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A306722
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Number of pairs of primes (p,q), p < q, which are a solution of the Diophantine equation (p-1)*(q-1) = (2n)^2.
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3
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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
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OFFSET
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1,3
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COMMENTS
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a(n) is also the number of semiprimes p*q whose totient is a square (A247129) and equal to (2*n)^2.
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
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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f[n_] := Length@ Select[ Divisors[ 4n^2], # < 2n && PrimeQ[# + 1] && PrimeQ[ 4n^2/# + 1] &]; Array[f, 81] (* Robert G. Wilson v, Mar 30 2019 *)
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PROG
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(PARI) 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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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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