Maxim Karimov - cp's OEIS frontend

A370108 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors in which the convex hull of a set of beads of any color A can be transformed by rotation into the convex hull of a set of beads of any other color B (n >= 1, k >= 1).

0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 6, 2, 2, 0, 0, 10, 8, 6, 0, 0, 0, 15, 20, 18, 0, 2, 0, 0, 21, 40, 50, 0, 10, 0, 0, 0, 28, 70, 120, 24, 28, 0, 4, 0, 0, 36, 112, 252, 144, 60, 0, 12, 0, 0, 0, 45, 168, 476, 504, 230, 0, 54, 8, 4, 0

Offset: 1

Maxim Karimov and Vladislav Sulima, Feb 10 2024

It is assumed that all beads lie on a circle and distance between any two adjacent is the same.

			n\k| 1 2  3  4   5   6    7    8     9 ...
---+----------------------------------
 1 | 0 0  0  0   0   0    0    0     0 ... A000007
 2 | 0 1  3  6  10  15   21   28    36 ... A000217
 3 | 0 0  2  8  20  40   70  112   168 ... A007290
 4 | 0 2  6 18  50 120  252  476   828 ... A062026
 5 | 0 0  0  0  24 144  504 1344  3024 ... A059593
 6 | 0 2 10 28  60 230 1022 3640 10488
 7 | 0 0  0  0   0   0  720 5760 25920 ... A153760
 8 | 0 4 12 54 190 510 1134 7252 49284
 9 | 0 0  8 32  80 160  280  448 40992
...

Cf. A000007, A000013, A000217, A007290, A062026, A059593, A153760.

T(n,2) = A000013(ceiling(n/2)) * [n mod 2 == 0], where [] is the Iverson bracket.

For prime p, T(p,k) = (p-1)! * binomial(k,p).

A369878 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors in which the convex hull of a set of beads of any color A cannot be transformed by rotation into the convex hull of a set of beads of any other color B (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 4, 1, 5, 4, 9, 4, 1, 6, 5, 16, 9, 8, 1, 7, 6, 25, 16, 27, 12, 1, 8, 7, 36, 25, 64, 93, 20, 1, 9, 8, 49, 36, 125, 304, 243, 32, 1, 10, 9, 64, 49, 216, 705, 928, 699, 60, 1, 11, 10, 81, 64, 343, 1356, 2405, 4384, 2019, 104, 1

Offset: 1

Maxim Karimov and Vladislav Sulima, Feb 03 2024

It is assumed that all beads lie on a circle and that the distance between any two adjacent beads is the same.

			n\k| 1  2    3     4     5      6      7      8       9 ...
---+---------------------------------------------------
 1 | 1  2    3     4     5      6      7      8       9 ... A000027
 2 | 1  2    3     4     5      6      7      8       9 ... A000027
 3 | 1  4    9    16    25     36     49     64      81 ... A000290
 4 | 1  4    9    16    25     36     49     64      81 ... A000290
 5 | 1  8   27    64   125    216    343    512     729 ... A000578
 6 | 1 12   93   304   705   1356   2317   3648    5409
 7 | 1 20  243   928  2405   5076   9415  15968   25353
 8 | 1 32  699  4384 15245  39216  84007 159104  275769
 9 | 1 60 2019 18256 72765 202236 457135 902784 1620441
...

Cf. A000012, A000027, A000290, A000578, A063776.

T(n,2) = A063776(n).

A369524 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors with black beads always occurring in runs of even length.

Original entry on oeis.org

0, 1, 1, 2, 2, 0, 3, 4, 2, 1, 4, 7, 6, 3, 0, 5, 11, 14, 11, 3, 1, 6, 16, 28, 34, 18, 5, 0, 7, 22, 50, 87, 81, 38, 5, 1, 8, 29, 82, 191, 276, 227, 70, 8, 0, 9, 37, 126, 373, 759, 983, 615, 151, 10, 1, 10, 46, 184, 666, 1782, 3301, 3500, 1789, 314, 15, 0, 11, 56, 258, 1109, 3717, 9180, 14545, 13007, 5206, 684, 19, 1

Offset: 1

Maxim Karimov and Vladislav Sulima, Jan 25 2024

Equivalently, black beads can be considered to have length 2, while all other beads have length 1.

Column k is the "CIK" (necklace, indistinct, unlabeled) transform of {k-1, 1, 0, 0, 0, ...} (see C. Bower link). - Andrew Howroyd, Jan 25 2024

			n\k| 1  2   3     4      5       6       7        8         9 ...
---+-----------------------------------------------------------------
 1 | 0  1   2     3      4       5       6        7         8 ...A001477
 2 | 1  2   4     7     11      16      22       29        37 ...A000124
 3 | 0  2   6    14     28      50      82      126       184 ...A033547
 4 | 1  3  11    34     87     191     373      666      1109
 5 | 0  3  18    81    276     759    1782     3717      7080
 6 | 1  5  38   227    983    3301    9180    22163     47997
 7 | 0  5  70   615   3500   14545   48210   135155    333400
 8 | 1  8 151  1789  13007   66166  260113   844691   2370229
 9 | 0 10 314  5206  48820  304970 1423790  5358934  17110376
10 | 1 15 684 15490 186195 1425453 7897006 34438104 125093109
...

C. G. Bower, Transforms (2).

Columns 1..2 are A000035(n-1), A000358.
Rows 1..3 are A001477(k-1), A000124(k-1), A033547(k-1).
Cf. A000010 (phi), A075195 (all beads of same length).

function [res] = num2(n,k)
res=0;
for d=divisors(n)
    s=(k-1)^d;
    for i=1:floor(d/2)
        s=s + nchoosek(d-i-1,i-1) * d/i * (k-1)^(d-2*i);
    end
    res= res + eulerPhi(n/d) * s;
end
res=res/n;
end

T(n,k) = sum(d=1, n, eulerphi(d)*polcoef(log(1/(1 - (k-1)*x^d - x^(2*d)) + O(x*x^n)), n)/d)  \\ Andrew Howroyd, Jan 25 2024

T(n,k) = (1/n) * Sum_{d|n} phi(n/d) * ((k-1)^d + Sum_{i=1..floor(d/2)} binomial(d-i-1,i-1) * d/i * (k-1)^(d-2*i)), where phi(n) = A000010.

G.f. of column k: Sum_{d>=1} (phi(d)/d) * log(1/(1 - (k-1)*x^d - x^(2*d))). - Andrew Howroyd, Jan 25 2024

A347351 Triangle read by rows: T(n,k) is the number of links of length k in a set of all necklaces A000358 of length n, 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 5, 1, 1, 0, 1, 6, 4, 2, 1, 0, 1, 7, 3, 2, 1, 1, 0, 1, 8, 8, 3, 3, 1, 1, 0, 1, 9, 8, 7, 3, 2, 1, 1, 0, 1, 10, 18, 9, 5, 4, 2, 1, 1, 0, 1, 11, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1, 12, 40, 24, 16, 8, 6, 3, 2, 1, 1, 0, 1, 13, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1

Offset: 0

Maxim Karimov and Vladislav Sulima, Aug 28 2021

Definitions:
1. A link is any 0 in any necklace from A000358 and all 1s following this 0 in this necklace to right until another 0 is encountered.
2. Length of the link is the number of elements in the link.
Sum of all elements n-row is Fibonacci(n-1)+n iff n=1 or n=p (follows from the identity for the sum of the Fibonacci numbers and the formula for the triangle T(n,k)).

			For k > 0:
   n\k |  1   2   3   4   5   6   7   8   9  10  ...
  -----+---------------------------------------
   1   |  1
   2   |  2   1
   3   |  3   0   1
   4   |  4   2   0   1
   5   |  5   1   1   0   1
   6   |  6   4   2   1   0   1
   7   |  7   3   2   1   1   0   1
   8   |  8   8   3   3   1   1   0   1
   9   |  9   8   7   3   2   1   1   0   1
  10   | 10  18   9   5   4   2   1   1   0   1
  ...
If we continue the calculation for nonpositive k, we get a table in which each row is a Fibonacci sequence, in which term(0) = A113166, term(1) = A034748.
For k <= 0:
   n\k |  0   -1   -2   -3   -4   -5   -6   -7   -8   -9 ...
  -----+------------------------------------------------
   1   |  0    1    1    2    3    5    8   13   21   34 ... A000045
   2   |  1    2    3    5    8   13   21   34   55   89 ... A000045
   3   |  1    4    5    9   14   23   37   60   97  157 ... A000285
   4   |  3    6    9   15   24   39   63  102  165  267 ... A022086
   5   |  3    9   12   21   33   54   87  141  228  369 ... A022379
   6   |  8   14   22   36   58   94  152  246  398  644 ... A022112
   7   |  8   19   27   46   73  119  192  311  503  814 ... A206420
   8   | 17   30   47   77  124  201  325  526  851 1377 ... A022132
   9   | 23   44   67  111  178  289  467  756 1223 1979 ... A294116
  10   | 41   68  109  177  286  463  749 1212 1961 3173 ... A022103
  ...

Cf. A000010, A000027, A000045, A000358, A113166, A034748.

function [res] = calcLinks(n,k)
if k==1
    res=n;
else
    d=divisors(n);
    res=0;
    for i=1:length(d)
        if d (i) >= k
            res=res+eulerPhi(n/d(i))*fiboExt(d(i)-k-1);
        end
    end
end
function [s] = fiboExt(m) % extended fibonacci function (including negative arguments)
m=sym(m); % for large fibonacci numbers
if m>=0 || mod(m,2)==1
    s=fibonacci(abs(m));
else
    s=fibonacci(abs(m))*(-1);
end

T(n, k) = if (k==1, n, sumdiv(n, d, if (d>=k, eulerphi(n/d)*fibonacci(d-k-1)))); \\ Michel Marcus, Aug 29 2021

If k=1, T(n,k)=n, otherwise T(n,k) = Sum_{d>=k, d|n} Phi(n/d)*Fibonacci(d-k-1), where Phi=A000010.

A340273 a(n) is the number of divisors d of n such that phi(n)/phi(lpf(n)) mod phi(n)/phi(d) = 0, where phi is Euler's totient function (A000010), and lpf(n) is the least prime factor of n (A020639).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 4, 2, 4, 1, 6, 1, 4, 3, 5, 1, 6, 1, 6, 3, 4, 1, 8, 2, 4, 3, 6, 1, 8, 1, 6, 3, 4, 2, 9, 1, 4, 3, 8, 1, 8, 1, 6, 5, 4, 1, 10, 2, 6, 3, 6, 1, 8, 2, 8, 3, 4, 1, 12, 1, 4, 5, 7, 3, 8, 1, 6, 3, 8, 1, 12, 1, 4, 5, 6, 2, 8, 1, 10, 4, 4, 1, 12, 3, 4

Offset: 1

Maxim Karimov, Jan 02 2021

nonn

This equivalence criterion splits the divisor set of n into two types of divisors and can be used to compute the number of links of length k on the set of Fibonacci necklaces (A000358) of length n. This counting is a combinatorial problem over the positive integers.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A000027, A000358.

n=100;
A=[];
for i=1:n
    d=divisors(i);
    t=0;
    for j=1:size(d,2)
        if checkCD(i,d(j))==1
            t=t+1;
        end
    end
    A=[A t];
end
function [res] = checkCD(n,d)
    if mod(n,d)==0 && mod(totient(n)/totient(min(factor(n))),totient(n)/totient(d))==0
        res=1;
    else
        res=0;
    end
end
function [res] = totient(n)
res=0;
    for i=1:n
        if gcd(i,n)==1
            res=res+1;
        end
    end
end

with(numtheory):
a:= n-> `if`(n=1, 1, (f-> nops(select(d-> irem(phi(n)/phi(f),
         phi(n)/phi(d))=0, divisors(n))))(min(factorset(n)))):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 12 2021

Table[Function[{e, f}, DivisorSum[n, 1 &, Mod[e, f/EulerPhi[#]] == 0 &]] @@ {#2/#1, #2} & @@ {EulerPhi[FactorInteger[n][[1, 1]]], EulerPhi[n]}, {n, 86}] (* Michael De Vlieger, Feb 12 2021 *)

lpf(n) = if (n==1, 1, factor(n)[1,1]);
a(n) = my(lp = lpf(n), t = eulerphi(n)); sumdiv(n, d, Mod(t/eulerphi(lp), t/eulerphi(d)) == 0); \\ Michel Marcus, Jan 03 2021

A340268 Composite numbers k>1 such that (s-1) | (d-1) for each d | k, where s = lpf(k) = A020639(k).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96

Offset: 1

Maxim Karimov, Jan 02 2021

nonn

Not a duplicate of A340058 because the complements A335902 and A340269 differ. - R. J. Mathar, Feb 16 2021

Cf. A000010, A000961, A020639, A340058, A335902, A340269 (complement).

Contains all composite terms of at least A003586, A003591, A003592, A003593, A003596.

n=300; % gives all terms of the sequence not exceeding n
A=[];
for i=2:n
    lpf=2;
    while mod(i,lpf)~=0
        lpf=lpf+1;
    end
    for d=1:floor(i/2)
        if mod(i,d)==0 && mod(d-1,lpf-1)~=0
            break
        elseif d==floor(i/2)
            A=[A i];
        end
    end
end

with(numtheory):
q:= n-> (f-> andmap(d-> irem(d-1, f)=0, divisors(n)))(min(factorset(n))-1):
select(not isprime and q, [$2..96])[];  # Alois P. Heinz, Feb 12 2021

Select[Range[2, 96], Function[{n, s}, And[! PrimeQ@ n, AllTrue[Divisors[n] - 1, Mod[#, s] == 0 &]]] @@ {#, FactorInteger[#][[1, 1]] - 1} &] (* Michael De Vlieger, Feb 12 2021 *)

isok(c) = if ((c>1) && !isprime(c), my(f=factor(c)[,1]); for (k=1, #f~, if ((f[k]-1) % (f[1]-1), return(0))); return(1)); \\ Michel Marcus, Jan 03 2021

A340269 Numbers k > 1 such that lpf(k)-1 does not divide d-1 for at least one divisor d of k, where lpf(k) is the least prime factor of k (A020639).

Original entry on oeis.org

35, 55, 77, 95, 115, 119, 143, 155, 161, 175, 187, 203, 209, 215, 221, 235, 245, 247, 253, 275, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 385, 391, 395, 403, 407, 413, 415, 437, 455, 473, 475, 493, 497, 515, 517, 527, 533, 535, 539, 551, 559, 575, 581

Offset: 1

Maxim Karimov, Jan 02 2021

nonn

No terms are divisible by 2 or 3; no terms are in A000961. - Robert Israel, Oct 10 2023

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000961, A020639, A335902, A340058, A340268.

n=300; % gives all terms of the sequence not exceeding n
A=[];
for i=2:n
    lpf=2;
    while mod(i,lpf)~=0
        lpf=lpf+1;
    end
    for d=1:i
        if mod(i,d)==0 && mod(d-1,lpf-1)~=0
            A=[A i];
            break
        end
    end
end

with(numtheory):
q:= n-> (f-> ormap(d-> irem(d-1, f)>0, divisors(n)))(min(factorset(n))-1):
select(q, [$2..600])[];  # Alois P. Heinz, Feb 12 2021

Select[Range[2, 600], Function[{d, k}, AnyTrue[d, Mod[#, k] != 0 &]] @@ {Divisors[#] - 1, FactorInteger[#][[1, 1]] - 1} &] (* Michael De Vlieger, Feb 12 2021 *)

A340058 Composite numbers c such that phi(c)/phi(mind(c)) mod phi(c)/phi(maxd(c)) = 0, where phi is the Euler function, mind(c) is the smallest nontrivial divisor of c, maxd(c) is the largest nontrivial divisor of c.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 98, 99

Offset: 1

Maxim Karimov, Dec 27 2020

nonn

This equivalence criterion splits a set of composite numbers into two classes and can be used to count certain combinatorial objects.

Cf. A000010, A002808, A335902.

n=100; % gives all terms of the sequence not exceeding n
A=[];
for i=1:n
   dn=divisors(i);
   if size(dn,2)>2 && mod(totient(i)/totient(dn(2)),totient(i)/totient(dn(end-1)))==0
      A=[A i];
   end
end
function [res] = totient(n)
res=0;
    for i=1:n
        if gcd(i,n)==1
            res=res+1;
        end
    end
end

isok(c) = if ((c>1) && !isprime(c), my(t=eulerphi(c), d=divisors(c)); ((t/eulerphi(d[2])) % (t/eulerphi(d[#d-1]))) == 0); \\ Michel Marcus, Dec 28 2020

A335902 Composite numbers c such that phi(c)/phi(mind(c)) mod phi(c)/phi(maxd(c)) <> 0, where phi is the Euler function, mind(c) is the smallest nontrivial divisor of c, maxd(c) is the largest nontrivial divisor of c.

Original entry on oeis.org

35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 245, 247, 253, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 391, 395, 403, 407, 413, 415, 437, 473, 493, 497, 515, 517, 527, 533, 535, 551, 559, 581, 583, 589, 605, 611, 623, 629, 635, 649

Offset: 1

Maxim Karimov, Dec 28 2020

nonn

This equivalence criterion splits a set of composite numbers into two classes and can be used to count certain combinatorial objects.

Cf. A000010, A002808, A340058.

n=500; % gives all terms of the sequence not exceeding n
A=[];
for i=1:n
    dn=divisors(i);
    if size(dn,2)>2 && mod (totient(i)/totient(dn(2)),totient(i)/totient(dn(end-1)))~=0
        A=[A i];
    end
end
function [res] = totient(n)
res=0;
    for i=1:n
        if gcd(i,n)==1
            res=res+1;
        end
    end
end

isok(c) = if ((c>1) && !isprime(c), my(t=eulerphi(c), d=divisors(c)); ((t/eulerphi(d[2])) % (t/eulerphi(d[#d-1]))) != 0); \\ Michel Marcus, Dec 28 2020

cp's OEIS Frontend

User: Maxim Karimov

A370108 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors in which the convex hull of a set of beads of any color A can be transformed by rotation into the convex hull of a set of beads of any other color B (n >= 1, k >= 1).

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A369878 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors in which the convex hull of a set of beads of any color A cannot be transformed by rotation into the convex hull of a set of beads of any other color B (n >= 1, k >= 1).

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A369524 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors with black beads always occurring in runs of even length.

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A347351 Triangle read by rows: T(n,k) is the number of links of length k in a set of all necklaces A000358 of length n, 1 <= k <= n.

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A340273 a(n) is the number of divisors d of n such that phi(n)/phi(lpf(n)) mod phi(n)/phi(d) = 0, where phi is Euler's totient function (A000010), and lpf(n) is the least prime factor of n (A020639).

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A340268 Composite numbers k>1 such that (s-1) | (d-1) for each d | k, where s = lpf(k) = A020639(k).

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A340269 Numbers k > 1 such that lpf(k)-1 does not divide d-1 for at least one divisor d of k, where lpf(k) is the least prime factor of k (A020639).

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A340058 Composite numbers c such that phi(c)/phi(mind(c)) mod phi(c)/phi(maxd(c)) = 0, where phi is the Euler function, mind(c) is the smallest nontrivial divisor of c, maxd(c) is the largest nontrivial divisor of c.

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