A143152 Inverse Möbius transform of the least prime factor of n: A051731 * A020639.
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, 10, 16, 30, 20, 32, 11, 18, 22, 18, 19, 38, 24, 20, 18, 42, 22, 44, 20, 18, 28, 48, 20, 15, 17, 24, 22, 54, 18, 22, 20, 26, 34, 60, 28, 62, 36, 20, 13, 24, 26, 68, 26, 30, 26, 72, 25, 74
Offset: 1
Keywords
Examples
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,...). 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.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..20000
Programs
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Maple
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
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PARI
A020639(n) = if(1==n,n,(factor(n)[1, 1])); A143152(n) = sumdiv(n,d,A020639(d)); \\ Antti Karttunen, Nov 12 2021
Formula
a(p) = (p+1) for prime p.
Row sums of triangle A143151.
a(n) = Sum_{d|n} A020639(d). - Antti Karttunen, Nov 12 2021
Extensions
Extended beyond a(14) by R. J. Mathar, Aug 11 2008
Name amended by Antti Karttunen, Nov 12 2021