A306440 -id:A306440 - cp's OEIS frontend

A307221 Positions of records in A306440.

0, 1, 6, 30, 36, 540, 2100, 3780, 15120, 37800, 85680, 120960, 126000, 131040, 196560, 302400, 332640, 786240, 917280, 1179360, 2702700, 3341520, 4324320, 8648640, 19656000, 19958400, 21205800, 43243200, 46781280, 57657600, 64864800, 122522400, 151351200

Offset: 1

Robert Israel, Mar 29 2019

nonn

Numbers k such that A306440(k) > A306440(j) for all j < k.

			a(3) = 6 is in the sequence because A306440(6) = 2 while A306440(j) < 2 for 0 <= j < 6.

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A306440.

A306440:= proc(n) local t;
    nops(select(t -> t^2<2*n and isprime(t+1) and isprime(2*n/t+1), numtheory:-divisors(2*n)))
end proc:
Res:= 0: vmax:= 0:
for n from 1 to 10^6 do
  v:= A306440(n);
  if v > vmax then
    Res:= Res, n; vmax:= v;
  fi
od:
Res;

A306440[n_] := Length[Select[Divisors[2n], #^2 < 2n && PrimeQ[#+1] && PrimeQ[2n/#+1]&]];
Res = {0}; vmax = 0;
For[n = 1, n <= 10^6 , n++, v = A306440[n]; If[v > vmax, Print[n]; AppendTo[Res, n]; vmax = v]];
Res (* Jean-François Alcover, Oct 17 2020, translated from Maple *)

a(24)-a(33) from Daniel Suteu, Mar 29 2019

A306722 Number of pairs of primes (p,q), p < q, which are a solution of the Diophantine equation (p-1)*(q-1) = (2n)^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 0, 3, 1, 1, 1, 1, 0, 3, 0, 3, 1, 1, 0, 3, 1, 1, 4, 3, 0, 3, 0, 1, 4, 0, 1, 3, 1, 0, 0, 3, 0, 3, 0, 1, 4, 0, 1, 3, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 0, 5, 0, 1, 4, 0, 1, 4, 1, 0, 0, 4, 0, 6, 1, 1, 4, 0, 0, 5, 0, 4, 1

Offset: 1

Bernard Schott, Mar 06 2019

nonn

a(n) is also the number of semiprimes p*q whose totient is a square (A247129) and equal to (2*n)^2.
From Robert G. Wilson v, Mar 30 2019, Mar 30 2019: (Start)
First occurrence of k=1,2,3,...: 1, 3, 10, 27, 60, 72, 120, 180, 270, 480, 252, 1155, 720, 792, 1260, 630, ..., . = A307245.
Start of table:
a(k_i) = n:
\i  1   2   3   4   5   6    7    8    9   10   11   12   13   14   15  ...
n\
0  11  17  19  23  29  31   34   38   39   41   43   46   49   51   53  ...
1   1   2   4   5   7   8    9   13   14   15   16   21   22   25   26  ...
2   3   6  54  56  58  87  100  115  116  123  138  148  160  170  176  ...
3  10  12  18  20  24  28   30   36   40   42   48   84   88   99  144  ...
4  27  33  45  63  66  70   75   80  112  126  135  153  156  162  165  ...
5  60  78  90 102 140 168  200  260  264  285  288  315  378  408  432  ...
6  72 105 108 130 150 306  348  357  450  495  528  560  672  696  708  ...
7 120 132 240 297 312 330  390  588  750  882  980 1140 1176 1190 1215  ...
8 180 198 210 280 396 468  540  612  648  700  810  910  945  960 1020  ...
9 270 420 660 858 918 990 1248 1620 1782 1920 2088 2184 2352 2376 2688  ...
... (End).
If n is a prime <> 3, then a(n) = 1 if n is in A052291 and 0 otherwise, and a(n^2) = 1 if 2*n+1 and 2*n^3+1 are primes and 0 otherwise. - Robert Israel, Apr 04 2019

			a(2) = 1 because (2*2)^2 = (2-1) * (17-1), also, phi(2*17) = 4^2.
a(3) = 2 because (2*3)^2 = (2-1) * (37-1) = (3-1) * (19-1), also, phi(2*37) = phi(3*19) = 6^2.
a(11) = 0  because (2*11)^2 can't be written as (p-1)*(q-1) with p < q.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A039770, A052291, A062732, A221284, A221285, A247129, A306440, A307245.

f:= proc(n) local w;
  w:= (2*n)^2;
  nops(select(t -> t < 2*n and isprime(t+1) and isprime(w/t + 1),  numtheory:-divisors(w)))
end proc:
map(f, [$1..100]); # Robert Israel, Apr 04 2019

f[n_] := Length@ Select[ Divisors[ 4n^2], # < 2n && PrimeQ[# + 1] && PrimeQ[ 4n^2/# + 1] &]; Array[f, 81] (* Robert G. Wilson v, Mar 30 2019 *)

a(n) = {my(nb = 0, nn = 4*n^2); fordiv(nn, d, if (d == 2*n, break); if (isprime(d+1) && isprime(nn/d+1), nb++);); nb;} \\ Michel Marcus, Mar 06 2019

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A307221 Positions of records in A306440.

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