A177803 The number of lines in the analog of Pratt primality certificate for the n-th semiprime.
1, 2, 1, 2, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 1, 5, 6, 7, 1, 3, 4, 5, 3, 3, 1, 7, 1, 2, 3, 4, 5, 5, 1, 2, 3, 5, 1, 8, 1, 8, 5, 6, 1, 3, 4, 7, 8, 6, 1, 4, 5, 3, 4, 5, 1, 8, 1, 2, 8, 2, 3, 10, 1, 5, 6, 9, 1, 5, 1, 2, 6, 3, 4, 8, 1, 5, 3, 4, 1, 9, 10, 11, 12, 8, 1, 9, 10, 7, 8, 9, 10, 3, 1, 5, 5
Offset: 4
Examples
a (5) = 2 = 1 + a(4) because 4 | (5-1) and 4 = 2*2 is a semiprime. a (6) = 1 because there is no semiprime that divides (6-1) = 5, a prime. a (7) = 2 = 1 + a(6) = 1+1 because 6 | (7-1) and 6 = 2*3 is a semiprime. a (8) = 1 because there is no semiprime that divides (8-1) = 7, a prime. a (9) = 2 = 1 + a(4) = 1+1 because 4 | (9-1). a(10) = 3 = 1 + a(9) = 1+2 because 9 | (10-1) and 9 is a semiprime. a(11) = 4 = 1 + a(10) = 1+3 because 10 | (11-1) and 10 = 2*5 is a semiprime. a(12) = 1 because there is no semiprime that divides (12-1) = 11, a prime. a(13) = 3 = 1 + a(4) + a(6) = 1+1+1 because both 4 and 6 divide into (13-1) = 12 and are semiprimes. a(14) = 1 because there is no semiprime that divides (14-1) = 13, a prime. a(15) = 2 = 1 + a(14) = 1+1 because 14 | (15-1). a(16) = 3 = 1 + a(15) = 1+2 because 15=3*5 is the only semiprime which divides 16-1. a(17) = 2 = 1 + a(4) = 1+1 because 4 | (17-1) and 4 is the only such semiprime.
Links
- Alois P. Heinz, Table of n, a(n) for n = 4..1000
Programs
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Maple
a:= proc(n) option remember; 1 +add (`if` (not isprime(k) and add (i[2], i=ifactors(k)[2])=2 and irem (n-1, k)=0, a(k), 0), k=4..n-1) end: seq (a(n), n=4..100); # Alois P. Heinz, Dec 12 2010
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Mathematica
a[n_] := a[n] = 1 + Sum[If[!PrimeQ[k] && Total@FactorInteger[k][[All, 2]] == 2 && Mod[n - 1, k] == 0, a[k], 0], {k, 4, n - 1}]; a /@ Range[4, 100] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)
Formula
a(4) = 1; a(n) = 1 + Sum a(k), k semiprime, k | n-1.
Extensions
More terms from Alois P. Heinz, Dec 12 2010