cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 13 2019

Keywords

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.

Examples

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

Links

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, A326620.

Programs

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

Extensions

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

Views

Author

Gus Wiseman, Jul 13 2019

Keywords

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.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Reinhard Zumkeller, Nov 20 2002

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

  • Haskell
    a078174 n = a078174_list !! (n-1)
a078174_list = filter (\x -> a008472 x `mod` a001221 x == 0) [2..]
-- Reinhard Zumkeller, Jun 01 2013
  • Mathematica
    Select[Range[2,200],IntegerQ[Mean[Transpose[FactorInteger[#]][[1]]]]&] (* Harvey P. Dale, Apr 18 2016 *)
  • PARI
    is(n)=my(f=factor(n)[,1]);sum(i=1,#f,f[i])%#f==0 \\ Charles R Greathouse IV, May 30 2013

Formula

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

Views

Author

Gus Wiseman, Jul 14 2019

Keywords

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.

Examples

			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.

Links

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.

Programs

  • Mathematica
    Table[Denominator[Mean[PrimePi/@First/@FactorInteger[n]]],{n,2,100}]
  • PARI
    A326620(n) = if(1==n,0,denominator(vecsum(apply(primepi,factor(n)[,1]))/omega(n))); \\ Antti Karttunen, Jan 28 2025

Extensions

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

Views

Author

Gus Wiseman, Jul 14 2019

Keywords

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.

Examples

			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.

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.

Programs

  • Mathematica
    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

Views

Author

Franklin T. Adams-Watters, Oct 02 2006

Keywords

Examples

			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.

Links

Crossrefs

Cf. A123529, A067629, A001414, A001222.

Programs

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

Formula

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

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

Views

Author

Cino Hilliard, Jan 06 2007

Keywords

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.

Programs

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

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

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

Views

Author

Gus Wiseman, Jul 18 2023

Keywords

Examples

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

Crossrefs

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.

Programs

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

Formula

Ceiling of A123528(n)/A123529(n).
Showing 1-8 of 8 results.

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