerg Arndt: trivial and not at all interesting
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| 16:50
| Michel Marcus: contemplate all integers ....
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