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A130849 a(n) is half the sum of the terms in the n-th 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

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

FORMULA

a(n) = 1/2 * Sum_{i=0..n-1} d(n-i, 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, (n-1)/2, multDist(n-i, 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

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Last modified June 19 11:17 EDT 2021. Contains 345127 sequences. (Running on oeis4.)