A188999 -id:A188999 - cp's OEIS frontend

A323342 Numbers k whose bi-unitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

704, 1458, 2394, 7544, 10184, 46400, 60416, 106434, 115182, 118098, 121014, 125000, 129762, 141426, 147258, 150174, 156006, 158922, 164754, 176418, 185166, 190998, 199746, 202662, 217242, 220158, 228906, 237654, 243486, 246402, 252234, 260982, 263898, 278478

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

The bi-unitary version of A171641.

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

Cf. A171641, A188999, A292982, A323341, A323343, A323344.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[n_] := Select[Divisors[n], Last@Intersection[f@#, f[n/#]] == 1 &]; fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[n_] := If[n==1, 1, Times @@ (fun @@@ FactorInteger[n])]; seq={}; Do[s=bsigma[n]; If[OddQ[s] || s<=2n, Continue[]]; div = bdiv[n]; If[Coefficient[Times @@ (1 + x^div) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 10000}]; seq

A324706 The sum of the tri-unitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144

Offset: 1

Amiram Eldar, Mar 11 2019

A divisor d of n is tri-unitary if the greatest common bi-unitary divisor of d and n/d is 1.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
Pentti Haukkanen, On the k-ary convolution of arithmetical functions, The Fibonacci Quarterly, Vol. 38, No. 5 (2000) pp. 440-445.

Cf. A000203, A034448, A049417, A072691, A188999, A322485, A324707, A324708, A324709.

f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; a[1]=1; a[n_]:= Times @@ f @@@ FactorInteger[n]; Array[a, 100]

A324706(n) = { my(f = factor(n)); prod(i=1, #f~, if(3==f[i,2], sigma(f[i,1]^f[i,2]), if(6==f[i,2], ((f[i,1]^8)-1)/((f[i,1]^2)-1), 1+(f[i,1]^f[i,2])))); }; \\ Antti Karttunen, Mar 12 2019

Multiplicative with a(p^3) = 1 + p + p^2 + p^3, a(p^6) = 1 + p^2 + p^4 + p^6, and a(p^e) = 1 + p^e otherwise.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^3 + 1/p^4 - 2/p^6 + 2/p^8 - 1/p^9 - 1/p^12 + 1/p^13) = 0.72189237802... . - Amiram Eldar, Nov 24 2022

A189000 Bi-unitary multiperfect numbers.

Original entry on oeis.org

1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240, 342720, 523776, 1028160, 1528800, 6168960, 7856640, 7983360, 14443520, 22932000, 23569920, 43330560, 44553600, 51979200, 57657600, 68796000, 133660800, 172972800, 779688000, 1476304896, 2339064000, 6840038400

Offset: 1

R. J. Mathar, Apr 15 2011

nonn

All entries greater than 1 are even [Hagis].
14443520 is the first (only?) composite term not divisible by 3.  Excluding the factor p=3, all composite terms <= 172972800 have nonincreasing exponents in the factorization (sorted by primes). - D. S. McNeil, Apr 15 2011
Wall shows that 6, 60, and 90 are the only bi-unitary perfect numbers. - Tomohiro Yamada, Apr 15 2017
McNeil's observation about exponents does not hold in general. Indeed, a(41) = 2^8 * 3^5 * 5^2 * 7 * 11 * 13^2 * 17. - Giovanni Resta, Apr 15 2017
a(43) > 4.66*10^12. - Giovanni Resta, Sep 07 2018
We include 1 here, although this is not "multi"-perfect. - R. J. Mathar, Sep 08 2018

			n=120 divides A188999(120)=360.
n=90 divides A188999(90)=180.
n=672 divides A188999(672)=2016.

Giovanni Resta, Table of n, a(n) for n = 1..42
Peter Hagis, Bi-Unitary amicable and multiperfect numbers, Fib. Quart. 25 (2) (1987) 144-151
Pentti Haukkanen and V. Sitaramaiah, Bi-unitary multiperfect numbers, I, Notes Number Theory Discrete Math. 26 (1) (2020) 93-171.
Michel Marcus, Unexhaustive list of terms
C. R. Wall, Bi-unitary perfect numbers, Proc. Amer. Math. Soc. 33 (1) (1972) 39-42.
Tomohiro Yamada, Determining all biunitary triperfect numbers of a certain form, arXiv:2406.19331 [math.NT], 2024.

Cf. A007691 (analog for sigma).

Cf. A188999 (bi-unitary sigma), A318175, A318781 (the k coefficients).

bsig[n_] := If[n == 1, 1, Block[{p, e}, Product[{p, e} = pe; (p^(e + 1) - 1)/(p - 1) - If[EvenQ[e], p^(e/2), 0], {pe, FactorInteger[n]}]]]; Select[Range[10^5], Mod[bsig[#], #] == 0 &] (* Giovanni Resta, Apr 15 2017 *)

a188999(n) = {my(f = factor(n)); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
isok(n) = ! frac(a188999(n)/n); \\ Michel Marcus, Sep 03 2018

{n | A188999(n)}.

a(18)-a(27) by D. S. McNeil, Apr 15 2011
a(28)-a(31) from Giovanni Resta, Apr 15 2017
a(1)=1 inserted by Giovanni Resta, Sep 07 2018

A302994 Number of bi-unitary abundant numbers < 10^n.

Original entry on oeis.org

0, 14, 147, 1553, 15450, 155395, 1549818, 15498814, 155079196, 1550331185, 15503061466, 155037242668, 1550370696100, 15503650949671, 155036854371220, 1550366484701654, 15503648102080675

Offset: 1

Amiram Eldar, Apr 17 2018

Cf. A188999, A292982, A302992.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=
DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; babQ[n_] := bsigma[n] > 2 n; c = 0; k = 1; seq={}; Do[While[k < 10^n, If[babQ[k], c++]; k++]; AppendTo[seq, c], {n, 1, 5}]; seq

biusigma(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
a(n) = sum(k=1, 10^n-1, biusigma(k) > 2*k); \\ Michel Marcus, Apr 17 2018

Conjecture: Lim_{n->oo} a(n)/10^n = 0.15... is the density of bi-unitary abundant numbers.

a(8)-a(17) from Hiroaki Yamanouchi, Aug 24 2018

A322485 The sum of the semi-unitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 11, 10, 18, 12, 20, 14, 24, 24, 19, 18, 30, 20, 30, 32, 36, 24, 44, 26, 42, 31, 40, 30, 72, 32, 39, 48, 54, 48, 50, 38, 60, 56, 66, 42, 96, 44, 60, 60, 72, 48, 76, 50, 78, 72, 70, 54, 93, 72, 88, 80, 90, 60, 120, 62, 96, 80, 71, 84, 144

Offset: 1

Amiram Eldar, Dec 11 2018

A semi-unitary divisor of n is defined as the largest divisor d of n such that the largest divisor of d that is a unitary divisor of n/d is 1 (see A322483).

			The semi-unitary divisors of 8 are 1, 2, 8 (4 is not semi-unitary divisor since the largest divisor of 4 that is a unitary divisor of 8/4 = 2 is 2 > 1), and their sum is 11, thus a(8) = 11.

J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Pentti Haukkanen, Basic properties of the bi-unitary convolution and the semi-unitary convolution, Indian J. Math, Vol. 40 (1998), pp. 305-315.
D. Suryanarayana and V. Siva Rama Prasad, Sum functions of k-ary and semi-k-ary divisors, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.
Laszlo Tóth, Sum functions of certain generalized divisors, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Vol. 41 (1998), pp. 165-180.

Cf. A000203, A005117, A034448, A183699, A188999, A322483.

f[p_, e_] := (p^Floor[(e+1)/2] - 1)/(p-1) + p^e; susigma[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Array[susigma, 100]

a(n) = {my(f = factor(n)); for (k=1, #f~, my(p=f[k,1], e=f[k,2]); f[k,1] = (p^((e+1)\2) - 1)/(p-1) + p^e; f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Dec 14 2018

Multiplicative with a(p^e) = sigma(p^floor((e-1)/2)) + p^e = (p^floor((e+1)/2) - 1)/(p-1) + p^e.
In particular a(p) = p + 1, a(p^2) = p^2 + 1, a(p^3) = p^3 + p + 1.
a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)*zeta(3)/2) * Product_{p prime} (1 - 2/p^3 + 1/p^5) = 0.7004703314... . - Amiram Eldar, Nov 24 2022

A324276 Bi-unitary untouchable numbers: numbers that are not the sum of aliquot bi-unitary divisors of any number.

Original entry on oeis.org

2, 3, 4, 5, 38, 68, 80, 96, 98, 128, 138, 146, 158, 164, 180, 188, 192, 206, 208, 210, 212, 222, 224, 248, 264, 278, 290, 300, 304, 308, 324, 326, 328, 338, 360, 374, 380, 390, 398, 416, 418, 420, 430, 432, 458, 476, 480, 488, 498, 516, 518, 530, 536, 542, 548

Offset: 1

Amiram Eldar, Feb 20 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3591 (terms below 30000)

Cf. A188999, A005114, A063948 (unitary), A324277 (infinitary), A324278 (exponential), A331970.

fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := bsigma[n] = Times @@ (fun @@@ FactorInteger[n]); untouchableQ[n_] := Catch[ Do[ If[n == bsigma[k]-k, Throw[True]], {k, 0, (n-1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Sow[n]], {n, 2, 550}]][[2, 1]] (* after Jean-François Alcover at A005114 *)

A332036 Number of integers whose bi-unitary divisors sum to n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 1, 1, 0, 0, 3, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 5, 0, 1, 0, 0, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

			a(12) = 2 since there are 2 solutions to bsigma(x) = 12 (bsigma is A188999): 6 and 11.

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

Cf. A054973, A063974, A188999, A332038, A332040.

fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); m = 100; v = Table[0, {m}]; Do[b = bsigma[k]; If[b <= m, v[[b]]++], {k, 1, m}]; v

A335215 Bi-unitary Zumkeller numbers: numbers whose set of bi-unitary divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

6, 24, 30, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 88, 90, 96, 102, 104, 114, 120, 138, 150, 160, 162, 168, 174, 186, 192, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390, 402

Offset: 1

Amiram Eldar, May 27 2020

nonn

			6 is a term since its set of bi-unitary divisors, {1, 2, 3, 6}, can be partitioned into 2 disjoint sets, whose sum is equal: 1 + 2 + 3 = 6.

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

The bi-unitary version of A083207.
Subsequence of A292982.
Cf. A188999, A222266, A290466, A323342, A335142, A335197.

uDivs[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; bDivs[n_] := Select[Divisors[n], Last @ Intersection[uDivs[#], uDivs[n/#]] == 1 &]; bzQ[n_] := Module[{d = bDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[10^3], bzQ]

A361782 Numerators of the harmonic means of the bi-unitary divisors of the positive integers.

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 64, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 64, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 32, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 384, 65, 11, 67, 136

Offset: 1

Amiram Eldar, Mar 24 2023

			Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.

Cf. A188999, A222266, A286324, A361783 (denominators).

Similar sequences: A099377, A103339, A361316.

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))]; a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); numerator(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))))); }

a(n) = numerator(n*A286324(n)/A188999(n)).

f(n) = a(n)/A361783(n) is multiplicative with f(p^e) = (e+1)*(p-1)/(p^(e+1)-1) if e is odd, and e/((p^(e+1)-1)/(p-1) - p^(e/2)) if e is even.

A361783 Denominators of the harmonic means of the bi-unitary divisors of the positive integers.

Original entry on oeis.org

1, 3, 2, 5, 3, 1, 4, 15, 5, 9, 6, 5, 7, 3, 2, 27, 9, 5, 10, 3, 8, 9, 12, 5, 13, 21, 10, 5, 15, 3, 16, 21, 4, 27, 12, 25, 19, 15, 14, 9, 21, 2, 22, 15, 1, 9, 24, 9, 25, 39, 6, 35, 27, 5, 18, 15, 20, 45, 30, 1, 31, 12, 20, 119, 21, 3, 34, 45, 8, 9, 36, 25, 37, 57

Offset: 1

Amiram Eldar, Mar 24 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.

Cf. A188999, A222266, A286324, A286325 (positions of 1's), A361782 (numerators).

Similar sequences: A099378, A103340, A361317.

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))]; a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); denominator(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))))); }

a(n) = denominator(n*A286324(n)/A188999(n)).

cp's OEIS Frontend

A323342 Numbers k whose bi-unitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

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A324706 The sum of the tri-unitary divisors of n.

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A189000 Bi-unitary multiperfect numbers.

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A302994 Number of bi-unitary abundant numbers < 10^n.

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A322485 The sum of the semi-unitary divisors of n.

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A324276 Bi-unitary untouchable numbers: numbers that are not the sum of aliquot bi-unitary divisors of any number.

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A332036 Number of integers whose bi-unitary divisors sum to n.

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A335215 Bi-unitary Zumkeller numbers: numbers whose set of bi-unitary divisors can be partitioned into two disjoint sets of equal sum.

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