

A130849


a(n) is half the sum of the terms in the nth antidiagonal of the table A130836.


1



0, 1, 1, 4, 2, 7, 4, 9, 8, 15, 6, 19, 13, 16, 13, 28, 15, 32, 17, 28, 27, 40, 16, 41, 34, 39, 30, 55, 28, 59, 34, 53, 50, 59, 32, 75, 57, 64, 41, 84, 47, 88, 55, 66, 72, 97, 42, 97, 71, 90, 70, 113, 65, 104, 67, 104, 97, 128, 56, 133, 103, 102, 82, 129, 89, 150, 99, 130, 100
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OFFSET

1,4


LINKS

Table of n, a(n) for n=1..69.


FORMULA

a(n) = 1/2 * Sum_{i=0..n1} d(ni, i+1) where, if m = Sum_{k} p_k^e^k, and n = Sum_{k} p_k^f^k, then d(m, n) = Sum_{i=1..k} abs(e_i  f_i), the multiplicative distance between m and n.


EXAMPLE

d(3, 1) = 1
d(2, 2) = 0
d(1, 3) = 1
So a(3) = 1/2 * (1 + 0 + 1) = 1


MATHEMATICA

MultDistance[m_, n_] := Module[{ mfac = FactorInteger[m], nfac = FactorInteger[ n]}, Plus @@ Map[(If[Length[ # ] == 1, #[[1, 2]], Abs[ #[[1, 2]]  #[[2, 2]]]]) &, Split[ Sort[Flatten[{mfac, nfac}, 1]], (#1[[1]] == #2[[1]]) &]]] DiagSum[n_] := 1/2 Sum[MultDistance[n  i, i + 1], {i, 0, n  1}] Table[DiagSum[j], {j, 1, 1000}]


PROG

(PARI) multDist(m, n) = {if (m==n, 0, my(f=vecsort(concat(factor(m)[, 1], factor(n)[, 1]), , 8)); sum(i=1, #f, abs(valuation(m, f[i])valuation(n, f[i]))))};
a(n)={sum(i=0, (n1)/2, multDist(ni, i+1))}; \\ edited by Michel Marcus, Sep 20 2018


CROSSREFS

Equals half the antidiagonal sums of A130836.
Sequence in context: A197026 A334417 A195779 * A138754 A021963 A131914
Adjacent sequences: A130846 A130847 A130848 * A130850 A130851 A130852


KEYWORD

nonn,easy


AUTHOR

Jacob Woolcutt (woolcutt(AT)gmail.com), Jul 21 2007


EXTENSIONS

Program and corrections by Charles R Greathouse IV, Sep 02 2009
Edited by Michel Marcus, Sep 20 2018


STATUS

approved



