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A306440
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Number of different ways of expressing 2*n as (p - 1)*(q - 1), where p < q are primes.
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3
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0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 1, 0, 1, 4, 0, 0, 1, 1, 1, 1, 0, 2, 0, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 0, 3, 0, 0, 1, 0, 1, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 1, 2, 0, 1, 0, 1, 1, 3, 0, 1, 0, 0, 1
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OFFSET
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0,7
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COMMENTS
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If 2*n-1 is an odd prime then a(n) > 0.
If a(n) > 0 then 2n+1 is in A323550.
Given any distinct odd primes p_1, ..., p_m, Dickson's conjecture implies that there are infinitely many m such that m/(p_i-1) + 1 is prime for all i. Thus the sequence should be unbounded. - Robert Israel, Mar 29 2019
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LINKS
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FORMULA
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a(6k+1) = 0 for k > 0 because 12k+2 can't be written as (p-1)(q-1) except for k = 0 with p = 2, q = 3: If q > 3, then q-1 is congruent to 0 or 4 (mod 6), and no p = 2, p = 3 (=> q-1 = 6k+1) or p > 3 is possible. - M. F. Hasler, Feb 25 2019
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EXAMPLE
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a(2) = 1 because 2*2 = 4 can only be expressed as (p - 1)*(q - 1) with primes p = 2 and q = 5.
a(6) = 2 because for 2*6 = 12, there are only two possible ordered pairs of distinct primes (p,q), (2,13) and (3,7), such that 12 = (p - 1)*(q - 1).
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MAPLE
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f:= 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:
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MATHEMATICA
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a[n_]:=Module[{k=0}, Do[Do[If[2n==(Prime[i]-1)*(Prime[j]-1), k++], {i, 1, j-1}], {j, 2, PrimePi[2n]+1}]; Return[k]];
Table[a[j], {j, 0, 128}]
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PROG
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(PARI) A306440(n, d, c)={forprime(p=2, sqrtint(-(n>0)+n*=2)+1, n%(p-1)==0 && isprime(n/(p-1)+1) && c++ && d && printf("%d-1=(%d-1)*(%d-1) [%d], ", n+1, p, n/(p-1)+1, c)); c} \\ Give 1 as 2nd optional arg (d=debug) to get a list of all decompositions. - M. F. Hasler, Feb 25 2019
(PARI) a(n) = if(n==0, return(0)); my(d=divisors(n<<1)); d+=vector(#d, i, 1); sum(i=1, #d\2, isprime(d[i]) && isprime(d[#d-i+1])) \\ for finding lots of terms or a(n) for large n. \\ David A. Corneth, Mar 18 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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