A053635 -id:A053635 - cp's OEIS frontend

A000031 Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

1, 2, 3, 4, 6, 8, 14, 20, 36, 60, 108, 188, 352, 632, 1182, 2192, 4116, 7712, 14602, 27596, 52488, 99880, 190746, 364724, 699252, 1342184, 2581428, 4971068, 9587580, 18512792, 35792568, 69273668, 134219796, 260301176, 505294128, 981706832

Offset: 0

N. J. A. Sloane

Also a(n)-1 is the number of 1's in the truth table for the lexicographically least de Bruijn cycle (Fredricksen).
In music, a(n) is the number of distinct classes of scales and chords in an n-note equal-tempered tuning system. - Paul Cantrell, Dec 28 2011
Also, minimum cardinality of an unavoidable set of length-n binary words (Champarnaud, Hansel, Perrin). - Jeffrey Shallit, Jan 10 2019
(1/n) * Dirichlet convolution of phi(n) and 2^n, n>0. - Richard L. Ollerton, May 06 2021
From Jianing Song, Nov 13 2021: (Start)
a(n) is even for n != 0, 2. Proof: write n = 2^e * s with odd s, then a(n) * s = Sum_{d|s} Sum_{k=0..e} phi((2^e*s)/(2^k*d)) * 2^(2^k*d-e) = Sum_{d|s} Sum_{k=0..e-1} phi(s/d) * 2^(2^k*d-k-1) + Sum_{d|s} phi(s/d) * 2^(2^e*d-e) == Sum_{k=0..e-1} 2^(2^k*s-k-1) + 2^(2^e*s-e) == Sum_{k=0..min{e-1,1}} 2^(2^k*s-k-1) (mod 2). a(n) is odd if and only if s = 1 and e-1 = 0, or n = 2.
a(n) == 2 (mod 4) if and only if n = 1, 4 or n = 2*p^e with prime p == 3 (mod 4).
a(n) == 4 (mod 8) if and only if n = 2^e, 3*2^e for e >= 3, or n = p^e, 4*p^e != 12 with prime p == 3 (mod 4), or n = 2s where s is an odd number such that phi(s) == 4 (mod 8). (End)

			For n=3 and n=4 the necklaces are {000,001,011,111} and {0000,0001,0011,0101,0111,1111}.
The analogous shift register sequences are {000..., 001001..., 011011..., 111...} and {000..., 00010001..., 00110011..., 0101..., 01110111..., 111...}.

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, pp. 120, 172.
May, Robert M. "Simple mathematical models with very complicated dynamics." Nature, Vol. 261, June 10, 1976, pp. 459-467; reprinted in The Theory of Chaotic Attractors, pp. 85-93. Springer, New York, NY, 2004. The sequences listed in Table 2 are A000079, A027375, A000031, A001037, A000048, A051841. - N. J. A. Sloane, Mar 17 2019
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).

Seiichi Manyama, Table of n, a(n) for n = 0..3333 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Verónica Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, arXiv:2308.16257 [math.CO], 2023. See references.
Joerg Arndt, Matters Computational (The Fxtbook), p. 151, pp. 379-383.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
J.-M. Champarnaud, G. Hansel, and D. Perrin, Unavoidable sets of constant length, Internat. J. Alg. Comput. 14 (2004), 241-251.
Vladimir Dotsenko and Irvin Roy Hentzel, On the conjecture of Kashuba and Mathieu about free Jordan algebras, arXiv:2507.00437 [math.RA], 2025. See p. 14.
James East and Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.
S. N. Ethier and J. Lee, Parrondo games with spatial dependence, arXiv preprint arXiv:1202.2609 [math.PR], 2012.
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.4.7.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see pages 18, 64.
H. Fredricksen, The lexicographically least de Bruijn cycle, J. Combin. Theory, 9 (1970) 1-5.
Harold Fredricksen, An algorithm for generating necklaces of beads in two colors, Discrete Mathematics, Volume 61, Issues 2-3, September 1986, Pages 181-188.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 2
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 130
Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
Abraham Lempel, On extremal factors of the de Bruijn graph, J. Combinatorial Theory Ser. B 11 1971 17--27. MR0276126 (43 #1874).
Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
Johannes Mykkeltveit, A proof of Golomb's conjecture for the de Bruijn graph, J. Combinatorial Theory Ser. B 13 (1972), 40-45. MR0323629 (48 #1985).
Matthew Parker, The first 25K terms (7-Zip compressed file) [a large file]
J. Riordan, Letter to N. J. A. Sloane, Jul. 1978
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Ville Salo, Universal gates with wires in a row, arXiv:1809.08050 [math.GR], 2018.
J. A. Siehler, The Finite Lamplighter Groups: A Guided Tour, College Mathematics Journal, Vol. 43, No. 3 (May 2012), pp. 203-211.
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, On single-deletion-correcting codes, arXiv:math/0207197 [math.CO], 2002; in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
David Thomson, Musical Polygons, Mathematics Today, Vol. 57, No. 2 (April 2021), pp. 50-51.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
A. M. Uludag, A. Zeytin and M. Durmus, Binary Quadratic Forms as Dessins, 2012.
Eric Weisstein's World of Mathematics, Necklace
Wolfram Research, Number of necklaces
Index entries for "core" sequences
Index entries for sequences related to necklaces

Column 2 of A075195.
Cf. A001037 (primitive solutions to same problem), A014580, A000016, A000013, A000029 (if turning over is allowed), A000011, A001371, A058766.
Rows sums of triangle in A047996.
Dividing by 2 gives A053634.
A008965(n) = a(n) - 1 allowing different offsets.
Cf. A008965, A053635, A052823, A100447 (bisection).
Cf. A000010.

a000031 0 = 1
a000031 n = (`div` n) $ sum $
   zipWith (*) (map a000010 divs) (map a000079 $ reverse divs)
   where divs = a027750_row n
-- Reinhard Zumkeller, Mar 21 2013

with(numtheory); A000031 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+phi(d)*2^(n/d); od; RETURN(s/n); fi; end; [ seq(A000031(n), n=0..50) ];

a[n_] := Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n/d), 0], {d, 1, n}]/n
a[n_] := Fold[#1 + 2^(n/#2) EulerPhi[#2] &, 0, Divisors[n]]/n (* Ben Branman, Jan 08 2011 *)
Table[Expand[CycleIndex[CyclicGroup[n], t] /. Table[t[i]-> 2, {i, 1, n}]], {n,0, 30}] (* Geoffrey Critzer, Mar 06 2011*)
a[0] = 1; a[n_] := DivisorSum[n, EulerPhi[#]*2^(n/#)&]/n; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2016 *)
mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-2*x^i]/i,{i,1,mx}],{x,0,mx}],x] (*Herbert Kociemba, Oct 29 2016 *)

{A000031(n)=if(n==0,1,sumdiv(n,d,eulerphi(d)*2^(n/d))/n)} \\ Randall L Rathbun, Jan 11 2002

from sympy import totient, divisors
def A000031(n): return sum(totient(d)*(1<Chai Wah Wu, Nov 16 2022

a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n/d) = A053635(n)/n, where phi is A000010.
Warning: easily confused with A001037, which has a similar formula.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 2*x^n)/n. - Herbert Kociemba, Oct 29 2016
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 2^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021
a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 2^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 06 2021
Dirichlet g.f.: f(s+1) * (zeta(s)/zeta(s+1)), where f(s) = Sum_{n>=1} 2^n/n^s. - Jianing Song, Nov 13 2021

There is an error in Fig. M3860 in the 1995 Encyclopedia of Integer Sequences: in the third line, the formula for A000031 = M0564 should be (1/n) sum phi(d) 2^(n/d).

A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 20, 33, 24, 5, 0, 0, 6, 30, 72, 96, 40, 6, 0, 0, 7, 42, 135, 280, 255, 84, 7, 0, 0, 8, 56, 228, 660, 1040, 780, 140, 8, 0, 0, 9, 72, 357, 1344, 3145, 4200, 2205, 288, 9, 0

Offset: 0

Alois P. Heinz, Aug 29 2013

Dirichlet convolution of phi(n) and k^n. - Richard L. Ollerton, May 07 2021

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,     0, ...
  0, 1,  2,   3,    4,     5,     6, ...
  0, 2,  6,  12,   20,    30,    42, ...
  0, 3, 12,  33,   72,   135,   228, ...
  0, 4, 24,  96,  280,   660,  1344, ...
  0, 5, 40, 255, 1040,  3145,  7800, ...
  0, 6, 84, 780, 4200, 15810, 46956, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0..10 give A000004, A001477, A053635, A054610, A054611, A054612, A054613, A054614, A054615, A054616, A054617.
Rows n=0..10 give A000004, A001477, A002378, A054602, A054603, A054604, A054605, A054606, A054607, A054608, A054609.
Main diagonal gives A228640.
Cf. A000010, A075195, A258170.

with(numtheory):
A:= (n, k)-> add(phi(d)*k^(n/d), d=divisors(n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[, 0] = a[0, ] = 0; a[n_, k_] := Sum[EulerPhi[d]*k^(n/d), {d, Divisors[n]}]; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)

A(n,k) = Sum_{d|n} phi(d)*k^(n/d).
A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258170(n,i). - Alois P. Heinz, May 22 2015
G.f. for column k: Sum_{n>=1} phi(n)*k*x^n/(1-k*x^n) for k >= 0. - Petros Hadjicostas, Nov 06 2017
From Richard L. Ollerton, May 07 2021: (Start)
A(n,k) = Sum_{i=1..n} k^gcd(n,i).
A(n,k) = Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)).
A(n,k) = A075195(n,k)*n for n >= 1, k >= 1.  (End)

A034738 Dirichlet convolution of b_n = 2^(n-1) with phi(n).

Original entry on oeis.org

1, 3, 6, 12, 20, 42, 70, 144, 270, 540, 1034, 2112, 4108, 8274, 16440, 32928, 65552, 131418, 262162, 524880, 1048740, 2098206, 4194326, 8391024, 16777300, 33558564, 67109418, 134226120, 268435484, 536888520, 1073741854, 2147516736

Offset: 1

Erich Friedman

nonn

Sum of GCD's of parts in all compositions of n. - Vladeta Jovovic, Aug 13 2003
From Petros Hadjicostas, Dec 07 2017: (Start)
It also equals the sum of all lengths of all cyclic compositions of n. This was proved in Perez (2008).
The bivariate g.f. for the number b(n,k) of all cyclic of compositions of n with k parts is Sum_{n,k>=1} b(n,k)*x^n*y^k = -Sum_{s>=1} (phi(s)/s)*log(1 - y^s*Sum_{t>=1} x^{s*t}) =  -Sum_{s>=1} (phi(s)/s)*log(1 - y^s*x^s/(1-x^s)). See, for example, Hadjicostas (2016). Differentiating w.r.t. y and setting y = 1, we get Sum_{n>=1} a(n)*x^n = Sum_{n>=1} (Sum_{k=1..n} b(n,k)*k)*x^n = Sum_{s>=1} phi(s)*x^s/(1-2*x^s).
(End)

			For the compositions of n=4 we have a(4) = gcd(4) + gcd(1,3) + gcd(3,1) + gcd(2,2) + gcd(2,1,1) + gcd(1,2,1) + gcd(1,1,2) + gcd(1,1,1,1) = 4 + 1 + 1 + 2 + 1 + 1 + 1 + 1 = 12. Also, for cyclic compositions of n=4, we have length(4) + length(1,3) + length(2,2) + length(1,1,2) + length(1,1,1,1) = 1 + 2 + 2 + 3 + 4 = 12.

Amiram Eldar, Table of n, a(n) for n = 1..3322
P. Hadjicostas, Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence, J. Integer Sequences 19 (2016), Article 16.8.2.
R. A. Perez, Compositions versus cyclic compositions, JP Journal of Algebra, Number Theory and Applications, Vol. 12, Issue 1 (2008), pp. 41-48.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See pp. 47-48, 50.

Cf. A000010, A000740, A034754, A053635, A078392.

Table[Sum[EulerPhi[d]*2^(n/d-1), {d, Divisors[n]}], {n, 1, 40}] (* Vaclav Kotesovec, Feb 07 2019 *)

a(n) = sum(k=1, n, 2^(gcd(k, n)-1)); \\ Seiichi Manyama, Apr 17 2021

a(n) = sumdiv(n, d, eulerphi(n/d)*2^(d-1)); \\ Seiichi Manyama, Apr 17 2021

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-2*x^k))) \\ Seiichi Manyama, Apr 17 2021

a(n) = A053635(n)/2.
a(n) = (1/2)* Sum_{d|n} phi(d)*2^(n/d), n >= 1.
G.f.: Sum_{s>=1} phi(s)*x^s/(1-2*x^s). - Petros Hadjicostas, Dec 07 2017
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Feb 07 2019
a(n) = Sum_{k=1..n} 2^(gcd(k, n) - 1). - Seiichi Manyama, Apr 17 2021
a(n) = Sum_{k=1..n} 2^(n/gcd(n,k) - 1)*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 06 2021

A053634 a(n) = Sum_{ d divides n } phi(d)*2^(n/d)/(2n).

Original entry on oeis.org

2, 3, 4, 7, 10, 18, 30, 54, 94, 176, 316, 591, 1096, 2058, 3856, 7301, 13798, 26244, 49940, 95373, 182362, 349626, 671092, 1290714, 2485534, 4793790, 9256396, 17896284, 34636834, 67109898, 130150588, 252647064, 490853416, 954440950

Offset: 3

N. J. A. Sloane, Mar 23 2000

nonn

Offset is 3 because a(2) is 3/2, not an integer. - Michel Marcus, Sep 11 2013

G. C. Greubel, Table of n, a(n) for n = 3..3000
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Cf. A000016, A000031, A053635, A063776.

a[n_] := DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2n); Table[a[n], {n, 3, 36}] (* Jean-François Alcover, Dec 07 2015 *)

a(n) = sumdiv (n, d, eulerphi(d)*2^(n/d)/(2*n));  \\ Michel Marcus, Sep 11 2013

a(n) = A000031(n)/2.

A053636 a(n) = Sum_{odd d|n} phi(d)*2^(n/d).

Original entry on oeis.org

0, 2, 4, 12, 16, 40, 72, 140, 256, 540, 1040, 2068, 4128, 8216, 16408, 32880, 65536, 131104, 262296, 524324, 1048640, 2097480, 4194344, 8388652, 16777728, 33554600, 67108912, 134218836, 268435552, 536870968, 1073744160, 2147483708

Offset: 0

N. J. A. Sloane, Mar 23 2000

nonn

			2*x + 4*x^2 + 12*x^3 + 16*x^4 + 40*x^5 + 72*x^6 + 140*x^7 + 256*x^8 + 540*x^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..3321

Cf. A000010, A001227, A053635, A063776, A182469.

a053636 0 = 0
a053636 n = sum $ zipWith (*) (map a000010 ods) (map ((2 ^) . (div n)) ods)
            where ods = a182469_row n
-- Reinhard Zumkeller, Sep 13 2013

a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] EulerPhi[ d] 2^(n / d), {d, Divisors[ n]}]] (* Michael Somos, May 09 2013 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (d % 2) * eulerphi(d) * 2^(n / d)))} /* Michael Somos, May 09 2013 */

from sympy import totient, divisors
def A053636(n): return (sum(totient(d)<>(~n&n-1).bit_length(),generator=True))<<1) # Chai Wah Wu, Feb 21 2023

a(n) = n * A063776(n).
a(n) = Sum_{k=1..A001227(n)} A000010(A182469(n,k)) * 2^(n/A182469(n, A001227(n)+1-k)). - Reinhard Zumkeller, Sep 13 2013
G.f.: Sum_{m >= 0} phi(2*m + 1)*2*x^(2*m + 1)/(1 - 2*x^(2*m + 1)). - Petros Hadjicostas, Jul 20 2019

A286237 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(phi(k), n/k), where P is sequence A000027 used as a pairing function N x N -> N, and phi is Euler totient function, A000010. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.

Original entry on oeis.org

1, 2, 1, 4, 0, 3, 7, 2, 0, 3, 11, 0, 0, 0, 10, 16, 4, 5, 0, 0, 3, 22, 0, 0, 0, 0, 0, 21, 29, 7, 0, 5, 0, 0, 0, 10, 37, 0, 8, 0, 0, 0, 0, 0, 21, 46, 11, 0, 0, 14, 0, 0, 0, 0, 10, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 67, 16, 12, 8, 0, 5, 0, 0, 0, 0, 0, 10, 79, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78, 92, 22, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 21, 106, 0, 17, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36

Offset: 1

Antti Karttunen, May 05 2017

Equally: square array A(n,k) = P(A000010(n), (n+k-1)/n) if n divides (n+k-1), 0 otherwise, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is sequence A000027 used as a pairing function N x N -> N.
When viewed as a triangular table, this sequence packs the values of phi(k) and quotient n/k (when it is integral) to a single value with the pairing function A000027. The two "components" can be accessed with functions A002260 & A004736, which allows us generate from this sequence various sums related to necklace enumeration (among other things).
For example, we have:
  Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 2^(A004736(a(i))) = A053635(n).
and
  Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 3^(A004736(a(i))) = A054610(n).

			The first fifteen rows of the triangle:
    1,
    2,  1,
    4,  0,  3,
    7,  2,  0,  3,
   11,  0,  0,  0, 10,
   16,  4,  5,  0,  0,  3,
   22,  0,  0,  0,  0,  0, 21,
   29,  7,  0,  5,  0,  0,  0, 10,
   37,  0,  8,  0,  0,  0,  0,  0, 21,
   46, 11,  0,  0, 14,  0,  0,  0,  0, 10,
   56,  0,  0,  0,  0,  0,  0,  0,  0,  0, 55,
   67, 16, 12,  8,  0,  5,  0,  0,  0,  0,  0, 10,
   79,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 78,
   92, 22,  0,  0,  0,  0, 27,  0,  0,  0,  0,  0,  0, 21,
  106,  0, 17,  0, 19,  0,  0,  0,  0,  0,  0,  0,  0,  0, 36
---------------------------------------------------------------
Note how triangle A286239 contains on each row the same numbers in the same "divisibility-allotted" positions, but in reverse order.
In the following examples: a = this sequence interpreted as a one-dimensional sequence, A = interpreted as a square array, T = interpreted as a triangular table, P = A000027 interpreted as a pairing function N x N -> N, phi = Euler totient function, A000010.
---
a(7) = A(1,4) = T(4,1) = P(phi(1),4/1) = P(1,4) = 1+(((1+4)^2 - 1 - (3*4))/2) = 7.
a(30) = A(2,7) = T(8,2) = P(phi(2),8/2) = P(1,4) (i.e., same as above) = 7.
a(10) = A(5,1) = T(5,5) = P(phi(5),5/5) = P(4,1) = 1+(((4+1)^2 - 4 - (3*1))/2) = 10.
a(110) = A(5,11) = T(15,5) = P(phi(5),15/5) = P(4,3) = 1+((4+3)^2 - 4 - (3*3))/2 = 19.

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Transpose: A286236.
Cf. A000124 (left edge of the triangle), A000217 (every number at the right edge is a triangular number).
Cf. A000010, A000027, A002024, A002260, A004736, A051731, A053635, A054610, A286235, A286239, A286247.

(* Based on Python script by Indranil Ghosh *)
T[n_, m_] := ((n + m)^2 - n - 3*m + 2)/2
t[n_, k_] := If[Mod[n, k] != 0, 0, T[EulerPhi[k], n/k]]
Table[t[n, k], {n, 1, 20}, {k, 1, n}]
(* David Radcliffe, Jun 12 2025 *)

from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def t(n, k): return 0 if n%k!=0 else T(totient(k), n//k)
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 10 2017

(define (A286237 n) (A286237bi (A002260 n) (A004736 n)))
(define (A286237bi row col) (if (not (zero? (modulo (+ row col -1) row))) 0 (let ((a (A000010 row)) (b (quotient (+ row col -1) row))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Alternatively, with triangular indexing:
(define (A286237 n) (A286237tr (A002024 n) (A002260 n)))
(define (A286237tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A000010 k)) (b (/ n k))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
;; Note that: (A286237tr n k) is equal to (A286237bi k (+ 1 (- n k))).

As a triangle (with n >= 1, 1 <= k <= n):
T(n,k) = 0 if k does not divide n, otherwise T(n,k) = (1/2)*(2 + ((A000010(k)+(n/k))^2) - A000010(k) - 3*(n/k)).
T(n,k) = A051731(n,k) * A286235(n,k).
Other identities. For all n >= 1:
T(prime(n),prime(n)) = A000217(prime(n)-1).

A286239 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A000010(n/k), k), where P is sequence A000027 used as a pairing function N x N -> N. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.

Original entry on oeis.org

1, 1, 2, 3, 0, 4, 3, 2, 0, 7, 10, 0, 0, 0, 11, 3, 5, 4, 0, 0, 16, 21, 0, 0, 0, 0, 0, 22, 10, 5, 0, 7, 0, 0, 0, 29, 21, 0, 8, 0, 0, 0, 0, 0, 37, 10, 14, 0, 0, 11, 0, 0, 0, 0, 46, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 10, 5, 8, 12, 0, 16, 0, 0, 0, 0, 0, 67, 78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79, 21, 27, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 92, 36, 0, 19, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 106

Offset: 1

Antti Karttunen, May 06 2017

This sequence packs the values of phi(n/k) and k (whenever k divides n) to a single value, with the pairing function A000027. The two "components" can be accessed with functions A002260 & A004736, which allows us generate from this sequence various sums related to necklace enumeration (among other things).
For example, we have:
  Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 2^(A004736(a(i))) = A053635(n).
and
  Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * A002260(a(i)) * 3^(A004736(a(i))) = A054610(n)
Triangle A286237 has the same property.

			The first fifteen rows of triangle:
   1,
   1,  2,
   3,  0,  4,
   3,  2,  0,  7,
  10,  0,  0,  0, 11,
   3,  5,  4,  0,  0, 16,
  21,  0,  0,  0,  0,  0, 22,
  10,  5,  0,  7,  0,  0,  0, 29,
  21,  0,  8,  0,  0,  0,  0,  0, 37,
  10, 14,  0,  0, 11,  0,  0,  0,  0, 46,
  55,  0,  0,  0,  0,  0,  0,  0,  0,  0, 56,
  10,  5,  8, 12,  0, 16,  0,  0,  0,  0,  0, 67,
  78,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 79,
  21, 27,  0,  0,  0,  0, 22,  0,  0,  0,  0,  0,  0, 92,
  36,  0, 19,  0, 17,  0,  0,  0,  0,  0,  0,  0,  0,  0, 106
   -------------------------------------------------------------
Note how triangle A286237 contains on each row the same numbers in the same "divisibility-allotted" positions, but in reverse order.

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array

Transpose: A286238.
Cf. A000124 (the right edge of the triangle).
Cf. A000010, A000027, A053635, A054610, A054523, A286156, A286237, A286249.

from sympy import totient
def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def t(n, k): return 0 if n%k!=0 else T(totient(n//k), k)
for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 09 2017

(define (A286239 n) (A286239tr (A002024 n) (A002260 n)))
(define (A286239tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A000010 (/ n k))) (b k)) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))

As a triangle (with n >= 1, 1 <= k <= n):

T(n,k) = 0 if k does not divide n, otherwise T(n,k) = (1/2)*(2 + ((A000010(n/k)+k)^2) - A000010(n/k) - 3*k).

cp's OEIS Frontend

A000031 Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

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A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A034738 Dirichlet convolution of b_n = 2^(n-1) with phi(n).

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A053634 a(n) = Sum_{ d divides n } phi(d)*2^(n/d)/(2n).

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A053636 a(n) = Sum_{odd d|n} phi(d)*2^(n/d).

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A286237 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(phi(k), n/k), where P is sequence A000027 used as a pairing function N x N -> N, and phi is Euler totient function, A000010. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.

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