A054481 -id:A054481 - cp's OEIS frontend

A054482 Denominator of lowest terms fraction from division of a highly composite number (1) by its predecessor.

1, 1, 2, 1, 1, 2, 3, 4, 1, 2, 3, 2, 1, 6, 2, 3, 2, 1, 2, 3, 2, 3, 4, 10, 11, 9, 10, 2, 3, 2, 3, 4, 5, 2, 9, 5, 12, 2, 3, 2, 3, 4, 5, 2, 9, 5, 4, 3, 5, 4, 2, 15, 17, 12, 5, 2, 9, 5, 4, 3, 5, 4, 2, 15, 19, 2, 15, 2, 19, 18, 5, 4, 3, 5, 4, 2, 3, 2, 3, 2, 6, 35, 92, 21, 23, 35, 92, 21, 2, 23, 21, 2

Offset: 2

Henry Bottomley, Mar 31 2000

Successive fractions are 2/1,2/1,3/2,2/1,2/1,3/2,4/3,5/4,2/1,3/2,4/3,3/2,2/1 a(n) is frequently A054483(n)-1, though not for example for n=26.

			a(7)=2 since A002182(7)=36, A002182(6)=24 and 36/24=3/2 in lowest terms.

Michael De Vlieger, Table of n, a(n) for n = 2..10000
Michael De Vlieger, Logarithmic scatterplot of A054483(n)/a(n), n = 2..4096.
Michael De Vlieger, Logarithmic scatterplot of A054483(n)/a(n), n = 2..779694, showing ratios for the entire Flammenkamp dataset of highly composite numbers.

Cf. A002182, A054483.

s = Import["https://oeis.org/A002182/b002182.txt", "Data"][[All, -1]]; Array[Denominator[s[[# + 1]]/s[[#]]] &, 120] (* Michael De Vlieger, Oct 11 2023 *)

a(n) = A002182(n-1)/A054481(n).

More terms from Jud McCranie, Jun 11 2005

A054483 Numerator of lowest terms fraction from division of a highly composite number (1) by its predecessor.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 4, 5, 2, 3, 4, 3, 2, 7, 3, 4, 3, 2, 3, 4, 3, 4, 5, 11, 18, 10, 11, 3, 4, 3, 4, 5, 6, 3, 10, 6, 13, 3, 4, 3, 4, 5, 6, 3, 10, 6, 5, 4, 6, 5, 3, 17, 20, 17, 6, 3, 10, 6, 5, 4, 6, 5, 3, 19, 20, 3, 19, 3, 20, 19, 6, 5, 4, 6, 5, 3, 4, 3, 4, 3, 7, 46, 105, 23, 28, 46, 105, 23, 3, 28, 23, 3

Offset: 2

Henry Bottomley, Mar 31 2000

Successive fractions are 2/1,2/1,3/2,2/1,2/1,3/2,4/3,5/4,2/1,3/2,4/3,3/2,2/1 a(n) is frequently A054482(n)+1, though not for example for n=26.

			a(7)=3 since A002182(7)=36, A002182(6)=24 and 36/24=3/2 in lowest terms.

Michael De Vlieger, Table of n, a(n) for n = 2..10000
Michael De Vlieger, Logarithmic scatterplot of a(n)/A054482(n), n = 2..4096.

Cf. A002182, A054482.

s = Import["https://oeis.org/A002182/b002182.txt", "Data"][[All, -1]]; Array[Numerator[s[[# + 1]]/s[[#]]] &, 120] (* Michael De Vlieger, Oct 11 2023 *)

a(n) = A002182(n)/A054481(n).

More terms from Jud McCranie, Jun 11 2005

A262501 First differences of A002182 (highly composite numbers, definition 1).

Original entry on oeis.org

1, 2, 2, 6, 12, 12, 12, 12, 60, 60, 60, 120, 360, 120, 420, 420, 840, 2520, 2520, 2520, 5040, 5040, 5040, 2520, 17640, 5040, 5040, 27720, 27720, 55440, 55440, 55440, 55440, 166320, 55440, 110880, 55440, 360360, 360360, 720720, 720720, 720720, 720720, 2162160, 720720, 1441440, 2162160, 3603600, 2882880, 4324320, 10810800, 4324320, 6486480, 18018000

Offset: 1

Antti Karttunen, Sep 24 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..999 (computed from the b-file of A002182 provided by _T. D. Noe_.)

Cf. A002182, A261100.

Cf. also A054481 (gives the gcds of successive terms of A002182, from which this sequence differs for the first time at n=25).

(define (A262501 n) (- (A002182 (+ 1 n)) (A002182 n)))

a(n) = A002182(n+1) - A002182(n).

A328520 GCD of terms in A002182 that have n prime factors counted with multiplicity.

Original entry on oeis.org

1, 2, 2, 12, 12, 12, 60, 120, 2520, 2520, 2520, 55440, 55440, 720720, 720720, 12252240, 36756720, 698377680, 3491888400, 80313433200, 160626866400, 160626866400, 9316358251200, 288807105787200, 2021649740510400, 74801040398884800, 74801040398884800, 3066842656354276800, 131874234223233902400

Offset: 0

David A. Corneth, Jan 04 2020

nonn

If for every term t > 1 in A002182 there exists a term in A002182 of the form t/p for some prime p|t (Cf. A328523) then (1): a(n+1) is a multiple of a(n) for n >= 1 and (2): this sequence can be used to find terms to A199337 and A330737.
Proof of (1): Let Generation(n) be the terms in A002182 with n prime factors counted with multiplicity. Then a(n) = GCD(Generation(n)). As each term in Generation(n) is of the form a(n) * t for some t and each term in Generation(n + 1) is p * g for some g in Generation(n), a(n + 1) is a multiple of a(n).
Proof of (2): As a(n + 1) is a multiple of a(n) we have that if m | a(n), we also have m | a(n + k), k >= 0 hence the largest number not divisible by m can have at most n - 1 prime factors counted with multiplicity.
Given Generation(n) we can find all candidates for Generation(n + 1) and from there on find terms to A002182, A199337 and A330737.

			The terms in A002182 with n = 4 prime divisors counted with multiplicity are 24, 36 and 60. Their GCD is 12 hence a(4) = 12.
Furthermore, If for every term t > 1 in A002182 there exists a term in A002182 of the form t/p for some prime p|t then we have that each term with more than 4 prime divisors counted with multiplicity is a multiple of at least one of 24, 36 or 60 hence is divisible by 12.

Cf. A001222, A002182, A199337, A328523, A330737

Cf. also A054481.

(* First, load the function f at A025487, then: *)
Block[{s = Union@ Flatten@ f@ 20, t}, t = DivisorSigma[0, s]; s = Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]]; t = PrimeOmega[s]; Drop[Array[GCD @@ s[[Position[t, #][[All, 1]] ]] &, Max@ t + 1, 0], -3] ] (* Michael De Vlieger, Jan 12 2020 *)

A053878 Difference between numerator and denominator of lowest terms fraction from division of a highly composite number (1) by its predecessor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 13, 2, 5, 11, 13, 2, 1, 5, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1

Offset: 2

Henry Bottomley, Mar 31 2000

nonn

Successive fractions are 2/1, 2/1, 3/2, 2/1, 2/1, 3/2, 4/3, 5/4, 2/1, 3/2, 4/3, 3/2, 2/1.

This sequence is not multiplicative.

			a(7)=1 since A002182(7)=36, A002182(6)=24, 36/24=3/2 in lowest terms and 3-2=1.

Cf. A002182.

a(n) = A054483(n)-A054482(n) =(A002182(n)-A002182(n-1))/A054481(n).

More terms from Jud McCranie, Jun 09 2005

cp's OEIS Frontend

A054482 Denominator of lowest terms fraction from division of a highly composite number (1) by its predecessor.

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A054483 Numerator of lowest terms fraction from division of a highly composite number (1) by its predecessor.

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A262501 First differences of A002182 (highly composite numbers, definition 1).

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A328520 GCD of terms in A002182 that have n prime factors counted with multiplicity.

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A053878 Difference between numerator and denominator of lowest terms fraction from division of a highly composite number (1) by its predecessor.

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