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Unitary harmonic numbers (those for which the unitary harmonic mean is an integer).
(Formerly M4248)
+10
26
1, 6, 45, 60, 90, 420, 630, 1512, 3780, 5460, 7560, 8190, 9100, 15925, 16632, 27300, 31500, 40950, 46494, 51408, 55125, 64260, 66528, 81900, 87360, 95550, 143640, 163800, 172900, 185976, 232470, 257040, 330750, 332640, 464940, 565488, 598500, 646425, 661500
COMMENTS
Let ud(n) and usigma(n) be number of and sum of unitary divisors of n; then the unitary harmonic mean of the unitary divisors is H(n) = n*ud(n)/usigma(n). - Emeric Deutsch, Dec 22 2004
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
If m is a term and omega(m) = A001221(m) = k, then m < 2^(k*2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020
MATHEMATICA
ud[n_] := 2^PrimeNu[n]; usigma[n_] := Sum[ If[ GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; uhm[n_] := n*ud[n]/usigma[n]; Reap[ Do[ If[ IntegerQ[uhm[n]], Print[n]; Sow[n]], {n, 1, 10^6}]][[2, 1]] (* Jean-François Alcover, May 16 2013 *)
PROG
(Haskell)
a006086 n = a006086_list !! (n-1)
a006086_list = filter ((== 1) . a103340) [1..]
(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
isok(n) = my(v=udivs(n)); denominator(n*#v/vecsum(v))==1; \\ Michel Marcus, May 07 2017 (PARI) is(n, f=factor(n))=(n<<(#f~))%sumdivmult([n, f], d, if(gcd(d, n/d)==1, d))==0 \\ Charles R Greathouse IV, Nov 05 2021 (PARI) list(lim)=my(v=List()); forfactored(n=1, lim\1, if((n[1]<<omega(n))%sumdivmult(n, d, if(gcd(d, n[1]/d)==1, d))==0, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2021
Numerator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.
+10
10
1, 4, 3, 8, 5, 2, 7, 16, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 8, 25, 52, 27, 14, 29, 10, 31, 64, 11, 68, 35, 72, 37, 38, 39, 80, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 28, 57, 116, 59, 4, 61, 31, 63, 128, 65, 11, 67, 136, 23, 35, 71, 16, 73
EXAMPLE
1, 4/3, 3/2, 8/5, 5/3, 2, ...
a(8) = 16 because the unitary divisors of 8 are {1,8} and 2/(1/1 + 1/8) = 16/9.
MAPLE
with(numtheory): udivisors:=proc(n) local A, k: A:={}: for k from 1 to tau(n) do if gcd(divisors(n)[k], n/divisors(n)[k])=1 then A:=A union {divisors(n)[k]} else A:=A fi od end: utau:=n->nops(udivisors(n)): usigma:=n->sum(udivisors(n)[j], j=1..nops(udivisors(n))): uH:=n->n*utau(n)/usigma(n):seq(numer(uH(n)), n=1..81);
MATHEMATICA
ud[n_] := 2^PrimeNu[n]; usigma[n_] := DivisorSum[n, If[GCD[#, n/#] == 1, #, 0]&]; a[1] = 1; a[n_] := Numerator[n*ud[n]/usigma[n]]; Array[a, 100] (* Jean-François Alcover, Dec 03 2016 *) a[n_] := Numerator[n * Times @@ (2 / (1 + Power @@@ FactorInteger[n]))]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 10 2023 *)
PROG
(Haskell)
import Data.Ratio ((%), numerator)
a103339 = numerator . uhm where uhm n = (n * a034444 n) % (a034448 n)
(Python)
from sympy import gcd
from sympy.ntheory.factor_ import udivisor_sigma
def A103339(n): return (lambda x, y: y*n//gcd(x, y*n))(udivisor_sigma(n), udivisor_sigma(n, 0)) # Chai Wah Wu, Oct 20 2021 (PARI) a(n) = {my(f = factor(n)); numerator(n * prod(i=1, #f~, 2/(1 + f[i, 1]^f[i, 2]))); } \\ Amiram Eldar, Mar 10 2023
Harmonic means of the infinitary divisors of the infinitary harmonic numbers.
+10
2
1, 2, 3, 4, 4, 6, 7, 7, 11, 13, 13, 10, 7, 15, 16, 15, 9, 20, 18, 14, 25, 24, 19, 25, 15, 27, 28, 30, 18, 36, 13, 21, 17, 29, 40, 33, 24, 28, 38, 31, 29, 45, 34, 27, 28, 44, 27, 60, 36, 52, 46, 26, 51, 42, 55, 33, 66, 40, 24, 37, 49, 29, 47, 57, 34, 68, 49, 44
COMMENTS
Each term appears a finite number of times in the sequence (Hagis and Cohen, 1990).
MATHEMATICA
f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; s[1] = 1; s[n_] := n * Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10^4], IntegerQ]
PROG
(PARI) ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
lista(kmax) = {my(ih); for(k = 1, kmax, ih = ihmean(k); if(denominator(ih) == 1, print1(ih, ", "))); }
a(n) is the number of distinct prime factors of the n-th unitary harmonic number.
+10
2
0, 2, 2, 3, 3, 4, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 4, 3, 5, 4, 5, 5, 5, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 4, 4, 4, 5, 6, 5, 6, 5, 5, 6, 6, 5, 5, 6, 6, 5, 5, 6, 6, 6, 6, 5, 6, 5, 6, 6, 6, 5, 6, 5, 6, 5, 6, 5, 6, 4, 5, 6, 6, 6, 6, 5, 6, 5, 6, 6, 6, 6, 5, 6
COMMENTS
Each term appears a finite number of times in the sequence (Hagis and Lord, 1975).
MATHEMATICA
uh[n_] := n * Times @@ (2/(1 + Power @@@ FactorInteger[n])); uh[1] = 1; PrimeNu[Select[Range[10^6], IntegerQ[uh[#]] &]]
PROG
(PARI) uhmean(n) = {my(f = factor(n)); n*prod(i=1, #f~, 2/(1+f[i, 1]^f[i, 2])); };
lista(kmax) = {my(uh); for(k = 1, kmax, uh = uhmean(k); if(denominator(uh) == 1, print1(omega(k), ", "))); }
Harmonic means the bi-unitary divisors of the bi-unitary harmonic numbers ( A286325).
+10
2
1, 2, 3, 4, 4, 6, 7, 7, 8, 11, 13, 13, 12, 10, 16, 7, 18, 16, 15, 24, 15, 20, 20, 18, 14, 22, 25, 24, 19, 25, 23, 27, 33, 31, 44, 32, 34, 30, 25, 36, 13, 46, 31, 21, 29, 40, 38, 33, 28, 40, 48, 38, 29, 45, 34, 47, 28, 32, 32, 44, 60, 27, 32, 28, 46, 26, 51
EXAMPLE
a(3) = 3 since A286325(3) = 45, the bi-unitary divisors of 45 are 1, 5, 9, and 45, and their harmonic mean is 3.
MATHEMATICA
f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10^5], IntegerQ]
PROG
(PARI) bhmean(n) = {my(f = factor(n), p, e); n * prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2)))); }
lista(kmax) = {my(bh); for(k = 1, kmax, bh = bhmean(k); if(denominator(bh) == 1, print1(bh, ", "))); }
a(n) is the number of "Fermi-Dirac prime" factors (or I-components) of the n-th infinitary harmonic number.
+10
1
0, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 4, 3, 5, 5, 5, 4, 6, 5, 5, 6, 6, 5, 6, 5, 6, 6, 6, 5, 7, 4, 5, 5, 6, 7, 6, 6, 6, 7, 6, 6, 7, 6, 6, 6, 7, 6, 8, 7, 7, 7, 6, 7, 7, 7, 6, 8, 6, 5, 6, 7, 6, 7, 7, 6, 8, 7, 7, 8, 7, 6, 7, 8, 7, 6, 8, 7, 7, 7, 7, 9, 6, 8, 6, 8, 8, 7
COMMENTS
Each term appears a finite number of times in the sequence (Hagis and Cohen, 1990).
MATHEMATICA
f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; ih[1] = 1; ih[n_] := n*Times @@ f @@@ FactorInteger[n]; ic[n_] := Plus @@ (DigitCount[Last /@ FactorInteger[n], 2, 1]); ic[1] = 0; ic /@ Select[Range[10^5], IntegerQ[ih[#]] &]
PROG
ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
lista(kmax) = {my(ih); for(k = 1, kmax, ih = ihmean(k); if(denominator(ih) == 1, print1( A064547(k), ", "))); }
Unitary harmonic numbers ( A006086) that are not unitary arithmetic numbers ( A103826).
+10
0
90, 40682250, 81364500, 105773850, 423095400, 1798155450, 14385243600
COMMENTS
There are 290 unitary harmonic numbers below 10^12, and only 7 of them are in this sequence.
EXAMPLE
90 is in the sequence since its unitary divisors are {1, 2, 5, 9, 10, 18, 45, 90}, their harmonic mean, 4, is an integer, but their arithmetic mean, 45/2, is not.
MATHEMATICA
q[n_] := Module[{f = FactorInteger[n], d, s}, d = 2^Length[f]; s = Times @@ (1 + Power @@@ f); IntegerQ[n*d/s] && !IntegerQ[s/d]]; Select[Range[5*10^7], q]
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