Displaying 1-10 of 304 results found.
page
1
2
3
4
5
6
7
8
9
10
... 31
a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.
+20
19
1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32, 108
COMMENTS
This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 8, although a(4) = 1 and a(27) = 4. See A344702. A more specific property holds: for prime p that does not divide n, a(p*n) = a(p) * a(n). In particular, on squarefree numbers ( A005117) this sequence coincides with sigma and psi, which are multiplicative. If prime p divides the squarefree part of n then p+1 divides a(n). (For example, 20 has square part 4 and squarefree part 5, so 5+1 divides a(20) = 6.) So a(n) = 1 only if n is square. The first square n with a(n) > 1 is a(196) = 21. See A344703. Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes). 5 does not appear as a term until a(366025) = 5, where 366025 = 5^2 * 11^4. At this point, the missing numbers less than 22 are 2, 10 and 17. 17 appears at the latest by a(17^2 * 103^16) = 17.
FORMULA
For prime p, a(p^e) = (p+1)^(e mod 2).
For prime p with gcd(p, n) = 1, a(p*n) = a(p) * a(n).
MATHEMATICA
Table[GCD[DivisorSigma[1, n], DivisorSum[n, MoebiusMu[n/#]^2*#&]], {n, 100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)
PROG
(PARI)
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
(Python 3.8+)
from math import prod, gcd
from sympy import primefactors, divisor_sigma
plist = primefactors(n)
return n*prod(p+1 for p in plist)//prod(plist)
Decimal expansion of the asymptotic mean of phi(k)/psi(k), where phi(k) is Euler totient function ( A000010) and psi(k) is Dedekind psi function ( A001615).
+20
15
4, 7, 1, 6, 8, 0, 6, 1, 3, 6, 1, 2, 9, 9, 7, 8, 6, 8, 0, 7, 5, 2, 3, 5, 6, 3, 3, 0, 8, 0, 4, 8, 2, 0, 8, 7, 4, 2, 5, 9, 2, 6, 3, 8, 2, 0, 0, 6, 9, 8, 6, 8, 8, 3, 6, 3, 5, 7, 3, 7, 2, 5, 5, 4, 1, 7, 7, 3, 2, 1, 1, 6, 7, 5, 9, 6, 8, 2, 7, 4, 4, 0, 9, 6, 2, 1, 0, 0, 2, 7, 3, 7, 6, 9, 4, 9, 0, 2, 3, 0, 3, 1, 3, 0, 1, 1
FORMULA
Equals lim_{m->oo} (1/m)*Sum_{k=1..m} phi(k)/psi(k).
Equals Product_{p prime} (1 - 2/(p * (p+1))).
EXAMPLE
0.47168061361299786807523563308048208742592638200698...
MATHEMATICA
$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-2, 1, 2}, {0, -4, 6}, m]; RealDigits[(2/3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
Dirichlet inverse of Dedekind's psi, A001615.
+20
12
1, -3, -4, 3, -6, 12, -8, -3, 4, 18, -12, -12, -14, 24, 24, 3, -18, -12, -20, -18, 32, 36, -24, 12, 6, 42, -4, -24, -30, -72, -32, -3, 48, 54, 48, 12, -38, 60, 56, 18, -42, -96, -44, -36, -24, 72, -48, -12, 8, -18, 72, -42, -54, 12, 72, 24, 80, 90, -60, 72, -62, 96, -32, 3, 84, -144, -68, -54, 96, -144, -72, -12, -74, 114, -24
FORMULA
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} psi(k) * A(x^k). - Ilya Gutkovskiy, Sep 04 2019 Multiplicative with a(p^e) = (-1)^e * (p+1).
Dirichlet g.f.: zeta(2*s)/(zeta(s-1)*zeta(s)). - Amiram Eldar, Dec 05 2022
MATHEMATICA
psi[n_] := If[n == 1, 1, n Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
a[n_] := a[n] = If[n == 1, 1, -Sum[psi[n/d] a[d], {d, Most@ Divisors[n]}]];
f[p_, e_] := (-1)^e * (p + 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)
PROG
(PARI)
A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
Dedekind psi function applied to the odd part of n: a(n) = A001615( A000265(n)).
+20
11
1, 1, 4, 1, 6, 4, 8, 1, 12, 6, 12, 4, 14, 8, 24, 1, 18, 12, 20, 6, 32, 12, 24, 4, 30, 14, 36, 8, 30, 24, 32, 1, 48, 18, 48, 12, 38, 20, 56, 6, 42, 32, 44, 12, 72, 24, 48, 4, 56, 30, 72, 14, 54, 36, 72, 8, 80, 30, 60, 24, 62, 32, 96, 1, 84, 48, 68, 18, 96, 48, 72, 12, 74, 38, 120, 20, 96, 56, 80, 6, 108, 42, 84, 32, 108
FORMULA
Multiplicative with a(2^e) = 1, a(p^e) = (p+1)*p^(e-1) for all odd primes p.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... ( A185199). - Amiram Eldar, Nov 19 2022 Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s))*(4^s-2^(s+1))/(4^s-1). - Amiram Eldar, Jan 04 2023
MATHEMATICA
f[p_, e_] := If[p == 2, 1, (p + 1)*p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2021 *)
PROG
(PARI) A347385(n) = if(1==n, n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, 132, 156, 180, 204, 222, 258, 278, 314, 346, 382, 406, 454, 484, 526, 562, 610, 640, 712, 744, 792, 840, 894, 942, 1014, 1052, 1112, 1168, 1240, 1282, 1378, 1422, 1494, 1566, 1638, 1686, 1782, 1838, 1928, 2000, 2084
REFERENCES
W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88; http://scientificadvances.co.in; DOI: http://dx.doi.org/10.18642/jantaa_7100121599
FORMULA
a(n) = Sum_{i=1..n} A001615(i) = Sum_{i=1..n} (n * Product_{p|n, p prime} (1 + 1/p)). a(n) = (1/2)*Sum_{k=1..n} mu(k)^2 * floor(n/k) * floor(1+n/k), where mu(k) is the Moebius function. - Daniel Suteu, Nov 19 2018 a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * ( A013928(1+floor(n/k)) - A013928(1+floor(n/(k+1)))) + Sum_{k=1..floor(n/(1+floor(sqrt(n))))} mu(k)^2 * floor(n/k) * floor(1+n/k))/2. - Daniel Suteu, Nov 23 2018
MAPLE
with(numtheory): a:=n->(1/2)*add(mobius(k)^2*floor(n/k)*floor(1+n/k), k=1..n); seq(a(n), n=1..55); # Muniru A Asiru, Nov 24 2018
MATHEMATICA
Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k], {k, 1, n}], {n, 60}] (* G. C. Greubel, Nov 23 2018 *) psi[n_] := If[n==1, 1, n*Times@@(1 + 1/FactorInteger[n][[;; , 1]])]; Accumulate[Array[psi, 50]] (* Amiram Eldar, Nov 23 2018 *)
PROG
(PARI)
S(n) = sum(k=1, sqrtint(n), moebius(k)*(n\(k*k))); \\ see: A013928a(n) = sum(k=1, sqrtint(n), k*(k+1) * (S(n\k) - S(n\(k+1))))/2 + sum(k=1, n\(1+sqrtint(n)), moebius(k)^2*(n\k)*(1+n\k))/2; \\ Daniel Suteu, Nov 23 2018 (Sage)
return add(k*mul(1+1/p for p in prime_divisors(k)) for k in (1..n))
(Magma) [(&+[MoebiusMu(k)^2*Floor(n/k)*Floor(1 + n/k): k in [1..n]])/2: n in [1..60]]; // G. C. Greubel, Nov 23 2018
CROSSREFS
Cf. A175836 (partial products of the Dedekind psi function).
a(n) = Product_{i=1..n} psi(i) where psi is the Dedekind psi function ( A001615).
+20
10
1, 3, 12, 72, 432, 5184, 41472, 497664, 5971968, 107495424, 1289945088, 30958682112, 433421549568, 10402117189632, 249650812551168, 5991619501228032, 107849151022104576, 3882569436795764736
COMMENTS
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = A060648(gcd(i,j)) for 1 <= i,j <= n, note that A060648 is the Inverse Möbius transform of A001615. - Enrique Pérez Herrero, Aug 12 2011
MATHEMATICA
JordanTotient[n_, k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/# ]&]/; (n>0)&&IntegerQ[n];
DedekindPsi[n_]:=JordanTotient[n, 2]/EulerPhi[n];
A175836[n_]:=Times@@DedekindPsi/@Range[n]
PROG
(PARI) a=direuler(p=2, 100, (1+X)/(1-p*X)); for(i=2, #a, a[i]*=a[i-1]); a
(Haskell)
a175836 n = a175836_list !! (n-1)
a175836_list = scanl1 (*) a001615_list
Partial alternating sums of the Dedekind psi function ( A001615): a(n) = Sum_{k=1..n} (-1)^(k+1) * psi(k).
+20
10
1, -2, 2, -4, 2, -10, -2, -14, -2, -20, -8, -32, -18, -42, -18, -42, -24, -60, -40, -76, -44, -80, -56, -104, -74, -116, -80, -128, -98, -170, -138, -186, -138, -192, -144, -216, -178, -238, -182, -254, -212, -308, -264, -336, -264, -336, -288, -384, -328, -418
FORMULA
a(n) = -(3/(2*Pi^3)) * n^2 + O(n * log(n)^(2/3)) (Tóth, 2017).
MATHEMATICA
psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Accumulate[Array[(-1)^(# + 1)*psi[#] &, 50]]
PROG
(PARI) f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615a(n) = sum(k=1, n, (-1)^(k+1) * f(k)); \\ Michel Marcus, Oct 15 2022
0, 1, 1, 2, 1, 6, 1, 4, 3, 8, 1, 12, 1, 10, 9, 8, 1, 18, 1, 16, 11, 14, 1, 24, 5, 16, 9, 20, 1, 42, 1, 16, 15, 20, 13, 36, 1, 22, 17, 32, 1, 54, 1, 28, 27, 26, 1, 48, 7, 40, 21, 32, 1, 54, 17, 40, 23, 32, 1, 84, 1, 34, 33, 32, 19, 78, 1, 40, 27, 74, 1, 72
COMMENTS
The sum of the proper divisors d of n such that n/d is squarefree. - Amiram Eldar, Sep 06 2019
FORMULA
a(n) = Sum_{d|n, d<n} (mu(n/d)^2 * d). - Amiram Eldar, Sep 06 2019 Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 - 1 = 0.519817... . - Amiram Eldar, Dec 08 2023
EXAMPLE
0 is a term because A001615(1) - 1 = 0. 1 is a term because A001615(2) - 2 = 1. 3 is a term because A001615(9) - 9 = 3.
MATHEMATICA
a[1] = 0; a[n_] := n * (Times @@ (1 + 1/FactorInteger[n][[;; , 1]]) - 1); Array[a, 100] (* Amiram Eldar, Sep 06 2019 *)
PROG
(PARI) a(n) = n*(sumdivmult(n, d, issquarefree(d)/d) - 1); \\ Michel Marcus, Mar 18 2019
a(n) = sigma(n) / gcd(sigma(n), A001615(n)).
+20
9
1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 10, 7, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 10, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 65, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 21, 1, 57, 13
COMMENTS
This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 35, although a(4) = 7 and a(27) = 10. See A344702.
PROG
(PARI)
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
1, 1, 1, 6, 1, 1, 1, 4, 12, 1, 1, 6, 1, 1, 1, 24, 1, 12, 1, 6, 1, 1, 1, 4, 30, 1, 9, 6, 1, 1, 1, 16, 1, 1, 1, 72, 1, 1, 1, 4, 1, 1, 1, 6, 12, 1, 1, 24, 56, 30, 1, 6, 1, 9, 1, 4, 1, 1, 1, 6, 1, 1, 12, 96, 1, 1, 1, 6, 1, 1, 1, 48, 1, 1, 30, 6, 1, 1, 1, 24, 108, 1, 1, 6, 1, 1, 1, 4, 1, 12, 1, 6, 1, 1, 1, 16, 1, 56, 12, 180
COMMENTS
This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 27, although a(4) = 6 and a(27) = 9. See A344702.
PROG
(PARI)
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
Search completed in 0.168 seconds
|