A187778 -id:A187778 - cp's OEIS frontend

A332881 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j).

1, 2, 3, 2, 5, 1, 7, 2, 3, 5, 11, 1, 13, 7, 5, 2, 17, 1, 19, 5, 21, 11, 23, 1, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 5, 23, 47, 1, 7, 5, 17, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 21, 2, 65, 11, 67, 17, 23, 35

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Denominator of sum of reciprocals of squarefree divisors of n.

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001615, A008683, A017666, A048250, A007947, A109395, A187778 (positions of 1's), A306695, A308443, A308462, A332880 (numerators), A332883.

a:= n-> denom(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Denominator

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
A332881(n) = denominator(A001615(n)/n);

Denominators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = denominator of Sum_{d|n} mu(d)^2/d.
a(n) = denominator of psi(n)/n.
a(p) = p, where p is prime.
a(n) = n / A306695(n) = n / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021

A323327 Numbers that start an unbounded aliquot-like sequence based on Dedekind psi function (A001615).

Original entry on oeis.org

318, 330, 498, 510, 534, 546, 636, 660, 786, 798, 942, 954, 978, 990, 996, 1020, 1068, 1092, 1110, 1122, 1254, 1272, 1320, 1398, 1410, 1470, 1494, 1506, 1518, 1530, 1572, 1596, 1602, 1614, 1626, 1638, 1734, 1884, 1908, 1938, 1950, 1956, 1980, 1992, 2040, 2046

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

Let t(k) = psi(k) - k = A001615(k) - k be the sum of aliquot divisors d of k, such that k/d is squarefree. Penney & Pomerance proposed a problem to show that the aliquot-like sequence related to this function, i.e., the trajectory of an integer k under the repeated application of the map k -> t(k), can be unbounded. Since t(m^j * k) = m^j * t(k) if m|k, then if in the sequence a_0 = k, a_1 = t(k), a_2 = t(t(k)), ... there is a term a_{i1} = m^j * a_0 such that m|k and j > 0 then a_{i+i1} = m^j * a_i for all i and thus the sequence is unbounded.

			318 is in the sequence since t(318) = psi(318) - 318 = 330, t(330) = 534, etc., and this repeated mapping yields an unbounded sequence.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 71, entry 318.

Kevin Brown and Charles Vanden Eynden, Pseudo-aliquot Sequences, Solution to Problem 10323, The American Mathematical Monthly, Volume 103, No. 8 (1996), pp. 697-698.
David E. Penney and Carl Pomerance, Problem 10323, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.
Eric Weisstein's World of Mathematics, Aliquot Sequence.
Wikipedia, Aliquot sequence.

Cf. A001615, A098007, A187778.

t[1]=0; t[n_] := (Times@@(1+1/Transpose[FactorInteger[n]][[1]])-1)*n; rt[n_] := Module[{f=FactorInteger[n]}, e=GCD@@f[[;;,2]]; Surd[n,e]]; divrootQ[n_, m_] := Divisible[n, rt[m]]; divQ[s_, n_] := If[n==0, 0, If[MemberQ[s, n], 1, If[ Length[Select[s, Divisible[n,#] && divrootQ[#, n/#] &]] > 0, 2, 3]]]; seqQ[n_] := Module[{n1=n}, s={}; While[divQ[s, n1] ==3, AppendTo[s, n1]; n1=t[n1]]; divQ[s, n1]] == 2; Select[Range[10000], seqQ]

A253628 Psi(n) mod n, where Psi is the Dedekind psi function (A001615).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 4, 3, 8, 1, 0, 1, 10, 9, 8, 1, 0, 1, 16, 11, 14, 1, 0, 5, 16, 9, 20, 1, 12, 1, 16, 15, 20, 13, 0, 1, 22, 17, 32, 1, 12, 1, 28, 27, 26, 1, 0, 7, 40, 21, 32, 1, 0, 17, 40, 23, 32, 1, 24, 1, 34, 33, 32, 19, 12, 1, 40, 27, 4, 1, 0, 1, 40, 45

Offset: 1

Tom Edgar, Jan 06 2015

nonn

a(n) = A054024(n) when n is squarefree.
Indices of 1 appear to be given by primes A000040 (see conjecture in A068494). The (weaker) statement that a(prime(i)) = 1 is a direct consequence of the multiplicity of A001615.
a(n) = 0 if n is a member of A187778.

			A001615(12) = 24 and 24 == 0 (mod 12) so a(12) = 0.
A001615(15) = 24 and 24 == 9 (mod 15) so a(15) = 9.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001615, A068494, A054024, A006093, A187778.

A253628 := proc(n)
    modp(A001615(n),n) ;
end proc: # R. J. Mathar, Jan 09 2015

a253628[n_] :=
Mod[DirichletConvolve[j, MoebiusMu[j]^2, j, #], #] & /@ Range@n; a253628[75] (* Michael De Vlieger, Jan 07 2015, after Jan Mangaldan at A001615 *)

[(n*mul(1+1/p for p in prime_divisors(n)))%n for n in [1..100]]

a(n) = A001615(n) mod n.

cp's OEIS Frontend

A332881 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A323327 Numbers that start an unbounded aliquot-like sequence based on Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A253628 Psi(n) mod n, where Psi is the Dedekind psi function (A001615).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula