A028233 -id:A028233 - cp's OEIS frontend

A333145 Number of unimodal negated permutations of the multiset of prime indices of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 2, 2, 4, 1, 3, 1, 2, 3, 2, 2, 4, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 09 2020

nonn

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.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
Also permutations of the multiset of prime indices of n avoiding the patterns (1,2,1), (1,3,2), and (2,3,1).
Also the number divisors of n not divisible by the least prime factor of n. The other divisors are counted by A069157. - Gus Wiseman, Apr 12 2022

			The a(n) permutations for n = 2, 6, 18, 30, 90, 162, 210, 450:
  (1)  (12)  (122)  (123)  (1223)  (12222)  (1234)  (12233)
       (21)  (212)  (213)  (2123)  (21222)  (2134)  (21233)
             (221)  (312)  (2213)  (22122)  (3124)  (22133)
                    (321)  (3122)  (22212)  (3214)  (31223)
                           (3212)  (22221)  (4123)  (32123)
                           (3221)           (4213)  (32213)
                                            (4312)  (33122)
                                            (4321)  (33212)
                                                    (33221)

Eric Weisstein's World of Mathematics, Unimodal Sequence
Wikipedia, Permutation pattern

Dominated by A008480.
The complementary divisors are counted by A069157.
The non-negated version is A332288.
A more interesting version is A332741.
The complement is counted by A333146.
A001523 counts unimodal compositions.
A007052 counts unimodal normal sequences.
A028233 gives the highest power of the least prime factor, quotient A028234.
A332578 counts compositions whose negation is unimodal.
A332638 counts partitions with unimodal negated run-lengths.
A332642 lists numbers with non-unimodal negated unsorted prime signature.
Cf. A056239, A112798, A115981, A124010, A328509, A332283, A332294, A332639, A332669, A332670, A332671.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[primeMS[n]],unimodQ[-#]&]],{n,30}]

a(n) + A333146(n) = A008480(n).
a(n) = A000005(A028234(n)). - Gus Wiseman, Apr 14 2022
a(n) = A000005(n) - A069157(n). - Gus Wiseman, Apr 14 2022

A336650 a(n) = p^e, where p is the smallest odd prime factor of n, and e is its exponent, with a(n) = 1 when n is a power of two.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 3, 1, 17, 9, 19, 5, 3, 11, 23, 3, 25, 13, 27, 7, 29, 3, 31, 1, 3, 17, 5, 9, 37, 19, 3, 5, 41, 3, 43, 11, 9, 23, 47, 3, 49, 25, 3, 13, 53, 27, 5, 7, 3, 29, 59, 3, 61, 31, 9, 1, 5, 3, 67, 17, 3, 5, 71, 9, 73, 37, 3, 19, 7, 3, 79, 5, 81, 41, 83, 3, 5, 43, 3, 11, 89, 9, 7, 23, 3

Offset: 1

Antti Karttunen, Jul 30 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A028233, A078701, A336362.

A336650(n) = if(!bitand(n,n-1),1,my(f=factor(n>>valuation(n,2))); f[1, 1]^f[1, 2]);

a(n) = A028233(A000265(n)).

A216972 a(4n+2) = 2, otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 12, 13, 2, 15, 16, 17, 2, 19, 20, 21, 2, 23, 24, 25, 2, 27, 28, 29, 2, 31, 32, 33, 2, 35, 36, 37, 2, 39, 40, 41, 2, 43, 44, 45, 2, 47, 48, 49, 2, 51, 52, 53, 2, 55, 56, 57, 2, 59, 60, 61, 2, 63, 64, 65, 2, 67, 68, 69, 2

Offset: 0

Paul Curtz, Sep 21 2012

For n>0, a(n) is the denominator of A214282(n)/(-A214283(n+1)):
1/1, 1/2, 1/3, 3/4, 3/5, 1/2, 3/7, 5/8, 5/9, ...
For n>0, a(n) is the denominator of A214283(n)/A214283(n+1):
0/1, 1/2, 2/3, 3/4, 2/5, 1/2, 4/7, 5/8, 4/9, ...
a(n), first and second differences:
0, 1, 2, 3,  4,  5,  2, 7,  8,  9,  2, 11,  12, ...
1, 1, 1, 1,  1, -3,  5, 1,  1, -7,  9,  1,   1, ...
0, 0, 0, 0, -4,  8, -4, 0, -8, 16, -8,  0, -12, ...

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A028233, A034684, A000265, A026741, A060819, A145979, A214682.

[n mod 4 eq 2 select 2 else n: n in [0..70]]; // Bruno Berselli, Sep 26 2012

a[n_] := If[Mod[n, 4] == 2, 2, n]; Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Sep 25 2012 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,1,2,3,4,5,2,7},80] (* Harvey P. Dale, Nov 06 2017 *)

makelist(expand(2+(4-(1+(-1)^n)*(1-%i^n))*(n-2)/4), n, 0, 70); /* Bruno Berselli, Sep 26 2012 */

def A216972(n): return 2 if n&3==2 else n # Chai Wah Wu, Jan 31 2024

a(n) = 2*a(n-4) - a(n-8).
a(n+4) - a(n) = 4*A152822(n).
a(2n) + a(2n+1) = |A141124(n)|.
a(4n) + a(4n+1) + a(4n+2) + a(4n+3) = 6*A005408(n) = A017593(n).
G.f.: (x+2*x^2+3*x^3+4*x^4+3*x^5-2*x^6+x^7) / (1-2*x^4+x^8). - Jean-François Alcover, Sep 25 2012
a(n) = 2+(4-(1+(-1)^n)*(1-i^n))*(n-2)/4, where i=sqrt(-1). - Bruno Berselli, Sep 26 2012
a(2n) = 2*|A009531(n)|, a(2n+1) = 2n+1. - Bruno Berselli, Sep 27 2012

A304181 If n = Product (p_j^k_j) then a(n) = min{p_j}^min{k_j}.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 2, 13, 2, 3, 16, 17, 2, 19, 2, 3, 2, 23, 2, 25, 2, 27, 2, 29, 2, 31, 32, 3, 2, 5, 4, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 49, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 64, 5, 2, 67, 2, 3, 2, 71, 4, 73, 2, 3, 2, 7, 2, 79, 2, 81, 2, 83, 2, 5

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

			a(72) = 4 because 72 = 2^3*3^2, min{2,3} = 2, min{3,2} = 2 and 2^2 = 4.

Eric Weisstein's World of Mathematics, Least Prime Factor
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A000961 (fixed points), A005117, A020639, A028233, A034684, A051904, A073481, A081811, A304180.

Table[(FactorInteger[n][[1, 1]])^(Min @@ Last /@ FactorInteger[n]), {n, 85}]

a(n) = A020639(n)^A051904(n).
a(p^k) = p^k where p is a prime.
a(A005117(k)) = A073481(k).

A323130 a(1) = 1, and for any n > 1, let p be the least prime factor of n, and e be its exponent, then a(n) = p^a(e).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 4, 13, 2, 3, 16, 17, 2, 19, 4, 3, 2, 23, 8, 25, 2, 27, 4, 29, 2, 31, 32, 3, 2, 5, 4, 37, 2, 3, 8, 41, 2, 43, 4, 9, 2, 47, 16, 49, 2, 3, 4, 53, 2, 5, 8, 3, 2, 59, 4, 61, 2, 9, 4, 5, 2, 67, 4, 3, 2, 71, 8, 73, 2, 3, 4, 7, 2, 79

Offset: 1

Rémy Sigrist, Jan 05 2019

nonn

This sequence is a recursive variant of A028233.

All terms belong to A164336.

			a(320) = a(2^6 * 5) = 2^a(6) = 2^a(2*3) = 2^2 = 4.

See A323129 for the variant involving the greatest prime factor.

Cf. A020639, A028233, A067029, A164336.

Nest[Append[#, First@ FactorInteger[Length[#] + 1] /. {p_, e_} :> p^#[[e]] ] &, {1}, 78] (* Michael De Vlieger, Jan 07 2019 *)

a(n) = if (n==1, 1, my (f=factor(n)); f[1,1]^a(f[1,2]))

a(n) <= n with equality iff n belong to A164336.

a(n) = A020639(n)^a(A067029(n)) for any n > 1.

A085234 (Greatest power of smallest prime factor of n) < square root(n).

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 50, 51, 52, 54, 55, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 114, 115

Offset: 1

Reinhard Zumkeller, Jun 22 2003

nonn

A028233(a(n))^2 < a(n);

a(n)=A085232(n) for n<69: a(69)=120, A085232(69)=122=a(70).

isA085234 := proc(n)
    if A028233(n)^2 < n then
        true;
    else
        false;
    end if;
end proc:
for n from 1 to 115 do
    if isA085234(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jul 09 2016

okQ[n_] := Power @@ FactorInteger[n][[1]] < Sqrt[n]; Select[Range[120], okQ] (* Jean-François Alcover, Feb 13 2018 *)

cp's OEIS Frontend

A333145 Number of unimodal negated permutations of the multiset of prime indices of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A336650 a(n) = p^e, where p is the smallest odd prime factor of n, and e is its exponent, with a(n) = 1 when n is a power of two.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A216972 a(4n+2) = 2, otherwise a(n) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

Python

Formula

A304181 If n = Product (p_j^k_j) then a(n) = min{p_j}^min{k_j}.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A323130 a(1) = 1, and for any n > 1, let p be the least prime factor of n, and e be its exponent, then a(n) = p^a(e).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A085234 (Greatest power of smallest prime factor of n) < square root(n).

Views

Author

Keywords

Comments

Programs

Maple

Mathematica