A363895 -id:A363895 - cp's OEIS frontend

A126594 Floor of the average of the prime factors of n with multiplicity.

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

Offset: 2

Cino Hilliard, Jan 06 2007

Cf. A067629 (rounding instead of flooring), A076690.
This is the floor of A123528/A123529.
Without multiplicity we have A363895.
For prime indices instead of factors we have A363943, triangle A363945.
Positions of first appearances are A364037.
The ceiling is A364156.
Positions of 2's are A364157, for prime indices A363949.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, ranks A316413.
A078175 lists numbers with integer mean of prime factors.
Cf. A026905, A326567/A326568, A327473, A327476, A363944, A363948, A363950.

Table[Floor[(Plus@@Times@@@FactorInteger[n])/PrimeOmega[n]], {n, 2, 90}] (* Alonso del Arte, May 21 2012 *)

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) }

a(p^n)=p, p prime, n >= 1. - Philippe Deléham, Nov 23 2008

a(n) = floor(A001414(n)/A001222(n)). - Philippe Deléham, Nov 24 2008

A364037 Least number k such that the floor of the average of the distinct prime factors of k is n, or -1 if no such number exists.

Original entry on oeis.org

2, 3, 14, 5, 22, 7, 39, 34, 38, 11, 46, 13, 115, 58, 62, 17, 155, 19, 111, 82, 86, 23, 94, 141, 235, 106, 159, 29, 118, 31, 183, 305, 134, 201, 142, 37, 219, 365, 158, 41, 166, 43, 415, 178, 267, 47, 623, 194, 291, 202, 206, 53, 214, 218, 327, 226, 339, 59, 791, 61

Offset: 2

Jean-Marc Rebert, Jul 02 2023

nonn

All terms are squarefree. - Jon E. Schoenfield, Jul 02 2023

			a(4) = 14, because 14 = 2 * 7, floor((2 + 7) / 2) = 4, and no lesser number satisfies this.

Cf. A363895.

seq[len_, kmax_] := Module[{s = Table[0, {len}], c = 0, k = 2, i}, While[c < len && k < kmax, i = Floor[Mean[FactorInteger[k][[;; , 1]]]] - 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = k]; k++]; s]; seq[60, 1000] (* Amiram Eldar, Jul 02 2023 *)

f(n) = my(p = factor(n)[, 1]); vecsum(p)\#p; \\ A363895
a(n) = my(k=2); while (f(k) != n, k++); k; \\ Michel Marcus, Jul 02 2023

a(p) = p for prime p. - David A. Corneth, Jul 02 2023

A364156 Ceiling of the mean of the prime factors of n (with multiplicity).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 18 2023

nonn

			The prime factors of 450 are {2,3,3,5,5}, with mean 18/5, so a(450) = 4.

For median of prime indices we have triangle A124944, low A124943.
The round version is A067629.
The floor version is A126594.
A027746 lists prime factors, indices A112798.
A078175 lists numbers with integer mean of prime factors.
A123528/A123529 gives mean of prime factors, A326567/A326568 prime indices.
Cf. A026905, A051293, A316413, A327473, A327476, A327482, A363895, A363943, A363948, A363950, A364037.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[If[n==1,0,Ceiling[Mean[prifacs[n]]]],{n,100}]

Ceiling of A123528(n)/A123529(n).

A364157 Numbers whose rounded-down (floor) mean of prime factors (with multiplicity) is 2.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 18 2023

nonn

			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

Without multiplicity we appear to have A007694.
Prime factors are listed by A027746, indices A112798.
Positions of 2's in A126594, positions of first appearances A364037.
For prime indices and ceiling we have A363950, counted by A026905.
For prime indices we have A363954 (or A363949), counted by A363745.
A078175 lists numbers with integer mean of prime factors.
A123528/A123529 gives mean of prime factors, indices A326567/A326568.
A316413 ranks partitions with integer mean, counted by A067538.
A363895 gives floor of mean of distinct prime factors.
A363943 gives floor of mean of prime indices, ceiling A363944.
Cf. A001222, A056239, A327473, A327476, A360013, A360015, A363488, A363945.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Floor[Mean[prifacs[#]]]==2&]

cp's OEIS Frontend

A126594 Floor of the average of the prime factors of n with multiplicity.

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A364037 Least number k such that the floor of the average of the distinct prime factors of k is n, or -1 if no such number exists.

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A364156 Ceiling of the mean of the prime factors of n (with multiplicity).

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A364157 Numbers whose rounded-down (floor) mean of prime factors (with multiplicity) is 2.

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