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Floor of the average of the prime factors of n with multiplicity.
+10
4
2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 4, 4, 2, 17, 2, 19, 3, 5, 6, 23, 2, 5, 7, 3, 3, 29, 3, 31, 2, 7, 9, 6, 2, 37, 10, 8, 2, 41, 4, 43, 5, 3, 12, 47, 2, 7, 4, 10, 5, 53, 2, 8, 3, 11, 15, 59, 3, 61, 16, 4, 2, 9, 5, 67, 7, 13, 4, 71, 2, 73, 19, 4, 7, 9, 6, 79, 2, 3, 21, 83, 3, 11, 22, 16, 4, 89, 3, 10
MATHEMATICA
Table[Floor[(Plus@@Times@@@FactorInteger[n])/PrimeOmega[n]], {n, 2, 90}] (* Alonso del Arte, May 21 2012 *)
PROG
(PARI) avg(n) = { local(x, j, ln) for(x=2, n, a=ifactor(x); ln=length(a); print1(floor(sum(j=1, ln, a[j])/ln)", ")) } ifactor(n) = \The vector of the prime factors of n with multiplicity. { local(f, j, k, flist); flist=[]; f=Vec(factor(n)); for(j=1, length(f[1]), for(k = 1, f[2][j], flist = concat(flist, f[1][j]) ); ); return(flist) }
CROSSREFS
Without multiplicity we have A363895. For prime indices instead of factors we have A363943, triangle A363945. Positions of first appearances are A364037. A078175 lists numbers with integer mean of prime factors.
Ceiling of the mean of the prime factors of n (with multiplicity).
+10
1
0, 2, 3, 2, 5, 3, 7, 2, 3, 4, 11, 3, 13, 5, 4, 2, 17, 3, 19, 3, 5, 7, 23, 3, 5, 8, 3, 4, 29, 4, 31, 2, 7, 10, 6, 3, 37, 11, 8, 3, 41, 4, 43, 5, 4, 13, 47, 3, 7, 4, 10, 6, 53, 3, 8, 4, 11, 16, 59, 3, 61, 17, 5, 2, 9, 6, 67, 7, 13, 5, 71, 3, 73, 20, 5, 8, 9, 6
EXAMPLE
The prime factors of 450 are {2,3,3,5,5}, with mean 18/5, so a(450) = 4.
MATHEMATICA
prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
Table[If[n==1, 0, Ceiling[Mean[prifacs[n]]]], {n, 100}]
CROSSREFS
A078175 lists numbers with integer mean of prime factors.
Cf. A026905, A051293, A316413, A327473, A327476, A327482, A363895, A363943, A363948, A363950, A364037.
Numbers whose rounded-down (floor) mean of prime factors (with multiplicity) is 2.
+10
1
2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 40, 48, 54, 64, 72, 80, 96, 108, 120, 128, 144, 160, 162, 192, 216, 224, 240, 256, 288, 320, 324, 360, 384, 432, 448, 480, 486, 512, 576, 640, 648, 672, 720, 768, 800, 864, 896, 960, 972, 1024, 1080, 1152, 1280, 1296, 1344
EXAMPLE
The terms together with their prime factors begin:
2 = 2
4 = 2*2
6 = 2*3
8 = 2*2*2
12 = 2*2*3
16 = 2*2*2*2
18 = 2*3*3
24 = 2*2*2*3
32 = 2*2*2*2*2
36 = 2*2*3*3
40 = 2*2*2*5
48 = 2*2*2*2*3
54 = 2*3*3*3
64 = 2*2*2*2*2*2
72 = 2*2*2*3*3
80 = 2*2*2*2*5
96 = 2*2*2*2*2*3
MATHEMATICA
prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100], Floor[Mean[prifacs[#]]]==2&]
CROSSREFS
Without multiplicity we appear to have A007694. A078175 lists numbers with integer mean of prime factors.
A363895 gives floor of mean of distinct prime factors.
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