A123529 -id:A123529 - cp's OEIS frontend

A326568 Denominator of the average of the multiset of prime indices of n.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 3, 1, 3, 3, 1, 1, 5, 1, 3, 2, 3, 1, 4, 1, 4, 1, 2, 1, 4, 1, 1, 3, 1, 2, 3, 1, 1, 2, 3, 1, 5, 1, 2, 3, 3, 2, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 2, 6, 1, 1, 1, 1, 1, 3, 1, 4, 1, 2, 1, 5

Offset: 2

Gus Wiseman, Jul 13 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 12 are {1,1,2}, with average 4/3, so a(12) = 3.

Antti Karttunen, Table of n, a(n) for n = 2..65537
Index entries for sequences related to prime indices in the factorization of n.

a(n) is a divisor of Omega(n) = A001222(n).
Positions of 1's are A316413.
Cf. A001414, A056239, A067629, A112798, A123528/A123529, A289508, A289509, A290103, A316428, A326567, A326620.

Table[Denominator[Sum[q[[2]]*PrimePi[q[[1]]],{q,FactorInteger[n]}]/PrimeOmega[n]],{n,2,100}]

A326568(n) = { my(f=factor(n)); denominator(sum(i=1,#f~,f[i,2]*primepi(f[i,1]))/bigomega(n)); }; \\ Antti Karttunen, Jan 28 2025

Starting offset corrected from 0 to 2 and data section extended to a(108) by Antti Karttunen, Jan 28 2025

A326567 Numerator of the average of the multiset of prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 3, 4, 1, 2, 2, 5, 4, 6, 5, 5, 1, 7, 5, 8, 5, 3, 3, 9, 5, 3, 7, 2, 2, 10, 2, 11, 1, 7, 4, 7, 3, 12, 9, 4, 3, 13, 7, 14, 7, 7, 5, 15, 6, 4, 7, 9, 8, 16, 7, 4, 7, 5, 11, 17, 7, 18, 6, 8, 1, 9, 8, 19, 3, 11, 8, 20, 7, 21, 13, 8, 10, 9, 3, 22, 7, 2, 7

Offset: 2

Gus Wiseman, Jul 13 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 12 are {1,1,2}, with average 4/3, so a(12) = 4.

Cf. A001222, A001414, A056239, A067629, A112798, A123528/A123529, A289508, A289509, A290103, A316413, A326568.

Table[Numerator[Sum[q[[2]]*PrimePi[q[[1]]],{q,FactorInteger[n]}]/PrimeOmega[n]],{n,2,100}]

A078174 Numbers with an integer arithmetic mean of distinct prime factors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 110, 111, 113, 114, 115

Offset: 1

Reinhard Zumkeller, Nov 20 2002

nonn

A008472(a(n)) == 0 modulo A001221(a(n)).

			42=2*3*7: (2+3+7)/3=4, therefore 42 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Union of A246655 and A070005.
Positions of 1's in A323172.
The version counting multiplicity is A078175.
The version for prime indices is A326621.
The average of the set of distinct prime factors is A323171/A323172.
The average of the multiset of prime factors is A123528/A123529.
Cf. A001221, A001414, A067629, A112798, A316413, A316428, A326567/A326568, A328966.

a078174 n = a078174_list !! (n-1)
a078174_list = filter (\x -> a008472 x `mod` a001221 x == 0) [2..]
-- Reinhard Zumkeller, Jun 01 2013

Select[Range[2,200],IntegerQ[Mean[Transpose[FactorInteger[#]][[1]]]]&] (* Harvey P. Dale, Apr 18 2016 *)

is(n)=my(f=factor(n)[,1]);sum(i=1,#f,f[i])%#f==0 \\ Charles R Greathouse IV, May 30 2013

a(n) << n log n/(log log n)^k for any k. - Charles R Greathouse IV, May 30 2013

A326620 Denominator of the average of the set of distinct prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 3, 1, 2, 1

Offset: 2

Gus Wiseman, Jul 14 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The distinct prime indices of 12 are {1,2}, with average 3/2, so a(12) = 2.
The sequence of fractions begins: 1, 2, 1, 3, 3/2, 4, 1, 2, 2, 5, 3/2, 6, 5/2, 5/2, 1, 7, 3/2, 8, 2, 3, 3, 9, 3/2, 3, 7/2, 2, 5/2, 10, 2.

Antti Karttunen, Table of n, a(n) for n = 2..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of 1's are A326621.
The average of the multiset of prime indices is A326567/A326568.
The average of the multiset of prime factors is A123528/A123529.
The average of the set of distinct prime indices is A326619/A326620.
The average of the set of distinct prime factors is A323171/A323172.
Cf. A001221, A067629, A078174, A078175, A112798, A316413.

Table[Denominator[Mean[PrimePi/@First/@FactorInteger[n]]],{n,2,100}]

A326620(n) = if(1==n,0,denominator(vecsum(apply(primepi,factor(n)[,1]))/omega(n))); \\ Antti Karttunen, Jan 28 2025

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

A326619 Numerator of the average of the set of distinct prime indices of n.

Original entry on oeis.org

1, 2, 1, 3, 3, 4, 1, 2, 2, 5, 3, 6, 5, 5, 1, 7, 3, 8, 2, 3, 3, 9, 3, 3, 7, 2, 5, 10, 2, 11, 1, 7, 4, 7, 3, 12, 9, 4, 2, 13, 7, 14, 3, 5, 5, 15, 3, 4, 2, 9, 7, 16, 3, 4, 5, 5, 11, 17, 2, 18, 6, 3, 1, 9, 8, 19, 4, 11, 8, 20, 3, 21, 13, 5, 9, 9, 3, 22, 2, 2, 7

Offset: 2

Gus Wiseman, Jul 14 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The distinct prime indices of 12 are {1,2}, with average 3/2, so a(12) = 3.
The sequence of fractions begins: 1, 2, 1, 3, 3/2, 4, 1, 2, 2, 5, 3/2, 6, 5/2, 5/2, 1, 7, 3/2, 8, 2, 3, 3, 9, 3/2, 3, 7/2, 2, 5/2, 10, 2.

The average of the multiset of prime indices is A326567/A326568.
The average of the multiset of prime factors is A123528/A123529.
The average of the set of distinct prime indices is A326619/A326620.
The average of the set of distinct prime factors is A323171/A323172.
Cf. A001221, A067629, A078174, A078175, A112798, A316413, A326621.

Table[Numerator[Mean[PrimePi/@First/@FactorInteger[n]]],{n,2,100}]

A123528 Numerator of average of prime factors of n.

Original entry on oeis.org

2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 7, 13, 9, 4, 2, 17, 8, 19, 3, 5, 13, 23, 9, 5, 15, 3, 11, 29, 10, 31, 2, 7, 19, 6, 5, 37, 21, 8, 11, 41, 4, 43, 5, 11, 25, 47, 11, 7, 4, 10, 17, 53, 11, 8, 13, 11, 31, 59, 3, 61, 33, 13, 2, 9, 16, 67, 7, 13, 14, 71, 12, 73, 39, 13, 23, 9, 6, 79, 13, 3, 43, 83, 7

Offset: 2

Franklin T. Adams-Watters, Oct 02 2006

			12 = 2 * 2 * 3, so a(12) = (2 + 2 + 3) / 3 = 7/3.
Sequence of fractions starts 2/1, 3/1, 2/1, 5/1, 5/2, 7/1, 2/1, 3/1, 7/2, 11/1, 7/3.

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A123529, A067629, A001414, A001222.

f[n_] := Plus @@ Times @@@ FactorInteger@n; f[1] = 0; Numerator[ Table[ f[n]/PrimeOmega[n], {n, 2, 50}]] (* G. C. Greubel, Oct 14 2017 *)

a(n) = sopfr(n) / bigomega(n) = A001414(n) / A001222(n).

A360459 Two times the median of the multiset of prime factors of n; a(1) = 2.

Original entry on oeis.org

2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 4, 26, 9, 8, 4, 34, 6, 38, 4, 10, 13, 46, 4, 10, 15, 6, 4, 58, 6, 62, 4, 14, 19, 12, 5, 74, 21, 16, 4, 82, 6, 86, 4, 6, 25, 94, 4, 14, 10, 20, 4, 106, 6, 16, 4, 22, 31, 118, 5, 122, 33, 6, 4, 18, 6, 134, 4, 26, 10, 142, 4, 146

Offset: 1

Gus Wiseman, Feb 14 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

			The prime factors of 60 are {2,2,3,5}, with median 5/2, so a(60) = 5.

The union is 2 followed by A014091, complement of A014092.
The prime factors themselves are listed by A027746, distinct A027748.
The version for divisors is A063655.
Positions of odd terms are A072978 (except 1).
For mean instead of twice median: A123528/A123529, distinct A323171/A323172.
Positions of even terms are A359913 (and 1).
The version for prime indices is A360005.
The version for distinct prime indices is A360457.
The version for distinct prime factors is A360458.
The version for prime multiplicities is A360460.
The version for 0-prepended differences is A360555.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A027336, A078174, A316413, A359907, A359908, A360006, A360007, A360248, A360552.

Table[2*Median[Join@@ConstantArray@@@FactorInteger[n]],{n,100}]

A323172 Denominator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2

Offset: 2

Antti Karttunen, Jan 05 2019

Antti Karttunen, Table of n, a(n) for n = 2..65539

Cf. A323171 (numerators).

Cf. A001221, A008472, A123529, A322591.

a[n_] := Denominator[Mean[FactorInteger[n][[;; , 1]]]]; Array[a, 100, 2] (* Amiram Eldar, Sep 17 2024 *)

A008472(n) = vecsum(factor(n)[, 1]); \\ From A008472
A323172(n) = (denominator(A008472(n)/omega(n)));

A359904 Numbers whose prime factors and prime signature have the same mean.

Original entry on oeis.org

1, 4, 27, 400, 3125, 9072, 10800, 14580, 24057, 35721, 50625, 73984, 117760, 134400, 158976, 181440, 191488, 389376, 452709, 544000, 583680, 664848, 731136, 774400, 823543, 878592, 965888

Offset: 1

Gus Wiseman, Jan 25 2023

nonn

The multiset of prime factors of n is row n of A027746.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime factors begin:
      1: {}
      4: {2,2}
     27: {3,3,3}
    400: {2,2,2,2,5,5}
   3125: {5,5,5,5,5}
   9072: {2,2,2,2,3,3,3,3,7}
  10800: {2,2,2,2,3,3,3,5,5}
  14580: {2,2,3,3,3,3,3,3,5}
  24057: {3,3,3,3,3,3,3,11}
  35721: {3,3,3,3,3,3,7,7}
  50625: {3,3,3,3,5,5,5,5}
  73984: {2,2,2,2,2,2,2,2,17,17}

The prime factors are A027746, mean A123528/A123529.
The prime signature is A124010, mean A088529/A088530.
For prime indices instead of factors we have A359903.
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A078175 = numbers whose prime factors have integer mean, indices A316413.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A360005 gives median of prime indices (times two).
Cf. A240219, A326622, A327473, A359905, A359908, A359913, A360008, A360068.

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Select[Range[1000],Mean[prifac[#]]==Mean[prisig[#]]&]

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

Original entry on oeis.org

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

cp's OEIS Frontend

A326568 Denominator of the average of the multiset of prime indices of n.

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A326567 Numerator of the average of the multiset of prime indices of n.

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A078174 Numbers with an integer arithmetic mean of distinct prime factors.

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A326620 Denominator of the average of the set of distinct prime indices of n.

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A326619 Numerator of the average of the set of distinct prime indices of n.

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A123528 Numerator of average of prime factors of n.

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A360459 Two times the median of the multiset of prime factors of n; a(1) = 2.

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A323172 Denominator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

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