A131457 a(n+1) is the next semiprime such that a(n+1)-1 divides (a(1)...a(n))^2.
4, 9, 10, 21, 22, 25, 26, 33, 34, 35, 46, 49, 51, 55, 57, 58, 65, 69, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 161, 166, 169, 177, 178, 183, 185, 187, 201, 202, 203, 205, 206, 209, 213
Offset: 1
Examples
a(1) = 4 because 4 = 2^2 is the first semiprime. a(2) = 9 because 9 = 3^2 is the next semiprime after 4, where 9-1=8 divides 4^2 = 16. a(3) = 10 because 10 = 2*5 is the next semiprime after 9 where 10-9=9 divides (4*9)^2. a(4) = 21 because 21 = 3*7 is the next semiprime after 10, where 10-1=9 divides (4*9*10)^2. a(5) = 22 because 22 = 2*11 is the next semiprime after 21, where 21-1=20 divides (4*9*10*21)^2.
Programs
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Maple
isA001358 := proc(n) if numtheory[bigomega](n) = 2 then true ; else false; fi ; end: A131457 := proc(n) option remember ; local a,prevpr; if n =1 then 4; else prevpr := (mul(A131457(i),i=1..n-1))^2 ; a := A131457(n-1)+1 ; while not isA001358(a) or prevpr mod (a-1) <> 0 do a := a+1 ; od; RETURN(a) ; fi ; end: seq(A131457(n),n=1..80) ; # R. J. Mathar, Oct 30 2007
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Mathematica
semiprimeQ[n_] := PrimeOmega[n] == 2; a[n_] := a[n] = Module[{k, prevpr}, If[n == 1, 4, prevpr = Product[a[i], {i, 1, n-1}]^2; k = a[n-1]+1; While[!semiprimeQ[k] || Mod[prevpr, k-1] != 0, k++]; Return[k]]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 28 2024, after R. J. Mathar *)
Extensions
Corrected and extended by R. J. Mathar, Oct 30 2007
Comments