A123528 -id:A123528

A326567

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

+10
120

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=2..82.

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn,frac

AUTHOR

Gus Wiseman, Jul 13 2019

STATUS

approved

A326568

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

+10
120

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..86.

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn,frac

AUTHOR

Gus Wiseman, Jul 13 2019

STATUS

approved

A078174

Numbers with an integer arithmetic mean of distinct prime factors.

+10
34

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

(Haskell)

a078174 n = a078174_list !! (n-1)

a078174_list = filter (\x -> a008472 x `mod` a001221 x == 0) [2..]

-- Reinhard Zumkeller, Jun 01 2013

CROSSREFS

Union of A000961 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.

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Nov 20 2002

STATUS

approved

A326619

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

+10
27

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=2..82.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn,frac

AUTHOR

Gus Wiseman, Jul 14 2019

STATUS

approved

A326620

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

+10
27

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=2..88.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn,frac

AUTHOR

Gus Wiseman, Jul 14 2019

STATUS

approved

A123529

Denominator of average of prime factors of n.

+10
16

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,5

COMMENTS

Prime factors counted with multiplicity. - Harvey P. Dale, Jun 20 2013

Positions of 1's are A078175. a(n) is a divisor of Omega(n) = A001222(n). The average of prime indices (as opposed to prime factors) of n is A326567(n)/A326568(n). - Gus Wiseman, Jul 18 2019

LINKS

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

MATHEMATICA

Table[Denominator[Mean[Flatten[Table[#[[1]], {#[[2]]}]&/@ FactorInteger[ n]]]], {n, 110}] (* Harvey P. Dale, Jun 20 2013 *)

CROSSREFS

See A123528 for more formulas and references.

Cf. A078174, A078175, A112798, A316413, A326567/A326568, A326619/A326620, A326621.

KEYWORD

frac,nonn

AUTHOR

Franklin T. Adams-Watters, Oct 02 2006

STATUS

approved

A360459

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

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..73.

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 14 2023

STATUS

approved

A323171

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

+10
12

2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 4, 2, 17, 5, 19, 7, 5, 13, 23, 5, 5, 15, 3, 9, 29, 10, 31, 2, 7, 19, 6, 5, 37, 21, 8, 7, 41, 4, 43, 13, 4, 25, 47, 5, 7, 7, 10, 15, 53, 5, 8, 9, 11, 31, 59, 10, 61, 33, 5, 2, 9, 16, 67, 19, 13, 14, 71, 5, 73, 39, 4, 21, 9, 6, 79, 7, 3, 43, 83, 4, 11, 45, 16, 13, 89, 10, 10, 25, 17, 49, 12, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

LINKS

Antti Karttunen, Table of n, a(n) for n = 2..20000

EXAMPLE

Fractions begins with 2, 3, 2, 5, 5/2, 7, 2, 3, 7/2, 11, 5/2, 13, ...

MATHEMATICA

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

PROG

(PARI)

A008472(n) = vecsum(factor(n)[, 1]); \\ From A008472

A323171(n) = (numerator(A008472(n)/omega(n)));

CROSSREFS

Cf. A323172 (denominators).

Cf. A001221, A008472, A123528.

KEYWORD

nonn,easy,frac

AUTHOR

Antti Karttunen, Jan 05 2019

STATUS

approved

A359904

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..27.

EXAMPLE

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}

MATHEMATICA

prifac[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];

prisig[n_]:=If[n==1, {}, Last/@FactorInteger[n]];

Select[Range[1000], Mean[prifac[#]]==Mean[prisig[#]]&]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 25 2023

STATUS

approved

A126594

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

LINKS

Table of n, a(n) for n=2..91.

FORMULA

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

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

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.

KEYWORD

easy,nonn

AUTHOR

Cino Hilliard, Jan 06 2007

STATUS

approved