A141341 -id:A141341

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A141340

Positive integers n such that A061358(n) = #{primes p | n/2 <= p < n-1}.

+10
1

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 24, 30, 36, 42, 48, 60, 90, 210

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

According to Brouwers et al., Deshouillers et al. showed that the maximum term of this sequence is 210. A141341 is a subsequence.

LINKS

Table of n, a(n) for n=1..21.

J-M. Deshouillers, A. Granville, W. Narkiewicz and C. Pomerance, An upper bound in Goldbach's problem, Math. Comp. 61 (1993), 209-213.

David van Golstein Brouwers, John Bamberg and Grant Cairns, Totally Goldbach numbers and related conjectures, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.

CROSSREFS

Cf. A061358, A141341.

KEYWORD

fini,full,nonn

AUTHOR

Rick L. Shepherd, Jun 25 2008

STATUS

approved

A323215

Numbers k such that row k of A322936 is not empty and has only primes as members.

+10
1

5, 8, 9, 10, 12, 18, 24, 30

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a is strongly prime to n if and only if a <= n is prime to n and a does not divide n-1. See the link to 'Strong Coprimality'. (Our terminology follows the plea of Knuth, Graham and Patashnik in Concrete Mathematics, p. 115.)

From Robert Israel, Apr 02 2019: (Start)

If there is at least one prime <= sqrt(n) that divides neither n nor n-1, then its square is strongly prime to n and not prime. If there does not exist such a prime, then the first Chebyshev function theta(sqrt(n)) = Sum_{p <= sqrt(n)} log(p) <= 2 log(n). Now it is known that theta(x) = x + O(x/log(x)), so this can't happen if n is sufficiently large. Thus the sequence is finite.

The largest n for which no such p exists appears to be 120. There are none between 121 and 10^7. It is possible that a sufficiently tight lower bound on theta together with a finite search can be used to prove that there are no other terms of the sequence. (End)

There are no more terms. See proof at A307345. - Robert Israel, Apr 03 2019

LINKS

Table of n, a(n) for n=1..8.

Peter Luschny, Strong Coprimality.

MAPLE

filter:= proc(n) local k, found;

found:= false;

for k from 2 to n-2 do

if igcd(k, n)=1 and (n-1) mod k <> 0 then

found:= true;

if not isprime(k) then return false fi;

od;

found

end proc:

select(filter, [$1..1000]); # Robert Israel, Apr 02 2019

MATHEMATICA

Select[Range[10^3], With[{n = #}, AllTrue[Select[Range[2, n], And[GCD[#, n] == 1, Mod[n - 1, #] != 0] &] /. {} -> {0}, PrimeQ]] &] (* Michael De Vlieger, Apr 01 2019 *)

PROG

(Sage) # uses[A322936row from A322936]

def isA323215(n):

return all(is_prime(p) for p in A322936row(n))

[n for n in (1..100) if isA323215(n)] # Peter Luschny, Apr 03 2019

CROSSREFS

Cf. A181830, A141341, A307345, A322936, A322937.

KEYWORD

nonn,fini,full

AUTHOR

Peter Luschny, Apr 01 2019

EXTENSIONS

Name corrected after a notice from Robert Israel by Peter Luschny, Apr 02 2019

STATUS

approved

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