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.

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A145204 Numbers whose representation in base 3 (A007089) ends in an odd number of zeros.

Original entry on oeis.org

0, 3, 6, 12, 15, 21, 24, 27, 30, 33, 39, 42, 48, 51, 54, 57, 60, 66, 69, 75, 78, 84, 87, 93, 96, 102, 105, 108, 111, 114, 120, 123, 129, 132, 135, 138, 141, 147, 150, 156, 159, 165, 168, 174, 177, 183, 186, 189, 192, 195, 201, 204, 210, 213, 216, 219, 222, 228, 231
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 04 2008

Keywords

Comments

Previous name: Complement of A007417.
Also numbers having infinitary divisor 3, or the same, having factor 3 in their Fermi-Dirac representation as product of distinct terms of A050376. - Vladimir Shevelev, Mar 18 2013
For n > 1: where even terms occur in A051064. - Reinhard Zumkeller, May 23 2013
If we exclude a(1) = 0, these are numbers whose squarefree part is divisible by 3, which can be partitioned into numbers whose squarefree part is congruent to 3 mod 9 (A055041) and 6 mod 9 (A055040) respectively. - Peter Munn, Jul 14 2020
The inclusion of 0 as a term might be viewed as a cultural preference: if we habitually wrote numbers enclosed in brackets and then used a null string of digits for zero, the natural number sequence in ternary would be [], [1], [2], [10], [11], [12], [20], ... . - Peter Munn, Aug 02 2020
The asymptotic density of this sequence is 1/4. - Amiram Eldar, Sep 20 2020

Links

Crossrefs

Subsequence of A008585, A028983.
Subsequences: A016051, A055040, A055041, A329575.
Cf. A007089, A007417 (complement), A050376, A182581 (characteristic function).
Positions of 0s in A014578.
Excluding 0: the positions of odd numbers in A007949; equivalently, of even numbers in A051064; symmetric difference of A003159 and A036668.
Related to A042964 via A052330.
Related to A036554 via A064614.

Programs

  • Haskell
    a145204 n = a145204_list !! (n-1)
a145204_list = 0 : map (+ 1) (findIndices even a051064_list)
-- Reinhard Zumkeller, May 23 2013
  • Maple
    isA145204 := proc(n) local d, c;
if n = 0 then return true fi;
d := A007089(n); c := 0;
while irem(d, 10) = 0 do c := c+1; d := iquo(d, 10) od;
type(c, odd) end:
select(isA145204, [$(0..231)]); # Peter Luschny, Aug 05 2020
  • Mathematica
    Select[ Range[0, 235], (# // IntegerDigits[#, 3]& // Split // Last // Count[#, 0]& // OddQ)&] (* Jean-François Alcover, Mar 18 2013 *)
Join[{0}, Select[Range[235], OddQ @ IntegerExponent[#, 3] &]] (* Amiram Eldar, Sep 20 2020 *)
  • Python
    import numpy as np
def isA145204(n):
    if n == 0: return True
    c = 0
    d = int(np.base_repr(n, base = 3))
    while d % 10 == 0:
        c += 1
        d //= 10
    return c % 2 == 1
print([n for n in range(231) if isA145204(n)]) # Peter Luschny, Aug 05 2020
  • Python
    from sympy import integer_log
def A145204(n):
    if n == 1: return 0
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+sum(((m:=x//9**i)-2)//3+(m-1)//3+2 for i in range(integer_log(x,9)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 15 2025

Formula

a(n) = 3 * A007417(n-1) for n > 1.
A014578(a(n)) = 0.
For n > 1, A007949(a(n)) mod 2 = 1. [Edited by Peter Munn, Aug 02 2020]
{a(n) : n >= 2} = {A052330(A042964(k)) : k >= 1} = {A064614(A036554(k)) : k >= 1}. - Peter Munn, Aug 31 2019 and Dec 06 2020

Extensions

New name using a comment of Vladimir Shevelev by Peter Luschny, Aug 05 2020

A036668 Hati numbers: of form 2^i*3^j*k, i+j even, (k,6)=1.

Original entry on oeis.org

1, 4, 5, 6, 7, 9, 11, 13, 16, 17, 19, 20, 23, 24, 25, 28, 29, 30, 31, 35, 36, 37, 41, 42, 43, 44, 45, 47, 49, 52, 53, 54, 55, 59, 61, 63, 64, 65, 66, 67, 68, 71, 73, 76, 77, 78, 79, 80, 81, 83, 85, 89, 91, 92, 95, 96, 97, 99, 100, 101, 102, 103, 107
Offset: 1

Views

Author

N. J. A. Sloane, Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)

Keywords

Comments

If n appears then 2n and 3n do not. - Benoit Cloitre, Jun 13 2002
Closed under multiplication. Each term is a product of a unique subset of {6} U A050376 \ {2,3}. - Peter Munn, Sep 14 2019

Links

Crossrefs

Cf. A003159, A007310, A014601, A036667, A050376, A052330, A325424 (complement), A325498 (first differences), A373136 (characteristic function).
Positions of 0's in A182582.
Subsequences: A084087, A339690, A352272, A352273.
Cf. also A121537, A145204, A225837, A307150, A329575, A334747, A334748, A339746.

Programs

  • Maple
    N:= 1000: # to get all terms up to N
A:= {seq(2^i,i=0..ilog2(N))}:
Ae,Ao:= selectremove(issqr,A):
Be:= map(t -> seq(t*9^j, j=0 .. floor(log[9](N/t))),Ae):
Bo:= map(t -> seq(t*3*9^j,j=0..floor(log[9](N/(3*t)))),Ao):
B:= Be union Bo:
C1:= map(t -> seq(t*(6*i+1),i=0..floor((N/t -1)/6)),B):
C2:= map(t -> seq(t*(6*i+5),i=0..floor((N/t - 5)/6)),B):
A036668:= C1 union C2; # Robert Israel, May 09 2014
  • Mathematica
    a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1,
Apply[Or, Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/2}],
IntegerQ]]] &]], {150}]; a  (* A036668 *)
(* Peter J. C. Moses, Apr 23 2019 *)
  • PARI
    twos(n) = {local(r,m);r=0;m=n;while(m%2==0,m=m/2;r++);r}
threes(n) = {local(r,m);r=0;m=n;while(m%3==0,m=m/3;r++);r}
isA036668(n) = (twos(n)+threes(n))%2==0 \\ Michael B. Porter, Mar 16 2010
  • PARI
    is(n)=(valuation(n,2)+valuation(n,3))%2==0 \\ Charles R Greathouse IV, Sep 10 2015
  • PARI
    list(lim)=my(v=List(),N);for(n=0,logint(lim\=1,3),N=if(n%2,2*3^n,3^n); while(N<=lim, forstep(k=N,lim,[4*N,2*N], listput(v,k)); N<<=2)); Set(v) \\ Charles R Greathouse IV, Sep 10 2015
  • Python
    from itertools import count
def A036668(n):
    def f(x):
        c = n+x
        for i in range(x.bit_length()+1):
            i2 = 1<x:
                    break
                m = x//k
                c -= (m-1)//6+(m-5)//6+2
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 28 2025

Formula

a(n) = 12/7 * n + O(log^2 n). - Charles R Greathouse IV, Sep 10 2015
{a(n)} = A052330({A014601(n)}), where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Sep 14 2019

Extensions

Offset changed by Chai Wah Wu, Jan 28 2025

A339746 Positive integers of the form 2^i*3^j*k, gcd(k,6)=1, and i == j (mod 3).

Original entry on oeis.org

1, 5, 6, 7, 8, 11, 13, 17, 19, 23, 25, 27, 29, 30, 31, 35, 36, 37, 40, 41, 42, 43, 47, 48, 49, 53, 55, 56, 59, 61, 64, 65, 66, 67, 71, 73, 77, 78, 79, 83, 85, 88, 89, 91, 95, 97, 101, 102, 103, 104, 107, 109, 113, 114, 115, 119, 121, 125, 127, 131, 133, 135
Offset: 1

Views

Author

Griffin N. Macris, Dec 15 2020

Keywords

Comments

From Peter Munn, Mar 16 2021: (Start)
The positive integers in the multiplicative subgroup of the positive rationals generated by 8, 6, and A215848 (primes greater than 3).
This subgroup, denoted H, has two cosets: 2H = (1/3)H and 3H = (1/2)H. It follows that the sequence is one part of a 3-part partition of the positive integers with the property that each part's terms are half the even terms of one of the other parts and also one third of the multiples of 3 in the remaining part.
(End)
Positions of multiples of 3 in A276085 (and in A276075). Because A276085 is completely additive, this is closed under multiplication: if m and n are in the sequence then so is m*n. - Antti Karttunen, May 27 2024
The coset sequences mentioned in Peter Munn's comment above are A373261 and A373262. - Antti Karttunen, Jun 04 2024

Links

Crossrefs

Sequences of positive integers in a multiplicative subgroup of positive rationals generated by a set S and A215848: S={}: A007310, S={6}: A064615, S={3,4}: A003159, S={2,9}: A007417, S={4,6}: A036668, S={3,8}: A191257, S={4,9}: A339690, S={6,8}: this sequence.
Positions of 0's in A373153, positions of multiples of 3 in A276085 and in A372576.
Cf. A372573 (characteristic function), A373261, A373262.
Subsequences: A064615, A373144, A373373, A373484, A373837, A374042, A374044, A374120, A377872.
Sequences giving positions of multiples of k in A276085, for k=2, 3, 4, 5, 8, 27, 3125: A003159, this sequence, A369002, A373140, A373138, A377872, A377878.
Cf. also A332820, A373992, A383288.

Programs

  • Maple
    N:= 1000: # for terms <= N
R:= {}:
for k1 from 0 to floor(N/6) do
  for k0 in [1,5] do
    k:= k0 + 6*k1;
    for j from 0 while 3^j*k <= N do
      for i from (j mod 3) by 3 do
        x:= 2^i * 3^j * k;
        if x > N then break fi;
        R:= R union {x}
od od od od:
sort(convert(R,list)); # Robert Israel, Apr 08 2021
  • Mathematica
    Select[Range[130], Mod[IntegerExponent[#, 2] - IntegerExponent[#, 3], 3] == 0 &]
  • PARI
    isA339746 = A372573; \\ Antti Karttunen, Jun 04 2024
  • Python
    from sympy import factorint
def ok(n):
  f = factorint(n, limit=4)
  i, j = 0 if 2 not in f else f[2], 0 if 3 not in f else f[3]
  return (i-j)%3 == 0
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(200)) # Michael S. Branicky, Mar 26 2021
  • Python
    from itertools import count
def A339746(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(x.bit_length()+1):
            i2 = 1<x:
                    break
                m = x//k
                c -= (m-1)//6+(m-5)//6+2
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 12 2025

Formula

a(n) ~ (91/43)*n.

A026225 Numbers of the form 3^i * (3k+1).

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 30, 31, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 55, 57, 58, 61, 63, 64, 66, 67, 70, 73, 75, 76, 79, 81, 82, 84, 85, 88, 90, 91, 93, 94, 97, 100, 102, 103, 106, 108, 109, 111, 112, 115
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Old name: a(n) = (1/3)*(s(n+1) - 1), where s = A026224.
Conjectures based on old name: these are numbers of the form (3*i+1)*3^j; see A182828, and they comprise the complement of A026179, except for the initial 1 in A026179.
From Peter Munn, Mar 17 2022: (Start)
Numbers with an even number of prime factors of the form 3k-1 counting repetitions.
Numbers whose squarefree part is congruent to 1 modulo 3 or 3 modulo 9.
The integers in an index 2 subgroup of the positive rationals under multiplication. As such the sequence is closed under multiplication and - where the result is an integer - under division; also for any positive integer k not in the sequence, the sequence's complement is generated by dividing by k the terms that are multiples of k.
Alternatively, the sequence can be viewed as an index 2 subgroup of the positive integers under the commutative binary operation A059897(.,.).
Viewed either way, the sequence corresponds to a subgroup of the quotient group derived in the corresponding way from A055047. (End)
The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022
Is this A026140 shifted right? - R. J. Mathar, Jun 24 2025

Links

Crossrefs

Elements of array A182828 in ascending order.
Union of A055041 and A055047.
Other subsequences: A007645 (primes), A352274.
Symmetric difference of A003159 and A225838; of A007417 and A189716.
Cf. A001222, A059897, A343430.

Programs

  • Mathematica
    a[b_] := Table[Mod[n/b^IntegerExponent[n, b], b], {n, 1, 160}]
p[b_, d_] := Flatten[Position[a[b], d]]
p[3, 1]  (* A026225 *)
p[3, 2] (* A026179 without initial 1 *)
(* Clark Kimberling, Oct 19 2016 *)
  • PARI
    isok(m) = core(m) % 3 == 1 || core(m) % 9 == 3; \\ Peter Munn, Mar 17 2022
  • Python
    from sympy import integer_log
def A026225(n):
    def f(x): return n+x-sum(((x//3**i)-1)//3+1 for i in range(integer_log(x,3)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 15 2025

Formula

From Peter Munn, Mar 17 2022: (Start)
{a(n) : n >= 1} = {m : A001222(A343430(m)) == 0 (mod 2)}.
{a(n) : n >= 1} = {A055047(m) : m >= 1} U {3*A055047(m) : m >= 1}.
{a(n) : n >= 1} = {A352274(m) : m >= 1} U {A352274(m)/10 : m >= 1, 10 divides A352274(m)}. (End)

Extensions

New name from Peter Munn, Mar 17 2022

A087787 a(n) = Sum_{k=0..n} (-1)^(n-k)*A000041(k).

Original entry on oeis.org

1, 0, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409
Offset: 0

Views

Author

Vladeta Jovovic, Oct 07 2003

Keywords

Comments

Essentially first differences of A024786 (see the formula). Also, a(n) is the number of 2's in the last section of the set of partitions of n+2 (see A135010). - Omar E. Pol, Sep 10 2008
From Gus Wiseman, May 20 2024: (Start)
Also the number of integer partitions of n containing an even number of ones, ranked by A003159. The a(0) = 1 through a(8) = 15 partitions are:
() . (2) (3) (4) (5) (6) (7) (8)
(11) (22) (32) (33) (43) (44)
(211) (311) (42) (52) (53)
(1111) (222) (322) (62)
(411) (511) (332)
(2211) (3211) (422)
(21111) (31111) (611)
(111111) (2222)
(3311)
(4211)
(22211)
(41111)
(221111)
(2111111)
(11111111)
Also the number of integer partitions of n + 1 containing an odd number of ones, ranked by A036554.
(End)

Links

Crossrefs

Cf. A000041, A024786, A135010, A138121, A141285.
The unsigned version is A000070, strict A036469.
For powers of 2 instead number of partitions we have A001045.
The strict or odd version is A025147 or A096765.
The ordered version (compositions instead of partitions) is A078008.
For powers of 2 instead of -1 we have A259401, cf. A259400.
A002865 counts partitions with no ones, column k=0 of A116598.
A072233 counts partitions by sum and length.
Cf. A000009, A003159, A026804, A027187, A027193, A036554, A058695, A101707.

Programs

  • Mathematica
    Table[Sum[(-1)^(n-k)*PartitionsP[k], {k,0,n}], {n,0,50}] (* Vaclav Kotesovec, Aug 16 2015 *)
(* more efficient program *) sig = 1; su = 1; Flatten[{1, Table[sig = -sig; su = su + sig*PartitionsP[n]; Abs[su], {n, 1, 50}]}] (* Vaclav Kotesovec, Nov 06 2016 *)
Table[Length[Select[IntegerPartitions[n], EvenQ[Count[#,1]]&]],{n,0,30}] (* Gus Wiseman, May 20 2024 *)
  • Python
    from sympy import npartitions
def A087787(n): return sum(-npartitions(k) if n-k&1 else npartitions(k) for k in range(n+1)) # Chai Wah Wu, Oct 25 2023

Formula

G.f.: 1/(1+x)*1/Product_{k>0} (1-x^k).
a(n) = 1/n*Sum_{k=1..n} (sigma(k)+(-1)^k)*a(n-k).
a(n) = A024786(n+2)-A024786(n+1). - Omar E. Pol, Sep 10 2008
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n) * (1 + (11*Pi/(24*sqrt(6)) - sqrt(3/2)/Pi)/sqrt(n) - (11/16 + (23*Pi^2)/6912)/n). - Vaclav Kotesovec, Nov 05 2016
a(n) = A000041(n) - a(n-1). - Jon Maiga, Aug 29 2019
Alternating partial sums of A000041. - Gus Wiseman, May 20 2024

A342051 Numbers k which have an even number of trailing zeros in their primorial base representation A049345(k).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 72, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 93, 95, 96, 97, 99, 101, 102, 103
Offset: 1

Views

Author

Amiram Eldar, Feb 26 2021

Keywords

Comments

Numbers k such that A276084(k) is even.
The number of terms not exceeding A002110(m) for m>=1 is A002110(m) * (1 - Sum_{k=1..m}(-1)^k/A002110(k)) = 1, 4, 19, 134, 1473, 19150, 325549 ...
The asymptotic density of this sequence is Sum_{k>=0} (-1)^k/A002110(k) = 0.637693... = 1 - A132120.
Also Heinz numbers of partitions with odd least gap. The least gap (mex or minimal excludant) of a partition is the least positive integer that is not a part. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. - Gus Wiseman, Apr 23 2021

Examples

			1 is a term since A049345(1) = 1 has 0 trailing zero.
6 is a term since A049345(6) = 100 has 2 trailing zeros.
From _Gus Wiseman_, Apr 23 2021: (Start)
The sequence of terms together with their prime indices begins:
     1: {}           25: {3,3}          51: {2,7}
     3: {2}          27: {2,2,2}        53: {16}
     5: {3}          29: {10}           54: {1,2,2,2}
     6: {1,2}        31: {11}           55: {3,5}
     7: {4}          33: {2,5}          57: {2,8}
     9: {2,2}        35: {3,4}          59: {17}
    11: {5}          36: {1,1,2,2}      61: {18}
    12: {1,1,2}      37: {12}           63: {2,2,4}
    13: {6}          39: {2,6}          65: {3,6}
    15: {2,3}        41: {13}           66: {1,2,5}
    17: {7}          42: {1,2,4}        67: {19}
    18: {1,2,2}      43: {14}           69: {2,9}
    19: {8}          45: {2,2,3}        71: {20}
    21: {2,4}        47: {15}           72: {1,1,1,2,2}
    23: {9}          48: {1,1,1,1,2}    73: {21}
    24: {1,1,1,2}    49: {4,4}          75: {2,3,3}
(End)

Links

Crossrefs

Cf. A002110, A049345, A132120, A276084.
Complement of A342050.
A099788 is subsequence.
Analogous sequences: A000201 (Zeckendorf representation), A003159 (binary), A007417 (ternary), A232744 (factorial base).
The version for reversed binary expansion is A121539.
Positions of odd terms in A257993.
A000070 counts partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A079067 counts gaps in prime indices.
A238709 counts partitions by sum and least difference.
A339662 gives greatest gap in prime indices.
Cf. A001223, A005408, A026794, A029707, A047235, A079068, A079523, A286469, A286470, A325351.

Programs

  • Mathematica
    seq[max_] := Module[{bases = Prime@Range[max, 1, -1], nmax}, nmax = Times @@ bases - 1; Select[Range[nmax], EvenQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[bases]], #1 == 0 &] &]]; seq[4]
Select[Range[100],OddQ[Min@@Complement[Range[PrimeNu[#]+1],PrimePi/@First/@FactorInteger[#]]]&] (* Gus Wiseman, Apr 23 2021 *)

A083346 Denominator of r(n) = Sum(e/p: n=Product(p^e)).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 3, 10, 11, 3, 13, 14, 15, 1, 17, 6, 19, 5, 21, 22, 23, 6, 5, 26, 1, 7, 29, 30, 31, 2, 33, 34, 35, 3, 37, 38, 39, 10, 41, 42, 43, 11, 15, 46, 47, 3, 7, 10, 51, 13, 53, 2, 55, 14, 57, 58, 59, 15, 61, 62, 21, 1, 65, 66, 67, 17, 69, 70, 71, 6, 73, 74, 15, 19, 77, 78
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 25 2003

Keywords

Comments

Multiplicative with a(p^e) = 1 iff p|e, p otherwise. For f(n) = A083345(n)/A083346(n), f(p^i*q^j*...) = f(p^i)+f(q^j)+ ... The denominator of each term is 1 or the prime, thus the denominator of the sum is the product of the denominators of the components. - Christian G. Bower, May 16 2005
n divided by the greatest common divisor of n and its arithmetic derivative, i.e., a(n) = n/gcd(n,n') = A000027(n)/A085731(n). - Giorgio Balzarotti, Apr 14 2011

Examples

			n=12 = 2*2*3 = 2^2 * 3^1 -> r(12) = 2/2 + 1/3 = (6+2)/6, therefore a(12)=3, A083345(12)=4;
n=18 = 2*3*3 = 2^1 * 3^2 -> r(18) = 1/2 + 2/3 = (3+4)/6, therefore a(18)=6, A083345(18)=7.

Links

Crossrefs

Cf. A083345 (numerator).
Cf. A035263 (parity of terms), A003159 (positions of odd terms), A036554 (of even terms).
Cf. A065463, A072873, A083347, A083348, A359588 (Dirichlet inverse).

Programs

  • Mathematica
    a[n_] := Product[Module[{p, e}, {p, e} = pe; If[Divisible[e, p], 1, p]], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Oct 06 2021 *)
  • PARI
    A083346(n) = { my(f=factor(n)); denominator(vecsum(vector(#f~,i,f[i,2]/f[i,1]))); }; \\ Antti Karttunen, Mar 01 2018

Formula

Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463 * Product_{p prime} (p^(2*p)*(p^2+p-1)-p^3)/((p^2+p-1)*(p^(2*p)-1)) = 0.3374565531... . - Amiram Eldar, Nov 18 2022

Extensions

Incorrect formula removed by Antti Karttunen, Jan 09 2023

A056832 All a(n) = 1 or 2; a(1) = 1; get next 2^k terms by repeating first 2^k terms and changing last element so sum of first 2^(k+1) terms is odd.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1
Offset: 1

Views

Author

Jonas Wallgren, Aug 30 2000

Keywords

Comments

Dekking (2016) calls this the Toeplitz sequence or period-doubling sequence. - N. J. A. Sloane, Nov 08 2016
Fixed point of the morphism 1->12 and 2->11 (1 -> 12 -> 1211 -> 12111212 -> ...). - Benoit Cloitre, May 31 2004
a(n) is multiplicative. - Christian G. Bower, Jun 03 2005
a(n) is the least k such that A010060(n-1+k) = 1 - A010060(n-1); the sequence {a(n+1)-1} is the characteristic sequence for A079523. - Vladimir Shevelev, Jun 22 2009
The squarefree part of the even part of n. - Peter Munn, Dec 03 2020

Examples

			1 -> 1,2 -> 1,2,1,1 -> 1,2,1,1,1,2,1,2 -> 1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1.
Here we have 1 element, then 2 elements, then 4, 8, 16, etc.

References

  • Manfred R. Schroeder, Fractals, Chaos, Power Laws, W. H. Freeman, NY, 1991; pp. 277-279.

Links

Crossrefs

Cf. A197911 (partial sums).
Essentially same as first differences of Thue-Morse, A010060. - N. J. A. Sloane, Jul 02 2015
See A035263 for an equivalent version.
Limit of A317956(n) for large n.
Row/column 2 of A059895.
Positions of 1s: A003159.
Positions of 2s: A036554.
A002425, A006519, A079523, A096268, A214682, A234957 are used in a formula defining this sequence.
A059897 is used to express relationship between terms of this sequence.

Programs

  • Haskell
    a056832 n = a056832_list !! (n-1)
a056832_list = 1 : f [1] where
   f xs = y : f (y : xs) where
          y = 1 + sum (zipWith (*) xs $ reverse xs) `mod` 2
-- Reinhard Zumkeller, Jul 29 2014
  • Mathematica
    Nest[ Function[l, {Flatten[(l /. {1 -> {1, 2}, 2 -> {1, 1}})]}], {1}, 7] (* Robert G. Wilson v, Mar 03 2005 *)
Table[Mod[-(-1)^(n + 1) (-1)^n Numerator[EulerE[2 n + 1, 1]], 3] , {n, 0, 120}] (* Michael De Vlieger, Aug 15 2016, after Jean-François Alcover at A002425 *)
  • PARI
    a(n)=numerator(2/n*(4^n-1)*bernfrac(2*n))%3
  • PARI
    a(n)=if(n<1, 0, valuation(n,2)%2+1) /* Michael Somos, Jun 18 2005 */
  • Python
    def A056832(n): return 1+((~n&n-1).bit_length()&1) # Chai Wah Wu, Jan 09 2023

Formula

a(n) = ((-1)^(n+1)*A002425(n)) modulo 3. - Benoit Cloitre, Dec 30 2003
a(1)=1, a(n) = 1 + ((Sum_{i=1..n-1} a(i)*a(n-i)) mod 2). - Benoit Cloitre, Mar 16 2004
a(n) is multiplicative with a(2^e) = 1 + (1-(-1)^e)/2, a(p^e)=1 if p > 2. - Michael Somos, Jun 18 2005
[a(2^n+1) .. a(2^(n+1)-1)] = [a(1) .. a(2^n-1)]; a(2^(n+1)) = 3 - a(2^n).
For n > 0, a(n) = 2 - A035263(n). - Benoit Cloitre, Nov 24 2002
a(n)=2 if n-1 is in A079523; a(n)=1 otherwise. - Vladimir Shevelev, Jun 22 2009
a(n) = A096268(n-1) + 1. - Reinhard Zumkeller, Jul 29 2014
From Peter Munn, Dec 03 2020: (Start)
a(n) = A007913(A006519(n)) = A006519(n)/A234957(n).
a(n) = A059895(n, 2) = n/A214682(n).
a(n*k) = (a(n) * a(k)) mod 3.
a(A059897(n, k)) = A059897(a(n), a(k)).
(End)
Asymptotic mean: lim_{m->oo} (1/m) * Sum__{k=1..m} a(k) = 4/3. - Amiram Eldar, Mar 09 2021

A345452 Positive integers with an even number of prime factors (counting repetitions) that sum to an even number.

Original entry on oeis.org

1, 4, 9, 15, 16, 21, 25, 33, 35, 36, 39, 49, 51, 55, 57, 60, 64, 65, 69, 77, 81, 84, 85, 87, 91, 93, 95, 100, 111, 115, 119, 121, 123, 129, 132, 133, 135, 140, 141, 143, 144, 145, 155, 156, 159, 161, 169, 177, 183, 185, 187, 189, 196, 201, 203, 204, 205, 209, 213, 215
Offset: 1

Views

Author

Peter Munn, Jun 20 2021

Keywords

Comments

Numbers with an even number of even prime factors and an even number of odd prime factors.
The representation (as defined in A206284) of polynomials with nonnegative integer coefficients that are in the ideal of the polynomial ring Z[x] generated by x^2+x and 2.
The above property arises because the sequence lists the integers in the multiplicative subgroup of positive rational numbers generated by the squares of primes (A001248) and the products of two consecutive odd primes (A006094\{6}).
The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 4 and 15. For example, A003961(4) = 9, A003961(9) = 25, A003961(15) = 35, 15 * 35 = 525, 525/25 = 21. Alternatively, the sequence may be defined as the closure of A046337 under multiplication by 4.
From the properties of subgroups of the positive rationals we know that if we take an absent positive integer m and divide all terms that are multiples of m by m, we get all the integers in the same subgroup coset as m, and we can expect some of the nice properties here to carry over to the resulting set. Specifically, dividing the even terms by 2 gives all numbers with an odd number of prime factors that sum to an even number; dividing all terms divisible by an odd prime p by p, gives all numbers with an odd number of prime factors that sum to an odd number. The positive integers satisfying the 4th of the 4 possibilities are generated similarly, dividing by 6 (for example).
Numbers whose squarefree part is in A056913.
Term by term, the sequence is one half of its complement within A036349.

Examples

			The definition specifies that we count repeated prime factors.
6 = 2 * 3; the sum of these prime factors is 2 + 3 = 5, an odd number; so 6 is not in the sequence.
50 = 2 * 5 * 5 has 3 prime factors and 3 is an odd number; so 50 is not in the sequence.
60 = 2 * 2 * 3 * 5 has 4 prime factors and 4 is an even number; the sum of these factors is 2 + 2 + 3 + 5 = 12, also an even number; so 60 is in the sequence.
1 has 0 prime factors, which sum to 0 (the empty sum). 0 is even, so 1 is in the sequence.

Links

Crossrefs

Cf. A001222, A001414, A003961, A059897, A307150.
Intersection of any 2 of A003159, A028260, A036349.
Other lists that have conditions on the number of odd prime factors: A046337, A072978.
Subsequences: A001248, A006094\{6}, A046315, A056913.

Programs

  • Mathematica
    {1}~Join~Select[Range@1000,(s=Flatten[Table@@@FactorInteger[#]];And@@EvenQ@{Length@s,Total@s})&] (* Giorgos Kalogeropoulos, Jun 24 2021 *)
  • PARI
    iseven(x) = ((x%2) == 0);
isok(m) = my(f=factor(m)); iseven(sum(k=1, #f~, f[k,1]*f[k,2])) && iseven(sum(k=1, #f~, f[k,2])); \\ Michel Marcus, Jun 24 2021
  • PARI
    is(n) = bigomega(n)%2 == 0 && valuation(n, 2)%2 == 0 \\ David A. Corneth, Jun 24 2021
  • Python
    from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values())%2 == 0 and sum(p*f[p] for p in f)%2 == 0
print(list(filter(ok, range(1, 216)))) # Michael S. Branicky, Jun 24 2021

Formula

{a(n) : n >= 1} = {m >= 1 : A001222(m) mod 2 = A001414(m) mod 2 = 0}.
{A036349(n) : n >= 1} = {a(n) : n >= 1} U {2 * a(n) : n >= 1}.
{A028260(n) : n >= 1} = {a(n) : n >= 1} U {A307150(a(n)) : n >= 1}.
For odd prime p, {A003159(n) : n >= 1} = {a(n) : n >= 1} U {A059897(a(n), p) : n >= 1}.

A300841 Fermi-Dirac factorization prime shift towards larger terms: a(n) = A052330(2*A052331(n)).

Original entry on oeis.org

1, 3, 4, 5, 7, 12, 9, 15, 11, 21, 13, 20, 16, 27, 28, 17, 19, 33, 23, 35, 36, 39, 25, 60, 29, 48, 44, 45, 31, 84, 37, 51, 52, 57, 63, 55, 41, 69, 64, 105, 43, 108, 47, 65, 77, 75, 49, 68, 53, 87, 76, 80, 59, 132, 91, 135, 92, 93, 61, 140, 67, 111, 99, 85, 112, 156, 71, 95, 100, 189, 73, 165, 79, 123, 116, 115, 117, 192, 81
Offset: 1

Views

Author

Antti Karttunen, Apr 12 2018

Keywords

Comments

With n having a unique factorization as A050376(i) * A050376(j) * ... * A050376(k), with i, j, ..., k all distinct, a(n) = A050376(1+i) * A050376(1+j) * ... * A050376(1+k).
Multiplicative because for coprime m and n the Fermi-Dirac factorizations of m and n are disjoint and their union is the Fermi-Dirac factorization of m * n. - Andrew Howroyd, Aug 02 2018

Examples

			For n = 6 = A050376(1)*A050376(2), a(6) = A050376(2)*A050376(3) = 3*4 = 12.
For n = 12 = A050376(2)*A050376(3), a(12) = A050376(3)*A050376(4) = 4*5 = 20.

Links

Crossrefs

Cf. A050376, A052330, A052331, A059897, A300840 (a left inverse).
Cf. also A003961.
Range of values is A003159.

Programs

  • Mathematica
    fdPrimeQ[n_] := Module[{f = FactorInteger[n], e}, Length[f] == 1 && (2^IntegerExponent[(e = f[[1, 2]]), 2] == e)];
nextFDPrime[n_] := Module[{k = n + 1}, While[! fdPrimeQ[k], k++]; k];
fd[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Table[If[b[[j]] > 0, p^(2^(m - j)), Nothing], {j, 1, m}]];
a[n_] := Times @@ nextFDPrime /@ Flatten[fd @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)
  • PARI
    up_to_e = 8192;
v050376 = vector(up_to_e);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A300841(n) = A052330(2*A052331(n));

Formula

a(n) = A052330(2*A052331(n)).
For all n >= 1, a(A050376(n)) = A050376(1+n).
For all n >= 1, A300840(a(n)) = n.
a(A059897(n,k)) = A059897(a(n), a(k)). - Peter Munn, Nov 23 2019
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