A275812 -id:A275812 - cp's OEIS frontend

A060129 Number of moved (non-fixed) elements in the permutation with rank number n in lists A060117 (or in A060118), i.e., the sum of the lengths of all cycles larger than one in that permutation.

0, 2, 2, 3, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 3, 4, 4, 4, 2, 3, 4, 4, 3, 4, 2, 4, 4, 5, 4, 5, 3, 5, 4, 5, 4, 5, 3, 4, 4, 5, 5, 5, 3, 4, 5, 5, 4, 5, 2, 4, 3, 4, 3, 4, 3, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 4, 5, 2, 3, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 3, 4, 4, 5, 5, 5, 4, 4, 5, 5, 5, 5, 2, 3, 4, 4, 3, 4, 4, 5, 5, 5, 4, 5, 4, 4, 5, 5, 5, 5, 3, 4, 5, 5, 4, 5, 2

Offset: 0

Antti Karttunen, Mar 05 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. also A000142, A055093, A060117, A060118, A060120, A060128, A060130, A084558, A275851.

a(n) = A060128(n) + A060130(n).
From Antti Karttunen, Aug 11 2016: (Start)
a(n) = A275812(A275725(n)).
a(n) = 1 + A084558(n) - A275851(n).
Other identities. For all n >= 0:
a(n) = A055093(A060120(n)).
a(A000142(n)) = 2.
(End)

A055093 Number of moved (non-fixed) elements in each permutation given in reversed colexicographic ordering A055089, i.e., the sum of their cycle lengths (excluding the 1-cycles, i.e., fixed elements).

Original entry on oeis.org

0, 2, 2, 3, 3, 2, 2, 4, 3, 4, 4, 3, 3, 4, 2, 3, 4, 4, 4, 3, 3, 2, 4, 4, 2, 4, 4, 5, 5, 4, 3, 5, 4, 5, 5, 4, 4, 5, 3, 4, 5, 5, 5, 4, 4, 3, 5, 5, 3, 5, 4, 5, 5, 4, 2, 4, 3, 4, 4, 3, 4, 5, 4, 5, 5, 5, 5, 4, 5, 4, 5, 5, 4, 5, 3, 4, 5, 5, 3, 4, 2, 3, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 4, 4, 5, 4, 4, 3, 5, 5, 4, 3, 3

Offset: 0

Antti Karttunen, Apr 04 2000

nonn

Also number of displacements for permutations in lexicographic order. - Joerg Arndt, Jan 22 2024

Antti Karttunen, Table of n, a(n) for n = 0..40320
Tilman Piesk, Permutations and partitions in the OEIS (Wikiversity)
Index entries for sequences related to factorial base representation

Cf. A055089, A055090, A055091, A055092, A060129, A275812, A290095.

A055093(n) = count_nonfixed(convert(PermRevLexUnrank(j), 'disjcyc')).
count_nonfixed := l -> convert(map(nops,l), `+`);
# Procedure PermRevLexUnrank given in A055089.

a(n) = A055090(n) + A055091(n).

a(n) = A275812(A290095(n)) = A060129(A060126(n)). - Antti Karttunen, Dec 30 2017

Entry revised by Antti Karttunen, Dec 30 2017

A290095 a(n) = A275725(A060126(n)); prime factorization encodings of cycle-polynomials computed for finite permutations listed in reversed colexicographic ordering.

Original entry on oeis.org

2, 4, 18, 8, 8, 12, 150, 100, 54, 16, 16, 24, 54, 16, 90, 40, 36, 16, 16, 24, 40, 60, 16, 36, 1470, 980, 882, 392, 392, 588, 750, 500, 162, 32, 32, 48, 162, 32, 270, 80, 108, 32, 32, 48, 80, 120, 32, 72, 750, 500, 162, 32, 32, 48, 1050, 700, 378, 112, 112, 168, 450, 200, 162, 32, 32, 72, 200, 300, 32, 48, 108, 32, 162, 32, 270, 80, 108, 32, 378, 112, 630, 280

Offset: 0

Antti Karttunen, Aug 17 2017

nonn

In this context "cycle-polynomials" are single-variable polynomials where the coefficients (encoded with the exponents of prime factorization of n) are equal to the lengths of cycles in the permutation listed with index n in table A055089 (A195663). See the examples.

			Consider the first eight permutations (indices 0-7) listed in A055089:
  1 [Only the first 1-cycle explicitly listed thus a(0) = 2^1 = 2]
  2,1 [One transposition (2-cycle) in beginning, thus a(1) = 2^2 = 4]
  1,3,2 [One fixed element in beginning, then transposition, thus a(2) = 2^1 * 3^2 = 18]
  3,1,2 [One 3-cycle, thus a(3) = 2^3 = 8]
  2,3,1 [One 3-cycle, thus a(4) = 2^3 = 8]
  3,2,1 [One transposition jumping over a fixed element, a(5) = 2^2 * 3^1 = 12]
  1,2,4,3 [Two 1-cycles, then a 2-cycle, thus a(6) = 2^1 * 3^1 * 5^2 = 150].
  2,1,4,3 [Two 2-cycles, not crossed, thus a(7) = 2^2 * 5^2 = 100].

Antti Karttunen, Table of n, a(n) for n = 0..40319
Index entries for sequences related to factorial base representation

Cf. A055090, A055091, A055092, A055093, A060126, A290096, A290097.

Cf. also A275725, A275734, A275735, A276076 and tables A055089, A195663.

a(n) = A275725(A060126(n)).
Other identities:
A046523(a(n)) = A290096(n).
A056170(a(n)) = A055090(n).
A046660(a(n)) = A055091(n).
A072411(a(n)) = A055092(n).
A275812(a(n)) = A055093(n).

A283484 Odd bisection of A283983; square root of the largest square dividing A277324.

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 15, 1, 3, 15, 45, 15, 15, 15, 105, 1, 3, 105, 225, 525, 1575, 1125, 1575, 105, 105, 525, 1575, 525, 105, 105, 1155, 1, 3, 1155, 1575, 3675, 7875, 275625, 55125, 5775, 17325, 275625, 4134375, 55125, 55125, 275625, 121275, 1155, 1155, 40425, 385875, 202125, 606375, 1929375, 606375, 5775, 8085, 40425, 121275, 40425, 1155, 1155, 15015, 1, 3

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2048

Cf. A000188, A001222, A277324, A283983, A284265.

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n]  (* after Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; A275812[n_]:= PrimeOmega[n] - If[n<2, 0,Count[Transpose[FactorInteger[n]][[2]], 1]]; A277324[n_]:=A260443[2n + 1];  A000188[n_]:=  Sum[Boole[Mod[i^2, n] == 0], {i, n}]; Table[A000188[A277324[n]], {n, 0, 50}] (* Indranil Ghosh, Mar 28 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1),  A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A277324(n) = A260443((2*n)+1);
A000188(n) = core(n, 1)[2]; \\ This function from Michel Marcus, Feb 27 2013
A283484(n) = A000188(A277324(n));

(define (A283484 n) (A000188 (A277324 n)))
(define (A283484 n) (A283983 (+ n n 1)))

a(n) = A283983((2*n)+1).
a(n) = A000188(A277324(n)).
A001222(a(n)) = A284265(n).

A372603 The maximal exponent in the prime factorization of the powerful part of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0

Offset: 1

Amiram Eldar, May 07 2024

First differs from A275812 at n = 36, and from A212172 at n = 37.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A057521, A087156, A368710.
Cf. A013661, A033150, A059956.
Cf. A212172, A275812.
Similar sequences: A007424, A368781, A372601, A372602, A372604.

f[n_] := If[n == 1, 0, n]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = if(n == 1, 0, n);
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A057521(n)).
a(n) = A087156(A051903(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - 1/zeta(2) + Sum_{i>=2} (1 - 1/zeta(i)) = A033150 - A059956 = 1.09728403825134113562... .

A366076 The number of prime factors of the largest divisor of n that is a cubefull number (A036966), counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Sep 28 2023

The sum of exponents larger than 2 in the prime factorization of n.

The number of distinct prime factors of the largest divisor of n that is a cubefull number is A295659(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Sums of omega(n) and Omega(n) over the k-free parts and k-full parts of some particular sequences, Integers, Vol. 22 (2022), Article #A113.

Cf. A001222, A004709, A036966, A046100, A085541, A152441, A295659, A360540, A366077.

Similar sequence: A275812 (number of prime factors of the powerful part).

f[p_, e_] := If[e < 3, 0, e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> if(x < 3, 0, x), factor(n)[, 2]));

a(n) = A001222(A360540(n)).
a(n) = A001222(n) - A366077(n).
Additive with a(p^e) = 0 if e <= 2, and a(p^e) = e for e >= 3.
a(n) >= 0, with equality if and only if n is cubefree (A004709).
a(n) <= A001222(n), with equality if and only if n is cubefull (A036966).
a(n) >= 3*A295659(n), with equality if and only if n is a biquadratefree number (A046100).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2/p^3 + 1/(p^2*(p-1))) = 2 * A085541 + A152441 = 0.67043452760761670220... .

A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 0, 0, 0, 0, 3, 0

Offset: 1

Amiram Eldar, Aug 12 2024

First differs from A212172, A275812 and A372603 at n = 36.
If n = m * p^e, such that m is squarefree, p is a prime that does not divide m and e >= 2, then a(n) = e, otherwise a(n) = 0.
By definition all the positive terms are larger than 1.
The asymptotic density of 0's in this sequence is 1 - Sum_{p prime} (1/(p^2-1)) / zeta(2) = 1 - A059956 * A154945 = 0.66461069244308962639... .
The asymptotic density of the occurrences of k >= 2 in this sequence is Sum_{p prime} (1/(p^(k-1)*(p+1))) / zeta(2). E.g., 0.200755... (A271971) for k = 2, 0.0741777... for k = 3, and 0.0320652... for k = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005361, A048109, A051903, A060687, A082293, A190641, A359466, A375341 (the positive terms).
Cf. A013661, A059956, A154945, A271971, A375340.
Cf. A212172, A275812, A372603.

a[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], 0]]; Array[a, 100]

a(n) = {my(e = select(x -> x > 1, factor(n)[,2])); if(#e == 1, e[1], 0);}

a(n) = A051903(n) * A359466(n).
a(n) = A005361(n) * A359466(n).
a(A190641(n)) >= 2.
a(n) = 2 if and only if n is in A060687.
a(n) = 3 if and only if n is in A048109.
a(n) <= 3 if and only if n is in A082293.
Asymptotic mean:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2*p-1)/((p-1)^2*(p+1)) / zeta(2) = A375340 / A013661 = 0.92105359989459565838... .
Asymptotic second raw moment:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)^2 = Sum_{p prime} (4*p^2-3*p+1)/((p-1)^3*(p+1)) / zeta(2) = 3.04027120804428071157... .
The asymptotic second central moment, or variance, is  - ^2 = 2.19193147416548680815... and the asymptotic standard deviation is sqrt( - ^2) = 1.48051729951577627898... .

cp's OEIS Frontend

A060129 Number of moved (non-fixed) elements in the permutation with rank number n in lists A060117 (or in A060118), i.e., the sum of the lengths of all cycles larger than one in that permutation.

Views

Author

Keywords

Links

Crossrefs

Formula

A055093 Number of moved (non-fixed) elements in each permutation given in reversed colexicographic ordering A055089, i.e., the sum of their cycle lengths (excluding the 1-cycles, i.e., fixed elements).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

Extensions

A290095 a(n) = A275725(A060126(n)); prime factorization encodings of cycle-polynomials computed for finite permutations listed in reversed colexicographic ordering.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A283484 Odd bisection of A283983; square root of the largest square dividing A277324.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

Formula

A372603 The maximal exponent in the prime factorization of the powerful part of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366076 The number of prime factors of the largest divisor of n that is a cubefull number (A036966), counted with multiplicity.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula