A101296 -id:A101296 - cp's OEIS frontend

A050358 Number of ordered factorizations of n with 3 levels of parentheses.

1, 1, 1, 5, 1, 9, 1, 25, 5, 9, 1, 65, 1, 9, 9, 125, 1, 65, 1, 65, 9, 9, 1, 425, 5, 9, 25, 65, 1, 121, 1, 625, 9, 9, 9, 605, 1, 9, 9, 425, 1, 121, 1, 65, 65, 9, 1, 2625, 5, 65, 9, 65, 1, 425, 9, 425, 9, 9, 1, 1145, 1, 9, 65, 3125, 9, 121, 1, 65, 9, 121, 1, 4825, 1, 9, 65, 65, 9, 121

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

The Dirichlet inverse is given by A050356, turning all but the first element of A050356 negative. - R. J. Mathar, Jul 15 2010

			6 = (((6))) = (((3*2))) = (((2*3))) = (((3)*(2))) = (((2)*(3))) = (((3))*((2))) = (((2))*((3))) = (((3)))*(((2))) = (((2)))*(((3))).

R. J. Mathar, Table of n, a(n) for n = 1..899

Cf. A002033, A050351-A050359. a(p^k)=5^(k-1). a(A002110)=A050353.

Dirichlet g.f.: (4-3*zeta(s))/(5-4*zeta(s)).
a(n) = A050359(A101296(n)). - R. J. Mathar, May 26 2017
Sum_{k=1..n} a(k) ~ -n^r / (16*r*Zeta'(r)), where r = 2.7884327053324956670606046076818023223650950899573090550836329583345... is the root of the equation Zeta(r) = 5/4. - Vaclav Kotesovec, Feb 02 2019

A050374 Number of ordered factorizations of n into composite factors.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 1, 0, 3, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 1, 1, 1, 0, 5, 1, 1, 1, 1, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 1, 5, 1, 1, 0, 1, 1, 1, 0, 7, 0, 1, 1, 1, 1, 1, 0, 5, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 1, 1, 1, 1, 10, 0, 1, 1, 4, 0, 1

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

The Dirichlet inverse is given by A005171, all but the first term in A005171 turned negative. - R. J. Mathar, Jul 15 2010

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A002033, A002808, A005171, A050370-A050375, A000045, A002110, A032032.

read(transforms):
[1, seq(-A005171(n), n=2..100)] ;
a050374 := DIRICHLETi(%) ; # R. J. Mathar, May 26 2017

A050374(n) = if(1==n,n,sumdiv(n,d,if(dA050374(d),0))); \\ Antti Karttunen, Oct 20 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of composite numbers.
a(n) = A050375(A101296(n)). - R. J. Mathar, May 26 2017
For n >= 1, a(p^n) = A000045(n-1), for any prime p.
For n >= 0, a(A002110(n)) = A032032(n).

A173675 Let d_1, d_2, d_3, ..., d_tau(n) be the divisors of n; a(n) = number of permutations p of d_1, d_2, d_3, ..., d_tau(n) such that p_(i+1)/p_i is a prime or 1/prime for i = 1,2,...,tau(n)-1 and p_1 <= p_tau(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 8, 1, 4, 4, 1, 1, 8, 1, 8, 4, 4, 1, 14, 1, 4, 1, 8, 1, 72, 1, 1, 4, 4, 4, 20, 1, 4, 4, 14, 1, 72, 1, 8, 8, 4, 1, 22, 1, 8, 4, 8, 1, 14, 4, 14, 4, 4, 1, 584, 1, 4, 8, 1, 4, 72, 1, 8, 4, 72, 1, 62, 1, 4, 8, 8, 4, 72, 1, 22, 1, 4

Offset: 1

N. J. A. Sloane, Nov 24 2010

nonn

Variant of A179926 in which the permutation of the divisors may start with any divisor but the first term may not be larger than the last term.
From Andrew Howroyd, Oct 26 2019: (Start)
Equivalently, the number of undirected Hamiltonian paths in a graph with vertices corresponding to the divisors of n and edges connecting divisors that differ by a prime.
a(n) depends only on the prime signature of n. See A295786. (End)

			a(1) = 1: [1].
a(2) = 1: [1,2].
a(6) = 4: [1,2,6,3], [1,3,6,2], [2,1,3,6], [3,1,2,6].
a(12) = 8: [1,2,4,12,6,3], [1,3,6,2,4,12], [1,3,6,12,4,2], [2,1,3,6,12,4], [3,1,2,4,12,6], [3,1,2,6,12,4], [4,2,1,3,6,12], [6,3,1,2,4,12].

Andrew Howroyd, Table of n, a(n) for n = 1..2048
V. Shevelev, Combinatorial minors of matrix functions and their applications, arXiv:1105.3154 [math.CO], 2011-2014.
V. Shevelev, Combinatorial minors of matrix functions and their applications, Zesz. Nauk. PS., Mat. Stosow., Zeszyt 4, pp. 5-16. (2014).
Robert G. Wilson v, Re: A combinatorial problem, SeqFan (Aug 02 2010)

Cf. A000005, A000040, A002110, A101296, A179926, A284673, A295786.

See A295557 for another version.

with(numtheory):
q:= (i, j)-> is(i/j, integer) and isprime(i/j):
b:= proc(s, l) option remember; `if`(s={}, 1, add(
     `if`(q(l, j) or q(j, l), b(s minus{j}, j), 0), j=s))
    end:
a:= proc(n) option remember; ((s-> add(b(s minus {j}, j),
       j=s))(divisors(n)))/`if`(n>1, 2, 1)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 26 2017

b[s_, l_] := b[s, l] = If[s == {}, 1, Sum[If[PrimeQ[l/j] || PrimeQ[j/l], b[s ~Complement~ {j}, j], 0], {j, s}]];
a[n_] := a[n] = Function[s, Sum[b[s ~Complement~ {j}, j], {j, s}]][ Divisors[n]] / If[n > 1, 2, 1];
Array[a, 100] (* Jean-François Alcover, Nov 28 2017, after Alois P. Heinz *)

From Andrew Howroyd, Oct 26 2019: (Start)
a(p^e) = 1 for prime p.
a(A002110(n)) = A284673(n).
a(n) = A295786(A101296(n)). (End)

Alois P. Heinz corrected and clarified the definition and provided more terms. - Nov 07 2014

A249770 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order n with k invariant factors (2 <= n, 1 <= k).

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

Offset: 2

Álvar Ibeas, Nov 06 2014

The length of n-th row is A051903(n) and its last element is A249773(A101296(n)).

T(n,k) depends only on k and the prime signature of n.

			First rows:
1;
1;
1,1;
1;
1;
1;
1,1,1;
1,1;
1;
1;
1,1;
1;
1;
1;
1,2,1,1;
1;
...

Álvar Ibeas, Rows n=2..58654, flattened

Refinement of A000688.

Cf. A008284, A051903, A026820, A101296, A249771, A249773, A046951, A264809.

f[{x_, y_}] := x^IntegerPartitions[y];
g[n_] := FactorInteger[n][[1, 1]];
h[list_] := Apply[Times,Map[PadRight[#, Max[Map[Length, SplitBy[list, g]]], 1] &,SplitBy[list, g]]]; t[list_] := Tally[Map[Length, list]][[All, 2]];
Map[t, Table[Map[h, Join @@@ Tuples[Map[f, FactorInteger[n]]]], {n, 2, 50}]] // Grid (* Geoffrey Critzer, Nov 26 2015 *)

T(n,k) = A249771(A101296(n),k).
T(n,1) = 1. If k > 1 and n = Product(p_i^e_i), T(n,k) = Sum(Product(A008284(e_i,k), i in I) * Product(A026820(e_i,k-1), i not in I)), where the sum is taken over nonempty subsets I of {1,...,omega(n)}.
If p is prime and gcd(p,n) = 1, T(pn,k) = T(n,k).
Dirichlet g.f. of column sums: zeta(s)zeta(2s)···zeta(ms) = 1 + Sum_{n >= 2} (Sum_{k=1..m} T(n,k)) / n^s.
T(n,1) + T(n,2) = A046951(n)

A297174 An auxiliary sequence for computing A300250. See comments and examples.

Original entry on oeis.org

0, 1, 1, 5, 1, 19, 1, 69, 5, 19, 1, 2123, 1, 19, 19, 4165, 1, 2131, 1, 2125, 19, 19, 1, 4228171, 5, 19, 69, 2125, 1, 526631, 1, 2101317, 19, 19, 19, 268706123, 1, 19, 19, 4228237, 1, 526643, 1, 2125, 2123, 19, 1, 550026380363, 5, 2131, 19, 2125, 1, 4229203, 19, 4228237, 19, 19, 1, 8798249190555, 1, 19, 2123, 17181970501, 19, 526643, 1, 2125

Offset: 1

Antti Karttunen, Mar 07 2018

nonn

In binary representation of a(n), the distances between successive 1's (one more than the lengths of intermediate 0-runs) from the right record the prime signature ranks (A101296) of successive divisors of n, as ordered from the smallest divisor (> 1) to the largest divisor (= n).

			a(1) = 0 by convention (as 1 has no prime divisors).
a(p) = 1 for any prime p.
For any n > 1, the least significant 1-bit is at rightmost position (bit-0), signifying the smallest prime factor of n, which is always the least divisor > 1.
For n = 4 = 2*2, the next divisor of 4 after 2 is 4, for which A101296(4) = 3, thus the second least significant 1-bit comes 3-1 = 2 positions left of the rightmost 1, thus a(4) = 2^0 + 2^(3-1) = 1+4 = 5.
For n = 6 with divisors d = 2, 3 and 6 larger than one, for which A101296(d)-1 gives 1, 1 and 3, thus a(6) = 2^(1-1) + 2^(1-1+1) + 2^(1-1+1+3) = 2^0 + 2^1 + 2^4 = 19.
For n = 12 with divisors d = 2, 3, 2*2, 2*3, 2*2*3 larger than one, A101296(d)-1 gives 1, 1, 2, 3 and 5 thus a(12) = 2^0 + 2^(0+1) + 2^(0+1+2) + 2^(0+1+2+3) + 2^(0+1+2+3+5) = 2123.
For n = 18 with divisors d = 2, 3, 2*3, 3*3, 2*3*3 larger than one, A101296(d)-1 gives 1, 1, 3, 2, and 5 thus a(18) = 2^0 + 2^(0+1) + 2^(0+1+3) + 2^(0+1+3+2) + 2^(0+1+3+2+5) = 2131.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences related to binary expansion of n

Cf. A101296, A300250 (restricted growth sequence transform of this sequence).

Cf. also A292258, A294897.

up_to = 4096;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523.
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A297174(n) = { my(s=0,i=-1); fordiv(n, d, if(d>1, i += (A101296(d)-1); s += 2^i)); (s); };

A305790 Filter-sequence combining prime signature of n (A046523) and similar signature obtained when (0,1)-polynomial encoded in the binary expansion of n is factored over Q (A304751).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 4, 8, 2, 9, 2, 7, 4, 4, 2, 10, 11, 4, 12, 7, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 10, 2, 13, 2, 7, 9, 4, 2, 16, 6, 17, 4, 7, 2, 18, 19, 10, 4, 4, 2, 20, 2, 4, 9, 21, 4, 13, 2, 7, 19, 13, 2, 22, 2, 4, 9, 7, 19, 13, 2, 16, 23, 4, 2, 20, 4, 4, 19, 10, 2, 24, 19, 7, 4, 4, 4, 25, 2, 9, 7, 26, 2, 13, 2, 10, 13

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

Restricted growth sequence transform of ordered pair [A046523(n), A304751(n)].

For all i, j: a(i) = a(j) => A305821(i) = A305821(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046523, A101296, A304751, A305821.

Cf. also A305789.

up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux304751(n) = { my(p=0, f=vecsort((factor(Pol(binary(n)))[, 2]), , 4)); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }
Aux305790(n) = [A046523(n), Aux304751(n)];
v305790 = rgs_transform(vector(up_to,n,Aux305790(n)));
A305790(n) = v305790[n];

A318890 Filter sequence combining the prime signature of n (A046523) with the prime signature of its conjugated prime factorization (A278221).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 10, 15, 16, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 14, 33, 34, 35, 36, 37, 38, 39, 40, 41, 18, 42, 43, 44, 45, 18, 46, 47, 48, 22, 31, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 39, 63, 64, 65, 66, 18, 67, 20, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 53, 36, 80, 81, 82, 83, 84, 85, 26, 86, 87, 88, 89, 90, 91, 39

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Restricted growth sequence transform of A286454.

For all i, j: a(i) = a(j) => A318891(i) = A318891(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046523, A101296, A278221, A286454, A286621, A318891.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278221(n) = A046523(A122111(n));
A318890aux(n) = [A046523(n), A278221(n)];
v318890 = rgs_transform(vector(up_to,n,A318890aux(n)));
A318890(n) = v318890[n];

A320004 Filter sequence combining the largest proper divisor of n (A032742) with n's residue modulo 4 (A010873), and a single bit (A319710) telling whether the smallest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 3, 7, 8, 9, 3, 10, 5, 11, 12, 13, 5, 14, 3, 15, 16, 17, 3, 18, 19, 20, 21, 22, 5, 23, 3, 24, 25, 26, 27, 28, 5, 29, 30, 31, 5, 32, 3, 33, 34, 35, 3, 36, 37, 38, 39, 40, 5, 41, 42, 43, 44, 45, 3, 46, 5, 47, 48, 49, 50, 51, 3, 52, 53, 54, 3, 55, 5, 56, 57, 58, 25, 59, 3, 60, 61, 62, 3, 63, 64, 65, 66, 67, 5, 68, 30, 69, 70, 71, 72, 73, 5, 74, 75, 76, 5, 77, 3

Offset: 1

Antti Karttunen, Oct 04 2018

nonn

Restricted growth sequence transform of triple [A010873(A020639(n)), A032742(n), A319710(n)], or equally, of ordered pair [A319714(n), A319710(n)].
Here any nontrivial equivalence classes (that is, when we exclude the singleton classes and two infinite classes of A002144 and A002145), like the example shown, may not contain any even numbers, nor any numbers from A283050. See additional comments in A319717 and A319994.
For all i, j:
  a(i) = a(j) => A024362(i) = A024362(j),
  a(i) = a(j) => A067029(i) = A067029(j),
  a(i) = a(j) => A071178(i) = A071178(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j).

			For n = 33 (3*11) and n = 77 (7*11), the modulo 4 residue of the smallest prime factor is 3, and the largest proper divisors (A032742) is also equal 11, and the smallest prime factor is unitary. Thus a(33) = a(77) (= 25, a running count value allotted by rgs-transform).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A319704, A319714, A319994.
Cf. also A319717 (analogous sequence for modulo 6 residues).
Cf. A002145 (positions of 3's), A002144 (positions of 5's).
Differs from A319704 for the first time at n=77, and from A319714 for the first time at n=49.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A286474(n) = if(1==n,n,(4*A032742(n) + (n % 4)));
A319710(n) = ((n>1)&&(factor(n)[1,2]>1));
v320004 = rgs_transform(vector(up_to,n,[A286474(n),A319710(n)]));
A320004(n) = v320004[n];

A355836 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A355442(i) = A355442(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 8, 5, 10, 3, 11, 3, 12, 5, 8, 3, 13, 14, 8, 15, 16, 3, 17, 3, 18, 19, 8, 20, 21, 3, 8, 5, 22, 3, 23, 3, 12, 24, 8, 3, 25, 14, 26, 5, 16, 3, 27, 20, 28, 5, 8, 3, 29, 3, 8, 30, 31, 20, 23, 3, 12, 5, 32, 3, 33, 3, 8, 34, 16, 19, 23, 3, 35, 36, 8, 3, 37, 20, 8, 5, 38, 3, 29, 19, 12, 19, 8, 20, 39, 3, 12, 9, 40, 3, 23, 3, 28, 41

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A355442(n)].

For all i, j: A355835(i) = A355835(j) => a(i) = a(j).

			a(6) = a(15) = a(21) = a(39) = a(51) = a(57) = a(69) = a(87) = a(111) = etc, for an infinite number of other indices k, because for all these k, A355442(k) = 5 and their prime signatures (A101296) are equal, as they are all squarefree semiprimes, A006881.
In contrast, powers of 2 (1, 2, 4, 8, 16, ..., A000079) obtain unique values in this sequence, and in general, for all proper prime powers k (A246547) for which A355442(k) > 1 [that are terms of A355822], the value a(k) is unique in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial base

Cf. A000079, A003961, A006881, A046523, A101296, A246547, A276086, A355442, A355822, A355835.

Cf. also A355000, A355830.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));
Aux355836(n) = [A046523(n), A355442(n)];
v355836 = rgs_transform(vector(up_to,n,Aux355836(n)));
A355836(n) = v355836[n];

A369031 LCM-transform of permutation induced by partition conjugation via Heinz numbers (A122111).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 12 2024

nonn

See discussion at A368900.

From the reduced formula it follows that for all i, j >= 1: A101296(i) = A101296(j) => a(i) = a(j), that is, the value of each a(n) is completely determined by its prime signature. Note that the same does not hold for related A369032.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A014963, A101296, A122111, A368900, A369032, A369033.

up_to = 2^18;
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
v369031 = LCMtransform(vector(up_to,i,A122111(i)));
A369031(n) = v369031[n];

A369031(n) = if(isprime(n),2, my(e=ispower(n,,&n)); if(e && isprime(n), prime(e), 1));

a(n) = lcm {1..A122111(n)} / lcm {1..A122111(n-1)}.
a(n) = A014963(A122111(n)). [A122111 satisfies the property S given in A368900]
If n = p^k, p prime, k >= 1, then a(n) = A000040(k), otherwise a(n) = 1.

cp's OEIS Frontend

A050358 Number of ordered factorizations of n with 3 levels of parentheses.

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A050374 Number of ordered factorizations of n into composite factors.

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A173675 Let d_1, d_2, d_3, ..., d_tau(n) be the divisors of n; a(n) = number of permutations p of d_1, d_2, d_3, ..., d_tau(n) such that p_(i+1)/p_i is a prime or 1/prime for i = 1,2,...,tau(n)-1 and p_1 <= p_tau(n).

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A249770 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order n with k invariant factors (2 <= n, 1 <= k).

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A297174 An auxiliary sequence for computing A300250. See comments and examples.

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A305790 Filter-sequence combining prime signature of n (A046523) and similar signature obtained when (0,1)-polynomial encoded in the binary expansion of n is factored over Q (A304751).

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A318890 Filter sequence combining the prime signature of n (A046523) with the prime signature of its conjugated prime factorization (A278221).

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A320004 Filter sequence combining the largest proper divisor of n (A032742) with n's residue modulo 4 (A010873), and a single bit (A319710) telling whether the smallest prime factor is unitary.

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