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A143152 Inverse Möbius transform of the least prime factor of n: A051731 * A020639.

%I A143152 #12 Nov 12 2021 22:33:13
%S A143152 1,3,4,5,6,8,8,7,7,10,12,12,14,12,12,9,18,13,20,14,14,16,24,16,11,18,
%T A143152 10,16,30,20,32,11,18,22,18,19,38,24,20,18,42,22,44,20,18,28,48,20,15,
%U A143152 17,24,22,54,18,22,20,26,34,60,28,62,36,20,13,24,26,68,26,30,26,72,25,74
%N A143152 Inverse Möbius transform of the least prime factor of n: A051731 * A020639.
%H A143152 Antti Karttunen, <a href="/A143152/b143152.txt">Table of n, a(n) for n = 1..20000</a>
%F A143152 a(p) = (p+1) for prime p.
%F A143152 Inverse Mobius transform of A020639, where A020639(n) = Lpf(n).
%F A143152 Row sums of triangle A143151.
%F A143152 a(n) = Sum_{d|n} A020639(d). - _Antti Karttunen_, Nov 12 2021
%e A143152 a(4) = 5 = (1, 1, 0, 1) dot (1, 2, 3, 2) = (1 + 2 + 0 + 2), where (1, 1, 0, 1) = row 4 of triangle A051731 and A010639 = (1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11,...).
%e A143152 Since a(n) = sum of least prime factors of the divisors of n, the divisors of 12 are recorded in triangle row 12 of A127093: (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12). Lpf of these terms = row 12 of triangle A143151: (1, 2, 3, 2, 0, 2, 0, 0, 0, 0, 0, 2); sum = 12.
%p A143152 read transforms : A020639 := proc(n) local i ; if n = 1 then 1; else for i from 1 do if n mod ithprime(i) = 0 then RETURN(ithprime(i)) ; fi; od: fi; end: a020639 := [seq(A020639(n),n=1..100)] : a143152 := MOBIUSi(a020639) : for i from 1 to nops(a143152) do printf("%d,",op(i,a143152)) ; od: # _R. J. Mathar_, Aug 11 2008
%o A143152 (PARI)
%o A143152 A020639(n) = if(1==n,n,(factor(n)[1, 1]));
%o A143152 A143152(n) = sumdiv(n,d,A020639(d)); \\ _Antti Karttunen_, Nov 12 2021
%Y A143152 Cf. A051731, A020639, A127093, A143151.
%K A143152 nonn
%O A143152 1,2
%A A143152 _Gary W. Adamson_, Jul 27 2008
%E A143152 Extended beyond a(14) by _R. J. Mathar_, Aug 11 2008
%E A143152 Name amended by _Antti Karttunen_, Nov 12 2021