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A143152
Inverse Möbius transform of the least prime factor of n: A051731 * A020639.
2
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
1,2
LINKS
FORMULA
a(p) = (p+1) for prime p.
Inverse Mobius transform of A020639, where A020639(n) = Lpf(n).
Row sums of triangle A143151.
a(n) = Sum_{d|n} A020639(d). - Antti Karttunen, Nov 12 2021
EXAMPLE
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.
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
PROG
(PARI)
A020639(n) = if(1==n, n, (factor(n)[1, 1]));
A143152(n) = sumdiv(n, d, A020639(d)); \\ Antti Karttunen, Nov 12 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jul 27 2008
EXTENSIONS
Extended beyond a(14) by R. J. Mathar, Aug 11 2008
Name amended by Antti Karttunen, Nov 12 2021
STATUS
approved