A171182 -id:A171182 - cp's OEIS frontend

A249223 Triangle read by rows: row n gives partial alternating sums of row n of A237048.

1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2

Offset: 1

Hartmut F. W. Hoft, Oct 23 2014

All entries in the triangle are nonnegative since the number of 1's in odd-numbered columns of A237048 prior to column j, 1 <= j <= row(n), is at least as large as the number of 1's in even-numbered columns through column j. As a consequence:
(a) The two adjacent symmetric Dyck paths whose legs are defined by adjacent rows of triangle A237593 never cross each other (see also A235791 and A237591) and the rows in this triangle describe the widths between the legs.
(b) Let legs(n) denote the n-th row of triangle A237591, widths(n) the n-th row of this triangle, and c(n) the rightmost entry in the n-th row of this triangle (center of the Dyck path). Then area(n) = 2 * legs(n) . width(n) - c(n), where "." is the inner product, is the area between the two adjacent symmetric Dyck paths.
(c) For certain sequences of integers, it is known that area(n) = sigma(n); see A238443, A245685, A246955, A246956 and A247687.
Right border gives A067742. - Omar E. Pol, Jan 21 2017
For a proof that T(n, k) = |{ d : d|n and k/2 < d <= k }|, for 1 <= k <= row(n), where row(n) is the length of row n, an identity suggested by Peter Munn, see the link. A corollary to it is that the number of divisors of n in the half-open interval (row(n)/2, row(n)] equals the width of the symmetric representation of n at the diagonal: T(n, row(n)) = | { d : d|n and row(n)/2 < d <= row(n) } |. See also the comments and conjectures of Michel Marcus in A067742 and A237593. - Hartmut F. W. Hoft, Jun 24 2024
From Omar E. Pol, Jul 24 2024: (Start)
Conjecture 1: Every column is a periodic sequence.
Conjecture 2: The periods of the columns 1..8 are respectively: 1, 2, 6, 12, 60, 60, 420, 840.
Question 1: Is the period of the column k equal to A003418(k)? (End).
From Omar E. Pol, Jul 26 2024: (Start)
Column 1 gives A000012.
Column 2 gives A000035.
Conjecture 3: Column 3 gives [2, 0] together with A115357, hence column 3 gives 2 together with A171182.
Question 2: Except the first nine terms of A337976, is the column 4 the same as A337976?
Question 3: Except the first 14 terms of A366981, is the column 5 the same as A366981? (End)
From Hartmut F. W. Hoft, Aug 01 2024: (Start)
Conjectures 1 and 2 are true and the answer to question 1 is affirmative.
By definition, each column k in triangle T237048(n, k) of sequence A237048 is a periodic sequence of period k. Since the k-th term in row n of the triangle T(n, k) = Sum_{i=1..k} (-1)^(i+1) * T237048(n, i), with 1 <= k <= A003056(n), each initial subsequence T(n, 1) .. T(n, k) of row n in this triangle is periodic of period lcm(1, .. , k) = A003418(k). This implies that each column k in this sequence has period A003418(k).
Conjecture 3 and Question 2 are true. Since T237048(n, 1) = 1, T237208(n, 2) = 1 if n odd and 0 if n even, T237048(n, 3) = 1 if 3|n and 0 otherwise, and T237048(n, 4) = 1 if 4|(n-2) and 0 otherwise, equations T249223(n, 3) = 1 - (n mod 2) + delta( n mod 3) and T249223(n, 4) = 1 - (n mod 2) + delta( n mod 3) - delta( (n-2) mod 4) hold where delta(k) = 1 if k = 0 and 0 otherwise. With the 3rd column starting at n = A000217(3) = 6, each period starting in a row that is a multiple of 6 is [ 2 0 1 1 1 0 ], and appropriate shifts yield A115357 and A171182. With the 4th column starting at n = A000217(4) = 10, each period starting in a row n with 12|(n+2) is [ 0 0 2 0 0 1 1 0 1 0 1 1 ], and with a shift of 9 yields the apparently periodic A337976(10), A337976(11), ... (End)

			Triangle begins:
---------------------------
   n \ k  1  2  3  4  5  6
---------------------------
   1 |    1;
   2 |    1;
   3 |    1, 0;
   4 |    1, 1;
   5 |    1, 0;
   6 |    1, 1, 2;
   7 |    1, 0, 0;
   8 |    1, 1, 1;
   9 |    1, 0, 1;
  10 |    1, 1, 1, 0;
  11 |    1, 0, 0, 0;
  12 |    1, 1, 2, 2;
  13 |    1, 0, 0, 0;
  14 |    1, 1, 1, 0;
  15 |    1, 0, 1, 1, 2;
  16 |    1, 1, 1, 1, 1;
  17 |    1, 0, 0, 0, 0;
  18 |    1, 1, 2, 1, 1;
  19 |    1, 0, 0, 0, 0;
  20 |    1, 1, 1, 1, 2;
  21 |    1, 0, 1, 1, 1, 0;
  22 |    1, 1, 1, 0, 0, 0;
  23 |    1, 0, 0, 0, 0, 0;
  24 |    1, 1, 2, 2, 2, 2;
  ...
The triangle shows that area(n) has width 1 for powers of 2 and that area(p) for primes p consists of only 1 horizontal leg of width 1 (and its symmetric vertical leg in the mirror symmetric duplicate of this triangle).

G. C. Greubel, Table of n, a(n) for the first 150 rows, flattened
Hartmut F. W. Hoft, Divisors d of n in half open interval (k/2, k]

Row n has length A003056(n).
Column k starts in row A000217(k).
Cf. A000012, A000035, A000203, A003418, A067742, A115357, A171182, A237048, A237270, A237271, A237593, A238443, A245685, A246955, A246956, A247687, A337976, A366981.

r := proc(n) floor((sqrt(1+8*n)-1)/2) ; end proc: # R. J. Mathar 2015 A003056
A237048:=proc(n,k) local i; global r;
if n<(k-1)*k/2 or k>r(n) then return(0); fi;
if (k mod 2)=1 and (n mod k)=0 then return(1); fi;
if (k mod 2)=0 and ((n-k/2) mod k) = 0 then return(1); fi;
return(0);
end;
A249223:=proc(n,k) local i; global r,A237048;
if n<(k-1)*k/2 or k>r(n) then return(0); fi;
add( (-1)^(i+1)*A237048(n,i),i=1..k);
end;
for n from 1 to 12 do lprint([seq(A249223(n,k),k=1..r(n))]); od; # N. J. A. Sloane, Jan 15 2021

cd[n_, k_] := If[Divisible[n, k], 1, 0]; row[n_] := Floor[(Sqrt[8 n + 1] - 1)/2]; a237048[n_, k_] := If[OddQ[k], cd[n, k], cd[n - k/2, k]];
a1[n_, k_] := Sum[(-1)^(j + 1)*a237048[n, j], {j, 1, k}];
a2[n_] := Drop[FoldList[Plus, 0, Map[(-1)^(# + 1) &, Range[row[n]]] a237048[n]], 1]; Flatten[Map[a2, Range[24]]] (* data *) (* Corrected by G. C. Greubel, Apr 16 2017 *)

t237048(n,k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0);
kmax(n) = (sqrt(1+8*n)-1)/2;
t(n,k) = sum(j=1, k, (-1)^(j+1)*t237048(n,j));
tabf(nn) = {for (n=1, nn, for (k=1, kmax(n), print1(t(n,k), ", ");); print(););} \\ Michel Marcus, Sep 20 2015

T(n, k) = Sum_{j=1..k} (-1)^(j+1)*A237048(n, j), for n>=1 and 1 <= k <= floor((sqrt(8*n + 1) - 1)/2). - corrected by Hartmut F. W. Hoft, Jan 25 2018

A210679 Number of distinct prime factors <= 7 of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 2, 0, 2, 2, 1, 0, 2, 0, 2, 2, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 2, 2, 0, 1, 1, 2, 0, 3, 0, 1, 2, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 2, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1

Offset: 1

Reinhard Zumkeller, Apr 01 2012

Periodic with period length 210. - Amiram Eldar, Sep 16 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-3,-5,-6,-6,-5,-3,0,3,5,6,6,5,3,1).

Cf. A001221, A002473, A008364, A027748, A080672, A165743.

Number of distinct prime factors <= p: A171182 (p=3), A178146 (p=5), this sequence (p=7).

a210679 = length . takeWhile (<= 7) . a027748_row

Join[{0},Table[Count[FactorInteger[n][[All,1]],?(#<8&)],{n,2,100}]] (* _Harvey P. Dale, Aug 18 2021 *)
a[n_] := PrimeNu[GCD[n, 210]]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

a(n) = omega(gcd(n, 210)); \\ Amiram Eldar, Sep 16 2023

a(n) <= 4.
a(A008364(n)) = 0; a(A080672(n)) > 0.
a(n) = A001221(n) iff n is 7-smooth: a(A002473(n)) = A001221(A002473(n)). [corrected by Amiram Eldar, Sep 16 2023]
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 7, and 0 otherwise.
a(n) = A001221(A165743(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 247/210. (End)

A115357 Period 6: repeat [1,1,1,0,2,0].

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0

Offset: 0

Paul Barry, Jan 21 2006

Diagonal sums of number triangle A115356.

Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).

Cf. A171182. [From Juri-Stepan Gerasimov, Dec 04 2009]

Cf. A115356.

PadRight[{},120,{1,1,1,0,2,0}] (* or *) LinearRecurrence[{-1,0,1,1},{1,1,1,0},120] (* Harvey P. Dale, Jan 28 2015 *)

G.f.: (1+2*x+2*x^2)/(1+x-x^3-x^4).
a(n) = -a(n-1)+a(n-3)+a(n-4).
a(n) = -cos(2*Pi*n/3+Pi/6)/sqrt(3)+sin(2*Pi*n/3+Pi/6)/3+(-1)^n/2+5/6.
Sequence shifted right by 2 is additive with a(p^e) = 1 if p = 2 or 3, 0 otherwise.

A178144 Sum of divisors d of n which are d=2, 3 or 5.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8, 2, 0, 5, 0, 7, 3, 2, 0, 5, 5, 2, 3, 2, 0, 10, 0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8, 2, 0, 5, 0, 7, 3, 2, 0, 5, 5, 2, 3, 2, 0, 10, 0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8, 2, 0, 5, 0, 7, 3, 2, 0, 5, 5, 2, 3, 2, 0, 10, 0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8

Offset: 1

Vladimir Shevelev, May 21 2010

The sequence is periodic with period length 30.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1,0,1,2,2,1).

Cf. A000203, A008472, A178143, A171182, A178142.

A178144 := proc(n)
    local a;
    a := 0 ;
    for d in {2,3,5} do
        if (n mod d) = 0 then
            a := a+d ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 23 2012

a[n_] := DivisorSum[n, Boole[MatchQ[#, 2|3|5]]*#&];
Array[a, 105] (* Jean-François Alcover, Nov 24 2017 *)
a[n_] := Sum[d * Boole[Divisible[n, d]], {d, {2, 3, 5}}]; Array[a, 100] (* Amiram Eldar, Dec 20 2024 *)

a(n) = sumdiv(n, d, if ((d==2) || (d==3) || (d==5), d)); \\ Michel Marcus, Nov 24 2017

a(n) = my(d = [2, 3, 5]); sum(k = 1, 3, d[k] * !(n % d[k])); \\ Amiram Eldar, Dec 20 2024

From R. J. Mathar, Jul 23 2012: (Start)
a(n) = -2*a(n-1) -2*a(n-2) -a(n-3) +a(n-5) +2*a(n-6) +2*a(n-7) +a(n-8).
G.f.: ( -x*(2+7*x+12*x^2+17*x^3+22*x^4+10*x^6+20*x^5) ) / ( (x-1)*(1+x)*(1+x+x^2)*(x^4+x^3+x^2+x+1) ). (End)

A178146 a(n) is the number of distinct prime factors <= 5 of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2

Offset: 1

Vladimir Shevelev, May 21 2010

The sequence is periodic with period {0 1 1 1 1 2 0 1 1 2 0 2 0 1 2 1 0 2 0 2 1 1 0 2 1 1 1 1 0 3} of length 30. There are 26 coincidences on the interval [1,30] with A156542.

Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009.
Index entries for linear recurrences with constant coefficients, signature (-2,-2,-1,0,1,2,2,1).

Cf. A000005, A001221, A156542, A171182, A178143, A178144, A178142.
Cf. A059841, A079978, A079998, A355582, A356006.
Number of distinct prime factors <= p: A171182 (p=3), this sequence (p=5), A210679 (p=7).

Rest@ CoefficientList[Series[-x^2*(3*x^6 + 6*x^5 + 7*x^4 + 6*x^3 + 5*x^2 + 3*x + 1)/((x - 1)*(x + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* G. C. Greubel, May 16 2017 *)
LinearRecurrence[{-2,-2,-1,0,1,2,2,1},{0,1,1,1,1,2,0,1},120] (* Harvey P. Dale, Sep 29 2021 *)
a[n_] := PrimeNu[GCD[n, 30]]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

my(x='x+O('x^50)); concat([0], Vec(-x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1)/((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)))) \\ G. C. Greubel, May 16 2017

a(n) = omega(gcd(n, 30)); \\ Amiram Eldar, Sep 16 2023

a(n) = a(n-2) + a(n-3) - a(n-7) - a(n-8) + a(n-10), a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 1, a(5) = 1, a(6) = 2, a(7) = 0, a(8) = 1, a(9) = 1, a(10) = 2.
G.f.: -x^2*(3*x^6+6*x^5+7*x^4+6*x^3+5*x^2+3*x+1) / ((x-1)*(x+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)). - Colin Barker, Mar 13 2013
From Amiram Eldar, Sep 16 2023: (Start)
Additive with a(p^e) = 1 if p <= 5, and 0 otherwise.
a(n) = A059841(n) + A079978(n) + A079998(n).
a(n) = A001221(gcd(n, 30)).
a(n) = A001221(A355582(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 31/30. (End)

Name edited by Amiram Eldar, Sep 16 2023

A240871 Number of partitions p of n into distinct parts such that max(p) = 3 + min(p).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0

Offset: 0

Clark Kimberling, Apr 15 2014

			a(7) counts these 2 partitions: 5+2, 4+2+1.

Antti Karttunen, Table of n, a(n) for n = 0..20000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (-1,-1,0,1,1,1).

Cf. A128508, A171182, A240872, A240873.

z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; Table[Count[f[n], p_ /; Max[p] == 2 + Min[p]], {n, 0, z}] (* A171182 *)
    Table[Count[f[n], p_ /; Max[p] == 3 + Min[p]], {n, 0, z}] (* A240871 *)
    Table[Count[f[n], p_ /; Max[p] == 4 + Min[p]], {n, 0, z}] (* A240872 *)
    Table[Count[f[n], p_ /; Max[p] == 5 + Min[p]], {n, 0, z}] (* A240873 *)

A240871aux(n, minp=0, maxp=0) = if(0==n, (minp>0 && (maxp==3+minp)), sum(i=1+maxp, min(n,3+minp), A240871aux(n-i, if(!minp,i,minp), i)));
A240871(n) = sum(i=1,floor(n/2),A240871aux(n-i,i,i)); \\ Antti Karttunen, Jan 13 2025

From Alois P. Heinz, Jan 13 2025: (Start)
G.f.: -x^5*(x^7+2*x^6+3*x^5+3*x^4+3*x^3+3*x^2+x+1)/((x-1)*(x+1)*(x^2+1)*(x^2+x+1)).
a(n) = a(n-12) for n>=19. (End)

More terms from Antti Karttunen, Jan 13 2025

A240872 Number of partitions p of n into distinct parts such that max(p) = 4 + min(p).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4

Offset: 0

Clark Kimberling, Apr 15 2014

			a(12) counts these 3 partitions:  84, 642, 5421.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,0,1,1,1,1).

Cf. A171182, A240871, A240873.

    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
    Table[Count[f[n], p_ /; Max[p] == 2 + Min[p]], {n, 0, z}] (* A171182 *)
    Table[Count[f[n], p_ /; Max[p] == 3 + Min[p]], {n, 0, z}] (* A240871 *)
    Table[Count[f[n], p_ /; Max[p] == 4 + Min[p]], {n, 0, z}] (* A240872 *)
    Table[Count[f[n], p_ /; Max[p] == 5 + Min[p]], {n, 0, z}] (* A240873 *)

G.f.: -x^6*(x^12+2*x^11+3*x^10+5*x^9+5*x^8+6*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+x+1) / ((x-1)*(x+1)*(x^2+1)*(x^4+x^3+x^2+x+1)). - Alois P. Heinz, Jun 18 2025

A240873 Number of partitions p of n into distinct parts such that max(p) = 5 + min(p).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 3, 3, 3, 4, 4, 4, 5, 4, 3, 6, 4, 4, 5, 3, 4, 6, 4, 4, 4, 4, 4, 6, 4, 3, 5, 4, 4, 6, 3, 4, 5, 4, 4, 5, 4, 4, 5, 4, 3, 6, 4, 4, 5, 3, 4, 6, 4, 4, 4, 4, 4, 6, 4, 3, 5, 4, 4, 6, 3, 4, 5, 4, 4, 5, 4, 4, 5, 4, 3

Offset: 0

Clark Kimberling, Apr 15 2014

			a(12) counts these 3 partitions:  732, 651, 6321.

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,0,1,1,1,1,1).

Cf. A171182, A240871, A240872.

z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] == 2 + Min[p]], {n, 0, z}] (* A171182 *)
Table[Count[f[n], p_ /; Max[p] == 3 + Min[p]], {n, 0, z}] (* A240871 *)
Table[Count[f[n], p_ /; Max[p] == 4 + Min[p]], {n, 0, z}] (* A240872 *)
Table[Count[f[n], p_ /; Max[p] == 5 + Min[p]], {n, 0, z}] (* A240873 *)

A171157 Number of distinct primes > 3 that divide n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Juri-Stepan Gerasimov, Dec 04 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001221, A171182.

omega := proc(n) nops(numtheory[factorset](n)) ; end proc:
A171182 := proc(n) op(1+ (n mod 6),[2,0,1,1,1,0]) ; end proc:
A171157 := proc(n) omega(n)-A171182(n) ; end proc: seq(A171157(n),n=1..120) ; # R. J. Mathar, Dec 09 2009

Table[PrimeNu[n] - (5 + 3*Cos[n*Pi] + 4*Cos[2*n*Pi/3])/6, {n, 1, 100}] (* G. C. Greubel, May 16 2017 *)
Table[Count[FactorInteger[n][[;;,1]],?(#>3&)],{n,110}] (* _Harvey P. Dale, Nov 16 2024 *)

for(n=1,100, print1(round(omega(n) - (5 + 3*cos(n*Pi) + 4*cos(2*n*Pi/3))/6), ", ")) \\ G. C. Greubel, May 16 2017

a(n) = A001221(n) - A171182(n).

cp's OEIS Frontend

A249223 Triangle read by rows: row n gives partial alternating sums of row n of A237048.

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A210679 Number of distinct prime factors <= 7 of n.

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A115357 Period 6: repeat [1,1,1,0,2,0].

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A178144 Sum of divisors d of n which are d=2, 3 or 5.

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A178146 a(n) is the number of distinct prime factors <= 5 of n.

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A240871 Number of partitions p of n into distinct parts such that max(p) = 3 + min(p).

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A240872 Number of partitions p of n into distinct parts such that max(p) = 4 + min(p).

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