A001615 -id:A001615 - cp's OEIS frontend

A357821 Denominators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

1, 3, 12, 4, 12, 6, 24, 8, 24, 72, 72, 18, 63, 504, 63, 504, 168, 504, 2520, 2520, 10080, 1120, 3360, 3360, 672, 224, 2016, 2016, 10080, 10080, 5040, 2520, 5040, 15120, 1890, 7560, 143640, 143640, 17955, 143640, 143640, 574560, 6320160, 6320160, 6320160, 6320160

Offset: 1

Amiram Eldar, Oct 14 2022

See A357820 for more details.

Cf. A001615, A173290, A357820 (numerators).

Similar sequence: A211178.

psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Denominator[Accumulate[1/Array[(-1)^(# + 1)*psi[#] &, 50]]]

f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = denominator(sum(k=1, n, (-1)^(k+1)/f(k))); \\ Michel Marcus, Oct 15 2022

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/psi(k)).

A031359 Bisection of A001615.

Original entry on oeis.org

1, 4, 6, 8, 12, 12, 14, 24, 18, 20, 32, 24, 30, 36, 30, 32, 48, 48, 38, 56, 42, 44, 72, 48, 56, 72, 54, 72, 80, 60, 62, 96, 84, 68, 96, 72, 74, 120, 96, 80, 108, 84, 108, 120, 90, 112, 128, 120, 98, 144, 102, 104, 192, 108, 110, 152, 114, 144, 168, 144, 132, 168

Offset: 1

N. J. A. Sloane

Number of coincidence site lattices of index 2n-1 in lattice Z^3.

			G.f. = x + 4*x^2 + 6*x^3 + 8*x^4 + 12*x^5 + 12*x^6 + 14*x^7 + 24*x^8 + ...
G.f. = q + 4*q^3 + 6*q^5 + 8*q^7 + 12*q^9 + 12*q^11 + 14*q^13 + 24*q^15 + ...

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Mathematics of Long-Range Aperiodic Order, Kluwer, 1997, pp. 9-44.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Michael Baake, Solution of coincidence problem in dimensions d<=4, arXiv:math/0605222 [math.MG], 2006.
Michael Baake and Peter A. B. Pleasants, Algebraic solution of the coincidence problem in two and three dimensions, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See page 715, the Dirichlet g.f. following Eq. (18).

Cf. A000010, A000203, A001615, A008683, A159634.

a031359 = a001615 . (subtract 1) . (* 2)
-- Reinhard Zumkeller, Jun 03 2013

A001615 := n -> mul((op(1,i)+1)*op(1,i)^(op(2,i)-1),i=op(2,numtheory[ifactors](n)));
A031359 := n -> A001615(2*n-1); # Peter Luschny, Oct 23 2010

a[n_] := (2n-1)*Sum[ MoebiusMu[d]^2/d, {d, Divisors[2n-1]}]; Table[a[n], {n, 1, 62}] (* Jean-François Alcover, Jan 18 2012, after Michael Somos *)
a[ n_] := If[ n < 1, 0, With[{m = 2 n - 1}, m Sum[ MoebiusMu[ d]^2 / d, {d, Divisors[m]}]]] (* Michael Somos, Nov 22 2013 *)

{a(n) = my(m); if( n<1, 0, m = 2*n - 1; m * sumdiv( m, d, moebius(d)^2 / d))} /* Michael Somos, Nov 22 2013 */

{a(n) = my(m); if( n<1, 0, m = 2*n - 1; direuler( p=2, m, (1 + X) / (1 - p*X))[ m])} /* Michael Somos, Nov 22 2013 */

{a(n) = my(A, p, e); if( n<1, 0, n = 2*n - 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, p^(e-1) * (p + 1)))))} /* Michael Somos, Nov 22 2013 */

a(n) = b(2*n - 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^(e-1) * (p+1) if p > 2. - Michael Somos, Nov 22 2013
Dirichlet series: Product (1+p^(-s))/(1-p^(1-s)); p != 2.
a(n) = A001615(2*n - 1).
From Peter Bala, Mar 19 2019: (Start)
a(n) = (2*n - 1)*Product_{p|(2*n-1), p prime} (1 + 1/p).
a(n) = Sum_{ d|(2*n-1) } mu(d)^2*(2*n-1)/d, where mu(n) = A008683(n) is the Möbius function.
a(n) = Sum_{ d^2|(2*n-1) } mu(d)*sigma((2*n-1)/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also
a(n) = Sum_{ d|(2*n-1) } 2^omega(d)*phi((2*n-1)/d), where omega(n) = A001221(n) is the number of different primes dividing n and phi(n) = A000010 is the Euler totient function.
O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^n*(1 + x^(2*n-1))/(1 - x^(2*n-1))^2.
Bisection of A159634. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 12/Pi^2 = 1.215854... . - Amiram Eldar, Nov 24 2022

Better description from Vladeta Jovovic, Jan 25 2002

More terms from Sascha Kurz, Mar 24 2002

A240111 Numbers for which the value of the Dedekind psi function (A001615) are less than the value of the infinitary Dedekind psi function (A049417).

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 440, 456, 459, 472, 480, 486

Offset: 1

Vladimir Shevelev, Apr 01 2014

nonn

Numbers k for which Product_{p|k} (1 + 1/p) < Product_{q is in Q_k} (1 + 1/q), where {p} are primes, {q} are terms of A050376 and Q_k is the set of distinct q's whose product is k.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 10, 108, 1072, 10679, 106722, 1067287, 10672851, 106728514, 1067285714, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1067285... . - Amiram Eldar, Feb 13 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Peter J. C. Moses)

Cf. A001615, A049417, A050376.

Complement of A240112 within the nonsquarefree numbers (A013929).

f1[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; f2[p_, e_] := (p+1)*p^(e-1); q[1] = False; q[n_] := Module[{fct = FactorInteger[n]}, Times @@ f2 @@@ fct < Times @@ f1 @@@ fct]; Select[Range[500], q] (* Amiram Eldar, Feb 13 2025 *)

isok(k) = {my(f = factor(k), b); prod(i=1, #f~, (f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)) < prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)));} \\ Amiram Eldar, Feb 13 2025

More terms from Peter J. C. Moses, Apr 02 2014

A291043 Numbers n such that psi(n) = psi(n+1), where psi(n) is Dedekind psi function (A001615).

Original entry on oeis.org

4, 8, 14, 15, 32, 44, 45, 62, 63, 75, 135, 188, 195, 567, 608, 663, 704, 825, 956, 957, 1023, 1034, 1275, 1334, 1484, 1634, 1845, 1935, 2223, 2534, 2685, 2751, 2871, 3195, 3404, 3843, 3915, 4994, 7004, 7315, 7544, 8024, 8055, 9207, 10695, 11205, 11984, 12032

Offset: 1

Amiram Eldar, Aug 16 2017

nonn

The only solutions to psi(n) = psi(n+1) = psi(n+2) below 10^8 are 14, 44, 62, 956.

In this sequence, smallest terms k such that k and k + 1 are both product of m + 1 distinct primes are 14, 1334, 84134, 3571905, 424152105 for 1 <= m <= 5. - Altug Alkan, Aug 17 2017

			4 is in the sequence since psi(4) = psi(5) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A001274, A001615.

psi[n_] := If[n < 1, 0, n Sum[MoebiusMu[d]^2 / d, {d, Divisors @ n}]];
Select[Range[12000], psi[#] == psi[# + 1] &]
SequencePosition[Table[If[n<1,0,n Sum[MoebiusMu[d]^2/d,{d,Divisors[n]}]],{n,13000}],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 22 2018 *)

a001615(n) = n*sumdivmult(n, d, issquarefree(d)/d);
isok(n) = a001615(n)==a001615(n+1) \\ Altug Alkan, Aug 17 2017, after Charles R Greathouse IV at A001615

A307055 Even k such that psi(m) = k has no solution, where psi is the Dedekind psi function A001615.

Original entry on oeis.org

2, 10, 16, 22, 26, 28, 34, 40, 46, 50, 52, 58, 64, 66, 70, 76, 78, 82, 86, 88, 92, 94, 100, 106, 116, 118, 122, 124, 130, 134, 136, 142, 146, 148, 154, 156, 166, 170, 172, 178, 184, 188, 190, 196, 202, 206, 208, 210, 214, 218, 220, 226, 232, 236, 238, 244, 246, 250

Offset: 1

Torlach Rush, Mar 21 2019

nonn

Analog of the nontotients A005277.

Contains 2*p if p is in A307390. - Robert Israel, Apr 17 2019

			2 is a term because there exists no m such that psi(m) = 2.
4 is not a term because 4 = 3*(3+1)/3.

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

Cf. A001615, A005277, A005382.

N:= 1000: # to get all terms <= N
psi:= proc(n) local p; n*mul(1+1/p, p=numtheory:-factorset(n)) end proc:
sort(convert({seq(i,i=2..N,2)} minus map(psi, {$1..N}), list)); # Robert Israel, Apr 17 2019

M = 1000; (* to get all terms <= M *)
psi[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}];
Range[2, M, 2] ~Complement~ (psi /@ Range[M]) (* Jean-François Alcover, Aug 01 2020, after Maple *)

dpsi(n) = = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
isok(n) = {if (!(n%2), for (k=1, n-1,  if (dpsi(k) == n, return(0));); return (1););} \\ Michel Marcus, Mar 22 2019

A307309 Self-composition of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 6, 26, 99, 348, 1194, 4000, 13326, 44058, 144066, 462504, 1459194, 4545588, 14068554, 43450848, 134213808, 414692130, 1280610858, 3948172380, 12142365042, 37235047770, 113844652986, 347103133068, 1055610536520, 3202944247674, 9697395164616, 29298206343284

Offset: 1

Ilya Gutkovskiy, Apr 02 2019

nonn

Eric Weisstein's World of Mathematics, Dedekind Function

Cf. A001615, A008683, A307308.

g[x_] := g[x] = Sum[MoebiusMu[k]^2 x^k/(1 - x^k)^2, {k, 1, 27}]; a[n_] := a[n] = SeriesCoefficient[g[g[x]], {x, 0, n}]; Table[a[n], {n, 27}]

G.f.: g(g(x)), where g(x) = Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2 is the g.f. of A001615.

A309324 Expansion of Sum_{k>=1} psi(k) * x^k/(1 + x^k), where psi = Dedekind psi function (A001615).

Original entry on oeis.org

1, 2, 5, 2, 7, 10, 9, 2, 17, 14, 13, 10, 15, 18, 35, 2, 19, 34, 21, 14, 45, 26, 25, 10, 37, 30, 53, 18, 31, 70, 33, 2, 65, 38, 63, 34, 39, 42, 75, 14, 43, 90, 45, 26, 119, 50, 49, 10, 65, 74, 95, 30, 55, 106, 91, 18, 105, 62, 61, 70, 63, 66, 153, 2, 105, 130, 69, 38, 125, 126, 73

Offset: 1

Ilya Gutkovskiy, Jul 23 2019

Dirichlet convolution of sum of odd divisors function with characteristic function of squarefree numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000593, A001615, A008966, A060648, A193356.

nmax = 71; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k]  x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[MoebiusMu[n/d]^2 Plus @@ Select[Divisors@ d, OddQ], {d, Divisors[n]}], {n, 1, 71}]
f[2, e_] := 2; f[p_, e_] := (p^e*(p+1)-2)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 01 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 2, (f[i,1]^f[i,2]*(f[i,1]+1)-2)/(f[i,1]-1)));} \\ Amiram Eldar, Nov 06 2022

a(n) = Sum_{d|n} (-1)^(n/d+1) * psi(d).
a(n) = Sum_{d|n} mu(n/d)^2 * A000593(d).
Multiplicative with a(2^e) = 2, and a(p^e) = (p^e*(p+1)-2)/(p-1) for odd primes p. - Amiram Eldar, Dec 01 2020
Sum_{k=1..n} a(k) ~ (5/8) * n^2. - Amiram Eldar, Nov 06 2022

A330703 Numbers k such that psi(k) = psi(k + 2) where psi(k) is the Dedekind psi function (A001615).

Original entry on oeis.org

6, 9, 12, 14, 18, 20, 33, 44, 62, 70, 92, 108, 116, 138, 164, 175, 212, 254, 280, 308, 320, 332, 348, 356, 452, 490, 524, 558, 572, 692, 716, 764, 833, 932, 956, 1004, 1105, 1124, 1172, 1188, 1436, 1496, 1562, 1593, 1676, 1724, 1772, 1964, 2002, 2036, 2088, 2132

Offset: 1

Amiram Eldar, Dec 26 2019

nonn

			6 is in the sequence since psi(6) = psi(8) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jozsef Sandor, On the composition of some arithmetic functions, II, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, No. 3 (2005), Article 73.

Cf. A001494, A001615, A007373, A062832, A291043.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[10^3], psi[#] == psi[# + 2] &]

A332042 Number of integers whose Dedekind psi function (A001615) values are n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

			a(6) = 2 since there are 2 solutions to psi(x) = 6: 4 and 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001615, A054973, A063974, A065463, A332036, A332038, A332040.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); m = 100; v = Table[0, {m}]; Do[i = psi[k]; If[i <= m, v[[i]]++], {k, 1, m}]; v

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.7044422... (A065463). - Amiram Eldar, Dec 25 2024

A344024 a(n) = A003415(A001615(n)).

Original entry on oeis.org

0, 1, 4, 5, 5, 16, 12, 16, 16, 21, 16, 44, 9, 44, 44, 44, 21, 60, 24, 60, 80, 60, 44, 112, 31, 41, 60, 112, 31, 156, 80, 112, 112, 81, 112, 156, 21, 92, 92, 156, 41, 272, 48, 156, 156, 156, 112, 272, 92, 123, 156, 124, 81, 216, 156, 272, 176, 123, 92, 384, 33, 272, 272, 272, 124, 384, 72, 216, 272, 384, 156, 384, 39, 101

Offset: 1

Antti Karttunen, May 09 2021

nonn

The first nonsquarefree numbers for which this sequence coincides with A342925 are 868, 920, 952, 4260, 4452, 4692, 5060, 5172, ...

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001615, A003415, A005117.

Cf. also A342925.

{0}~Join~Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[# Product[1 + 1/p, {p, FactorInteger[#][[All, 1]]}] ] &, 73, 2] (* Michael De Vlieger, May 16 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A344024(n) = A003415(A001615(n));

a(A005117(n)) = A342925(A005117(n)) for all n.

cp's OEIS Frontend

A357821 Denominators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A031359 Bisection of A001615.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Formula

Extensions

A240111 Numbers for which the value of the Dedekind psi function (A001615) are less than the value of the infinitary Dedekind psi function (A049417).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A291043 Numbers n such that psi(n) = psi(n+1), where psi(n) is Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A307055 Even k such that psi(m) = k has no solution, where psi is the Dedekind psi function A001615.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A307309 Self-composition of the Dedekind psi function (A001615).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A309324 Expansion of Sum_{k>=1} psi(k) * x^k/(1 + x^k), where psi = Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs