A054523 -id:A054523 - cp's OEIS frontend

A126246 a(n) is the number of Fibonacci numbers among (F(1),F(2),F(3),...,F(n)) which are coprime to F(n), where F(n) is the n-th Fibonacci number.

1, 2, 2, 3, 4, 4, 6, 6, 6, 8, 10, 6, 12, 12, 8, 12, 16, 12, 18, 12, 12, 20, 22, 12, 20, 24, 18, 18, 28, 16, 30, 24, 20, 32, 24, 18, 36, 36, 24, 24, 40, 24, 42, 30, 24, 44, 46, 24, 42, 40, 32, 36, 52, 36, 40, 36, 36, 56, 58, 24, 60, 60, 36, 48, 48, 40, 66, 48, 44, 48, 70, 36, 72

Offset: 1

Leroy Quet, Mar 08 2007

			F(12) = 144. The six Fibonacci numbers which are coprime to 144 and are <= 144 are F(1) = 1, F(2) = 1, F(5) = 5, F(7) = 13, F(10) = 55 and F(11) = 89. So a(12) = 6.
The six numbers k = 1, 2, 5, 7, 10 and 11 are <= 12 and satisfy gcd(k,12) divides 2. So a(12) = 6. - _Peter Bala_, Dec 31 2023

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000010, A000045, A054523, A080512, A345082.

with(combinat): a:=proc(n) local ct,i: ct:=0: for i from 1 to n do if gcd(fibonacci(i),fibonacci(n))=1 then ct:=ct+1 else ct:=ct fi: od: ct: end: seq(a(n),n=1..90); # Emeric Deutsch, Mar 24 2007
# alternative program based on the above
with(numtheory): a := proc(n) local ct, i: ct := 0: for i from 1 to n do if gcd(i,n) in divisors(2) then ct := ct + 1 else ct := ct fi: od: ct: end: seq(a(n), n = 1..90); # Peter Bala, Dec 31 2023

Table[Count[CoprimeQ[Fibonacci[n],#]&/@Fibonacci[Range[n]],True],{n,80}] (* Harvey P. Dale, Mar 09 2013 *)
a[n_] := {1, 2, 1, 3/2}[[Mod[n, 4, 1]]]*EulerPhi[n]; Array[a, 100] (* Amiram Eldar, Aug 21 2023 *)

a(n) = sum(k=1, n, gcd(fibonacci(k), fibonacci(n)) == 1); \\ Michel Marcus, Nov 13 2017

Equals A054523 * (1, 1, 0, 0, 0, ...). - Gary W. Adamson, Apr 17 2007
From Jud McCranie, Nov 11 2017: (Start)
Multiplicative with a(p^e) = phi(p^e) = p^(e-1)*(p - 1), except when p = 2, then a(2) = 2, because F(1) = F(2) = 1 and a(2^e) = 3*(2^(e-2)), (e > 1, all smaller Fibonacci numbers are coprime, except ones that are multiples of 3, i.e., every 4th one).
If n is odd, then a(n) = phi(n) (Euler's totient function).
If n is a multiple of 4 then a(n) = 3*phi(n)/2.
If n is congruent to 2 mod 4 then a(n) = 2*phi(n). (End)
From Amiram Eldar, Aug 21 2023: (Start)
Dirichlet g.f.: (1 + 1/2^s) * zeta(s-1)/zeta(s).
Sum_{k = 1..n} a(k) ~ c * n^2, where c = 15/(4*Pi^2) = 0.379954... . (End)
From Peter Bala, Dec 31 2023: (Start)
a(n) = Sum_{k = 1..n, gcd(k,n) = 1 or 2} 1 (since gcd(F(k),F(n)) = F(gcd(k,n)) = 1 iff gcd(k,n) = 1 or 2). Cf. phi(n) = A000010(n) = Sum_{k = 1..n, gcd(k,n) = 1} 1. See also A345082.
Sum_{d divides n} a(d) = n if n is odd, else 3*n/2 if n is even. See A080512.
The Lambert series Sum_{n >= 1} a(n)*x^n/(1 - x^n) = (1 + 3*x + x^2)/(1 - x^2)^2.
If n divides m then a(n) divides 2*a(m). (End)
a(n) = Sum_{d|gcd(n,2)} phi(n/d). - Ridouane Oudra, May 06 2025

More terms from Emeric Deutsch, Mar 24 2007

More terms from Gary W. Adamson, Apr 17 2007

A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).

Original entry on oeis.org

1, 2, 2, 6, 0, 6, 8, 8, 0, 24, 20, 0, 0, 0, 120, 12, 36, 48, 0, 0, 720, 42, 0, 0, 0, 0, 0, 5040, 32, 64, 0, 384, 0, 0, 0, 40320, 54, 0, 324, 0, 0, 0, 0, 0, 362880, 40, 200, 0, 0, 3840, 0, 0, 0, 0, 3628800, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800, 48, 144

Offset: 1

N. J. A. Sloane

T(n,k) = A054523(n,k) * A010766(n,k)^A002260(n,k) * A166350(n,k). - Reinhard Zumkeller, Jan 20 2014

			1; 2,2; 6,0,6; 8,8,0,24; 20,0,0,0,120; 12,36,48,0,0,720; ...

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

A064649 gives the row sums.
Cf. A002618 (left edge), A000142 (right edge), A049820 (zeros per row), A000005 (nonzeros per row).
See also A247917, A047918, A047919.

import Data.List (zipWith4)
a047916 n k = a047916_tabl !! (n-1) !! (k-1)
a047916_row n = a047916_tabl !! (n-1)
a047916_tabl = zipWith4 (zipWith4 (\x u v w -> x * v ^ u * w))
               a054523_tabl a002260_tabl a010766_tabl a166350_tabl
-- Reinhard Zumkeller, Jan 20 2014

a[n_, k_] := If[Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!, 0]; Flatten[ Table[ a[n, k], {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, May 04 2012 *)

a(n,k)=if(n%k, 0, eulerphi(n/k)*(n/k)^k*k!) \\ Charles R Greathouse IV, Feb 09 2017

A038045 Shifts left under transform T where Ta is phi DCONV a.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 17, 23, 33, 43, 58, 68, 97, 109, 138, 168, 211, 227, 298, 316, 401, 459, 537, 559, 723, 771, 892, 988, 1173, 1201, 1507, 1537, 1791, 1947, 2190, 2324, 2825, 2861, 3195, 3437, 3999, 4039, 4794, 4836, 5481, 5971, 6552, 6598, 7775

Offset: 1

Christian G. Bower

Eigensequence of triangle A054523; i.e., multiplication by A054523 shifts the sequence to (1, 2, 4, ...). - Gary W. Adamson, Apr 26 2009

Cf. A000010, A054523.

a[1] = 1; a[n_] := a[n] = Sum[EulerPhi[(n-1)/d]*a[d], {d, Divisors[n-1]}]; Table[a[n], {n, 1, 100}] (* Vaclav Kotesovec, Feb 23 2020 *)

a(1) = 1; a(n+1) = Sum_{d|n} phi(n/d) * a(d). - Ilya Gutkovskiy, Feb 23 2020

A212355 Coefficients for the cycle index polynomial for the dihedral group D_n multiplied by 2n, n>=1, read as partition polynomial.

Original entry on oeis.org

2, 2, 2, 2, 3, 1, 2, 0, 3, 2, 1, 4, 0, 0, 0, 5, 0, 1, 2, 0, 0, 2, 0, 0, 4, 0, 3, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 1, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 4, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Wolfdieter Lang, Jun 02 2012

The partitions are ordered like in Abramowitz-Stegun (for the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used).
The row lengths sequence is A000041. The number of nonzero entries in row nr. n is 1 for n=1, 2 for n=2 and A000005(n)+1 otherwise. This is the sequence A212356.
The cycle index (multivariate polynomial) for the dihedral group D_n (of order 2*n), called Z(D_n), is for odd n given by (Z(C_n) + x[1]*x[2]^((n-1)/2))/2 and for even n by (2*Z(C_n) + x[2] ^(n/2) + x[1]^2*x[2]^((n-2)/2))/4, where Z(C_n) is the cycle index for the cyclic group C_n. For the coefficients of Z(C_n) see A054523 or A102190. See the Harary and Palmer reference.

			n\k   1  2  3  4  5  6  7  8  9 10 11 ...
1:    2
2:    2  2
3:    2  3  1
4:    2  0  3  2  1
5:    4  0  0  0  5  0  1
6:    2  0  0  2  0  0  4  0  3  0  1
...
See the link for rows n=1..8 and the corresponding Z(D_n) polynomials for n=1..15.
n=6: Z(D_6) = (2*x[6] + 2*x[3]^2 +  4*x[2]^3 + 3*x[1]^2*x[2]^2 + x[1]^6)/12, because the relevant partitions of 6 appear for k=1: 6, k=4: 3^2, k=7: 2^3 and k=11: 1^6

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 37, (2.2.11).

Andrew Howroyd, Table of n, a(n) for n = 1..2713 (rows 1..20)
Wolfdieter Lang, Cycle index polynomials Z(D_n), n=1..15

Cf. A054523, A102190, A212356.

C(v)={my(n=vecsum(v), r=#v); if(v[1]==v[r], eulerphi(v[1])) + if(v[r]<=2 && 2*r <= n+2, if(n%2, n, n/2)) }
row(n)=[C(Vec(p)) | p<-Vec(partitions(n))]
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 02 2022

The cycle index polynomial for the dihedral group D_n is Z(D_n) = (a(n,k)*x[1]^(e[k,1])*x[2]^(e[k,2])*...*x[n]^(e[k,n]))/(2*n), n>=1, if the k-th partition of n in Abramowitz-Stegun order is 1^(e[k,1]) 2^(e[k,2]) ... n^(e[k,n]), where a part j with vanishing exponent e[k,j] has to be omitted. The n dependence of the exponents has been suppressed. See the comment above for the Z(D_n) formula and the link for these polynomials for n=1..15.

a(n,k) is the coefficient the term of 2*n*Z(D_n) corresponding to the k-th partition of n in Abramowitz-Stegun order. a(n,k) = 0 if there is no such term in Z(D_n).

Terms a(67) and beyond from Andrew Howroyd, Feb 02 2022

A130029 a(n) = Sum_{d|n} phi(n/d) * prime(d).

Original entry on oeis.org

2, 5, 9, 14, 19, 28, 29, 40, 45, 60, 51, 88, 65, 90, 105, 114, 91, 150, 103, 178, 161, 160, 127, 252, 181, 202, 215, 268, 165, 352, 187, 306, 289, 278, 331, 462, 229, 320, 357, 506, 259, 542, 275, 474, 537, 392, 303, 706, 413, 586, 495, 590, 345, 720, 571, 764, 565, 520

Offset: 1

Gary W. Adamson, May 02 2007

nonn

Old name: A054523 * A000040.

			a(4) = 14 = dot product of row 4 of A054523, (2, 1, 0, 1) and primes (2, 3, 5, 7) = (4 + 3 + 0 + 7) = 14.

Cf. A000010, A000040, A054523, A129691, A130030.

A130029 := proc(n)
add( A054523(n,k)*ithprime(k),k=1..n) ;
end proc: # R. J. Mathar, Apr 04 2012

a(n) = sumdiv(n, d, eulerphi(n/d)*prime(d)); \\ Michel Marcus, Mar 22 2020

A054523 as an infinite lower triangular matrix * A000040 (the primes) as a vector.

a(n) = Sum_{k=1..n} prime(gcd(n,k)). - Ilya Gutkovskiy, Mar 22 2020

New name and more terms from Ilya Gutkovskiy, Mar 22 2020

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).

A212357 Coefficients for the cycle index polynomial for the cyclic group C_n multiplied by n, n>=1, read as partition polynomial.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 0, 1, 0, 1, 4, 0, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Wolfdieter Lang, Jun 04 2012

The partitions are ordered like in Abramowitz-Stegun (for the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used).
The row lengths sequence is A000041. The number of nonzero entries in row nr. n is  A000005(n).
The cycle index (multivariate polynomial) for the cyclic group C_n, called Z(C_n), is (sum(phi(k)*x_k^{n/k} ,k divides n))/n, n>=1, with Euler's totient function phi(n)= A000010(n). See the Harary and Palmer reference. For the coefficients of Z(C_n) in different tabulations see also A054523 and A102190.

			n\k  1 2 3 4 5 6 7 8 9 10 11 ...
1:   1
2:   1 1
3:   2 0 1
4:   2 0 1 0 1
5:   4 0 0 0 0 0 1
6:   2 0 0 2 0 0 1 0 0  0  1
...
See the link for rows n=1..8 and the Z(C_n) polynomials for n=1..15.
n=6: Z(C_6) = (2*x[6] + 2*x[3]^2 + 1*x[2]^3 + x[1]^6)/6, because the relevant partitions of 6 appear for k=1: 6, k=4: 3^2, k=7: 2^3 and k=11: 1^6

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.10).

Wolfdieter Lang, Cycle index Z(C_n), n=1..15.

Cf. A000005, A000010, A054523, A102190.

The cycle index polynomial for the cyclic group C_n is Z(C_n) = (a(n,k)*x[1]^(e[k,1])*x[2]^(e[k,2])*...*x[n]^(e[k,n]))/n, n>=1, if the k-th partition of n in Abramowitz-Stegun order is 1^(e[k,1]) 2^(e[k,2]) ... n^(e[k,n]), where a part j with vanishing exponent e[k,j] has to be omitted. The n dependence of the exponents has been suppressed. See the comment above for the Z(C_n) formula and the link for these polynomials for n=1..15.

a(n,k) is the coefficient the term of n*Z(C_n) corresponding to the k-th partition of n in Abramowitz-Stegun order. a(n,k) = 0 if there is no such term in Z(C_n).

A087623 Square array A(n,k) = the cardinality of the set {x in [1,k-1] : gcd(x,k)=n}, read by rising antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 6

Offset: 1

Michele Dondi (bik.mido(AT)tiscalinet.it), Sep 14 2003

Triangle read by rows: T(m,n) is the cardinality of the set {k in [1,n-1] : gcd(k,n)=m}. - The original definition.

A generalization of Euler's phi function: the n-th term of topmost row = A000010(n), for n > 1.

			The top left corner of the array:
n\k| 1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18
---+------------------------------------------------------------------------
1  | 0,  1,  2,  2,  4,  2,  6,  4,  6,  4, 10,  4, 12,  6,  8,  8, 16,  6,
2  | 0,  0,  0,  1,  0,  2,  0,  2,  0,  4,  0,  2,  0,  6,  0,  4,  0,  6,
3  | 0,  0,  0,  0,  0,  1,  0,  0,  2,  0,  0,  2,  0,  0,  4,  0,  0,  2,
4  | 0,  0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  2,  0,  0,  0,  2,  0,  0,
5  | 0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  2,  0,  0,  0,
6  | 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0,  2,
7  | 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,
8  | 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  0,  0,
9  | 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,
etc.
A(1,4) = 2 and A(2,4) = 1 because gcd(1,4)=1, gcd(2,4)=2, gcd(3,4)=1.
A(1,12) = 4, A(2,12) = A(3,12) = A(4,12) = 2, and A(6,12) = 1 because gcd(1,12) = gcd(5,12) = gcd(7,12) = gcd(9,12) = 1, gcd(2,12) = gcd(10,12) = 2, gcd(3,12) = gcd(9,12) = 3, gcd(4,12) = gcd(8,12) = 4 and gcd(6,12) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals of the array (rows of the triangle)

Cf. A000010.

Cf. also A054523.

up_to = 105;
A087623sq(n, k) = sum(x=1,k-1,gcd(x,k)==n);
A087623list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A087623sq((a-(col-1)), col))); (v); };
v087623 = A087623list(up_to);
A087623(n) = v087623[n]; \\ Antti Karttunen, Jan 17 2025

Definition rephrased in terms of square array instead of triangular table, and data section extended up to 105 terms by Antti Karttunen, Jan 17 2025

A127527 Triangle T(n,k)= tau(k)*phi(n/k) if k|n, else T(n,k)=0.

Original entry on oeis.org

1, 1, 2, 2, 0, 2, 2, 2, 0, 3, 4, 0, 0, 0, 2, 2, 4, 2, 0, 0, 4, 6, 0, 0, 0, 0, 0, 2, 4, 4, 0, 3, 0, 0, 0, 4, 6, 0, 4, 0, 0, 0, 0, 0, 3, 4, 8, 0, 0, 2, 0, 0, 0, 0, 4

Offset: 1

Gary W. Adamson, Jan 17 2007

Tau is the number of divisors A000005, and phi the Euler totient A000010.

			First few rows of the triangle are:
1;
1, 2;
2, 0, 2;
2, 2, 0, 3;
4, 0, 0, 0, 2;
2, 4, 2, 0, 0, 4;
6, 0, 0, 0, 0, 0, 2;
4, 4, 0, 3, 0, 0, 0, 4;
...

Cf. A054523, A000005, A000010, A000203.

A127527 := proc(n,k) if n mod k = 0 then numtheory[tau](k)*numtheory[phi](n/k) ; else 0; end if; end proc: # R. J. Mathar, Apr 11 2011

T(n,k) = A000005(k)*A054523(n,k).
T(n,1) = A000010(n).
T(n,n) = A000005(n).
Sum_{k=1..n} T(n,k) = A000203(n).

Definition clarified by R. J. Mathar, Apr 11 2011

A127528 Triangle T(n,k) read by rows: tau(n)*phi(n/k) if k|n, else 0.

Original entry on oeis.org

1, 2, 2, 4, 0, 2, 6, 3, 0, 3, 8, 0, 0, 0, 2, 8, 8, 4, 0, 0, 4, 12, 0, 0, 0, 0, 0, 2, 16, 8, 0, 4, 0, 0, 0, 4, 18, 0, 6, 0, 0, 0, 0, 0, 3, 16, 16, 0, 0, 4, 0, 0, 0, 0, 4, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 24, 12, 12, 12, 0, 6, 0, 0, 0, 0, 0, 6, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 24, 24

Offset: 1

Gary W. Adamson, Jan 17 2007

			First few rows of the triangle are:
1;
2, 2;
4, 0, 2;
6, 3, 0, 3;
8, 0, 0, 0, 2;
8, 8, 4, 0, 0, 4;
12, 0, 0, 0, 0, 0, 2;
...

Cf. A054523, A127527, A062355, A038040, A000005.

A127528 := proc(n,k) if n mod k = 0 then numtheory[tau](n)*numtheory[phi](n/k) ; else  0; end if; end proc:
seq(seq(A127528(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Feb 05 2011

T(n,k) = A000005(n) * A054523(n,k).
T(n,1) = A062355(n).
T(n,n) = A000005(n).
sum_{k=1..n} T(n,k) = A038040(n).

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