A001088 -id:A001088 - cp's OEIS frontend

A345144 Product_{p primes, k>=1} ((p^(k+1) - 1)/(p^(k+1) - p))^(1/p^k).

1, 5, 6, 1, 5, 9, 6, 8, 4, 6, 9, 3, 1, 0, 2, 4, 1, 6, 4, 3, 2, 6, 9, 6, 7, 8, 8, 9, 1, 4, 4, 5, 5, 5, 6, 4, 4, 3, 6, 4, 7, 3, 7, 6, 4, 6, 8, 2, 2, 2, 3, 2, 1, 6, 9, 9, 4, 5, 8, 6, 6, 4, 5, 7, 0, 9, 6, 8, 3, 5, 7, 8, 4, 9, 4, 9, 0, 9, 5, 3, 9, 8, 8, 9, 4, 2, 4, 4, 3, 0, 1, 0, 8, 6, 8, 0, 9, 1, 0, 3, 2, 1, 4, 3, 7

Offset: 1

Vaclav Kotesovec, Jun 09 2021

			1.561596846931024164326967889144555644364737646822232169945866457...

Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20), constant c.

Cf. A001088, A066843, A066780, A124175.

$MaxExtraPrecision = 1000; m = 500; prod = 1; Do[Clear[f]; f[p_] := ((p^(k + 1) - 1)/(p^(k + 1) - p))^(1/p^k); cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; prod *= f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 100]]; Print[prod], {k, 1, 200}]

Equals exp(1) * lim_{n->infinity} (A066780(n)^(1/n)) / n.

A085244 Permanent of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j) for 1 <= i,j <= n.

Original entry on oeis.org

1, 3, 14, 112, 872, 14372, 154480, 3098480, 59710816, 1688186176, 27925409152, 1327833590272, 25675495200768, 1017195720916224, 47444016840290304, 2267031138313024512, 56480432945454004224, 4051971981329937580032

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 11 2003

nonn

Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007, Table of n, a(n) for n = 1..23

Cf. A001088, A005326.

permRWNb(a)=n=matsize(a)[1];if(n==1,return(a[1,1]));sg=1;in=vectorv(n);x=in;x=a[,n]-sum(j=1,n,a[,j])/2;p=prod(i=1,n,x[i]);for(k=1,2^(n-1)-1,sg=-sg;j=valuation(k,2)+1;z=1-2*in[j];in[j]+=z;x+=z*a[,j];p+=prod(i=1,n,x[i],sg));return(2*(2*(n%2)-1)*p) for(n=1,26,a=matrix(n,n,i,j,gcd(i,j));print1(permRWNb(a)",")) - Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

More terms from Vladeta Jovovic, Aug 13 2003

A063770 Numbers k such that Sum_{j=1..k} sigma(j) divides Product_{j=1..k} phi(j).

Original entry on oeis.org

1, 28, 52, 53, 55, 63, 76, 159, 166, 176, 219, 230, 289, 302, 303, 318, 321, 327, 348, 360, 364, 365, 381, 383, 402, 417, 430, 434, 438, 444, 451, 452, 454, 465, 469, 478, 504, 512, 522, 530, 531, 557, 559, 570, 572, 584, 595, 613, 629, 631, 641, 645, 684

Offset: 1

Jason Earls, Aug 15 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001088, A024916.

Module[{nn=700,s,p},s=Accumulate[DivisorSigma[1,Range[nn]]];p=FoldList[ Times,EulerPhi[ Range[nn]]]; Position[ Thread[{s,p}],?(Mod[#[[2]],#[[1]]]==0&),1,Heads->False]]//Flatten (* _Harvey P. Dale, Feb 28 2024 *)

j=[]; for(n=1,1300,a=sum(k=1,n, sigma(k)); b=prod(k=1,n,eulerphi(k)); if(Mod(b,a)==0,j=concat(j,n))); j

{ n=0; s=0; p=1; for (m=1, 10^9, s+=sigma(m); p*=eulerphi(m); if(p%s == 0, write("b063770.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 30 2009

A067578 a(n) = Product_{i=1..n} phi(i) * Sum_{i=1..n} 1/phi(i) where phi is the Euler totient function A000010(n).

Original entry on oeis.org

1, 2, 5, 12, 52, 120, 752, 3200, 19968, 84480, 863232, 3637248, 44384256, 275152896, 2254307328, 18459131904, 298743496704, 1846819160064, 33568893960192, 274421835104256, 3340027488632832, 33963860494909440, 752840786973818880, 6146715129678397440, 123926213264670720000

Offset: 1

Benoit Cloitre, Jan 30 2002

nonn

Cf. A000010, A001088, A028415, A048049.

a(n) = prod(i=1, n, eulerphi(i)) * sum(i=1, n, 1/eulerphi(i)); \\ Michel Marcus, Jan 09 2021

More terms from Michel Marcus, Jan 09 2021

A177066 Determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(2i-1,2j-1) for 1 <= i,j <= n.

Original entry on oeis.org

1, 2, 8, 48, 288, 2880, 34560, 276480, 4423680, 79626240, 955514880, 21021327360, 420426547200, 7567677849600, 211894979788800, 6356849393664000, 127136987873280000, 3051287708958720000, 109846357522513920000

Offset: 1

John W. Layman, Dec 09 2010

nonn

It appears, but has not been proved, that the ratios a(n+1)/a(n) give phi(2n+1) (A037225).

See A001088, A059381, and A059382 for determinants of matrices M defined by M(i,j) = gcd(i,j), gcd(i^2,j^2), and gcd(i^3,j^3), respectively.

Cf. A001088, A037225, A059381, A059382.

A177066 := proc(n) M := Matrix(n) ; for i from 1 to n do for j from 1 to n do M[i,j] := igcd(2*i-1,2*j-1) ; end do: end do: LinearAlgebra[Determinant](M) ; end proc: # R. J. Mathar, Dec 10 2010

A321613 Partial products of the unitary totient function (A047994): a(n) = Product_{k=1..n} uphi(k).

Original entry on oeis.org

1, 1, 2, 6, 24, 48, 288, 2016, 16128, 64512, 645120, 3870720, 46448640, 278691840, 2229534720, 33443020800, 535088332800, 4280706662400, 77052719923200, 924632639078400, 11095591668940800, 110955916689408000, 2441030167166976000, 34174422340337664000

Offset: 1

Amiram Eldar, Dec 19 2018

nonn

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = ugcd(i,j) for 1 <= i,j <= n, where ugcd(i,j) in the greatest common unitary divisor of i and j (A165430).

The unitary version of A001088.

			a(4) = uphi(1) * uphi(2) * uphi(3) * uphi(4) = 1 * 1 * 2 * 3 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..481
H. Jager, The unitary analogues of some identities for certain arithmetical functions, Nederl. Akad. Wetensch. Proc. Ser. A, Vol. 64 (1961), pp. 508-515.

Cf. A001088, A047994, A165430, A177754.

uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); FoldList[ Times, uphi /@ Range[50]]

uphi(n) = my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1); \\ A047994
a(n) = prod(k=1, n, uphi(k)); \\ Michel Marcus, Dec 19 2018

A349741 a(n) = Product_{k=1..n-1} phi(gcd(n,k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 4, 4, 1, 32, 1, 6, 256, 16, 1, 96, 1, 1024, 2304, 10, 1, 16384, 256, 12, 2304, 13824, 1, 524288, 1, 2048, 102400, 16, 5308416, 14155776, 1, 18, 589824, 134217728, 1, 63700992, 1, 1024000, 86973087744, 22, 1, 8589934592, 46656, 1310720

Offset: 1

Ilya Gutkovskiy, Nov 28 2021

nonn

Cf. A000010, A001088, A029935, A029940, A046022 (positions of 1's), A051190.

Table[Product[EulerPhi[GCD[n, k]], {k, 1, n - 1}], {n, 1, 50}]

a(n) = prod(k=1, n-1, eulerphi(gcd(n, k))); \\ Michel Marcus, Nov 28 2021

a(n) = Product_{d|n, d < n} phi(d)^phi(n/d).

A379149 Specialization of the Elementary Symmetric Functions e(n) at x_i -> Euler phi(i).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 5, 2, 1, 6, 13, 12, 4, 1, 10, 37, 64, 52, 16, 1, 12, 57, 138, 180, 120, 32, 1, 18, 129, 480, 1008, 1200, 752, 192, 1, 22, 201, 996, 2928, 5232, 5552, 3200, 768, 1, 28, 333, 2202, 8904, 22800, 36944, 36512, 19968, 4608, 1, 32, 445, 3534, 17712, 58416, 128144, 184288, 166016, 84480, 18432

Offset: 0

Wouter Meeussen, Dec 16 2024

Triangular table with alternating signed sum equal to 0 for n>0,
  1
  1,-1
  1,-2,1
  1,-4,5,-2
  1,-6,13,-12,4
  ..
and with alternating signed weighted sum (first moment) also equal to 0 for n>1,
  0
  0,-1
  0,-2,2
  0,-4,10,-6
  0,-6,26,-36,16
  ..
also when shifting the weights to start at 1,
  1
  1,-2
  1,-4,3
  1,-8,15,-8
  1,-12,39,-48,20

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  5,   2;
  1,  6, 13,  12,   4;
  1, 10, 37,  64,  52,  16;
  1, 12, 57, 138, 180, 120, 32;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Mathematics Stack Exchange, Specializations of elementary symmetric polynomials

Columns k=0-1 give: A000012, A002088.
Main diagonal gives A001088.
T(n,n-1) gives A067578.
Cf. A000010.

b:= proc(n) option remember; `if`(n=0, 1,
       b(n-1)*(1+x*numtheory[phi](n)))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Dec 16 2024

Table[CoefficientList[Expand@Product[z EulerPhi[k]+1,{k,0,n}],z,n+1],{n,0,10}]

row(n) = Vecrev(prod(k=1, n, 1 + 'x * eulerphi(k))) \\ Andrew Howroyd, Dec 16 2024

T(n,k) = [x^k] Product_{j=1..n} (1 + x*phi(j)). - Andrew Howroyd, Dec 16 2024

A092416 Determinant of the n X n matrix with entries gcd(X,Y)^gcd(X,Y).

Original entry on oeis.org

1, 1, 3, 78, 19656, 61405344, 2863085569344, 2357871215948696448, 39557911075122642360238080, 15325544184478930809864207383592960, 153255393906487099048546500580688904121221120

Offset: 0

Jon Perry, Mar 22 2004

nonn

			for n = 3:
[1,1,1]
[1,4,1]
[1,1,27]
with det 78.

Cf. A000178:super-factorials, A001088:product(A000010(i)).

for(i=0,10,m=matrix(i,i,X,Y,gcd(X,Y)^gcd(X,Y));print1(","matdet(m)))

A187748 Determinant of the n X n matrix m_(i,j) = gcd(2^i-1, 2^j-1).

Original entry on oeis.org

1, 2, 12, 144, 4320, 233280, 29393280, 7054387200, 3555411148800, 3519857037312000, 7201627498340352000, 28950542543328215040000, 237104943429858081177600000, 3853903750508913251460710400000, 126138269754156730720309051392000000, 8234306249551351381421774874869760000000, 1079270520128695625562952032849179443200000000, 282311265573183686952254740944556962034483200000000

Offset: 1

Benoit Cloitre, Jan 03 2013

nonn

Cf. A001088, A027375.

b[n_] := DivisorSum[n, MoebiusMu[n/#]*2^#& ]; a[n_] := a[n] = If[n == 1, 1, a[n-1]*b[n]]; Array[a, 18] (* Jean-François Alcover, Dec 18 2015 *)
Table[Det[Table[GCD[2^i-1,2^j-1],{i,n},{j,n}]],{n,20}] (* Harvey P. Dale, Sep 23 2022 *)

a(n)=if(n<1,0,(1/2)*prod(k=1,n,sumdiv(k,d,moebius(d)*2^(k/d))))

a(n+1)/a(n) = A027375(n+1).
a(n) = (1/2)*Product_{k=1..n} Sum_{d|k} moebius(d)*2^(k/d).
a(n) ~ c * 2^(n*(n+1)/2), where c = 0.09412540696949274854160062245002977344042957885767746756023904566838799439... - Vaclav Kotesovec, Apr 19 2024

cp's OEIS Frontend

A345144 Product_{p primes, k>=1} ((p^(k+1) - 1)/(p^(k+1) - p))^(1/p^k).

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A085244 Permanent of the symmetric n X n matrix M defined by M(i,j) = gcd(i,j) for 1 <= i,j <= n.

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A063770 Numbers k such that Sum_{j=1..k} sigma(j) divides Product_{j=1..k} phi(j).

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PARI

A067578 a(n) = Product_{i=1..n} phi(i) * Sum_{i=1..n} 1/phi(i) where phi is the Euler totient function A000010(n).

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A177066 Determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(2i-1,2j-1) for 1 <= i,j <= n.

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A321613 Partial products of the unitary totient function (A047994): a(n) = Product_{k=1..n} uphi(k).

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A349741 a(n) = Product_{k=1..n-1} phi(gcd(n,k)).

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A379149 Specialization of the Elementary Symmetric Functions e(n) at x_i -> Euler phi(i).

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A092416 Determinant of the n X n matrix with entries gcd(X,Y)^gcd(X,Y).