A188999 -id:A188999 - cp's OEIS frontend

A331970 The sum of the proper bi-unitary divisors of n.

0, 1, 1, 1, 1, 6, 1, 7, 1, 8, 1, 8, 1, 10, 9, 11, 1, 12, 1, 10, 11, 14, 1, 36, 1, 16, 13, 12, 1, 42, 1, 31, 15, 20, 13, 14, 1, 22, 17, 50, 1, 54, 1, 16, 15, 26, 1, 60, 1, 28, 21, 18, 1, 66, 17, 64, 23, 32, 1, 60, 1, 34, 17, 55, 19, 78, 1, 22, 27, 74, 1, 78, 1

Offset: 1

Amiram Eldar, Feb 03 2020

nonn

First differs from A126168 at n = 16.

			a(6) = 6 since A188999(6) - 6 = 12 - 6 = 6.

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

Cf. A001065, A034460, A126164, A126168, A188999, A222266.

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]); bs[n_] := bsigma[n] - n; Array[bs, 100]

a(n) = A188999(n) - n.

A332037 Indices of records in A332036.

Original entry on oeis.org

1, 12, 24, 60, 120, 240, 360, 720, 1440, 2160, 2880, 4320, 5760, 7200, 8640, 12960, 14400, 17280, 21600, 25920, 28800, 30240, 34560, 40320, 43200, 51840, 60480, 86400, 120960, 172800, 181440, 241920, 259200, 302400, 362880, 483840, 518400, 604800, 725760, 907200

Offset: 1

Amiram Eldar, Feb 05 2020

nonn

Numbers k such that bsigma(x) = k has more solutions x than any smaller k, where bsigma(x) is the sum of bi-unitary divisors of x (A188999).
The bi-unitary version of A145899.
The corresponding number of solutions for each term is 1, 2, 3, 5, 7, 12, 13, 20, ... (see the link for more values).

			There are 3 solutions to bsigma(x) = 24: bsigma(14) = bsigma(15) = bsigma(23) = 24. For all m < 24 there are 2 or fewer solutions to bsigma(x) = m, thus 24 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..58
Amiram Eldar, Table of n, a(n), A332036(a(n)) for n = 1..58

Cf. A145899, A188999, A289132, A332036, A332037, A332039.

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 = 10000; v = Table[0, {m}]; Do[b = bsigma[k]; If[b <= m, v[[b]]++], {k, 1, m}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 1, m}]; s

A376888 The sum of divisors of n that are products of factors of the form p^(k!) with multiplicities not larger than their multiplicity in n, where p is a prime and k >= 1, when the factorization of n is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 21, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 63, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 84, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144

Offset: 1

Amiram Eldar, Oct 08 2024

See A376885 for details about this factorization.
First differs from A188999 at n = 16.
The number of these divisors is given by A376887(n).

			For n = 12 = 2^2 * 3^1, the representation of 2 in factorial base is 10, i.e., 2 = 2!, so 12 = (2^(2!))^1 * (3^(1!))^1 and a(12) is the sum of the 4 divisors 1 + 3 + 4 + 12 = 20.

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

Cf. A188999, A376885, A376886, A376887.

Similar sequences: A049417, A049418, A074847, A097863, A331107, A331110.

ff[q_, s_] := (q^(s + 1) - 1)/(q - 1); f[p_, e_] := Module[{k = e, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, AppendTo[s, {p^(m - 1)!, r}];]; m++]; Times @@ ff @@@ s]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

fdigits(n) = {my(k = n, m = 2, r, s = []); while([k, r] = divrem(k, m); k != 0 || r != 0, s = concat(s, r); m++); s;}
a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2], d); prod(i = 1, #p, prod(j = 1, #d=fdigits(e[i]), (p[i]^(j!*(d[j]+1)) - 1)/(p[i]^j! - 1)));}

Multiplicative: if e = Sum_{k>=1} d_k * k! (factorial base representation), then a(p^e) = Product_{k>=1} (p^(k!*{d_k+1}) - 1)/(p^(k!) - 1).

A200723 The sum of integers k from 1 to n such that the greatest common unitary divisor of k and n is 1.

Original entry on oeis.org

1, 1, 3, 6, 10, 10, 21, 28, 36, 32, 55, 53, 78, 66, 69, 120, 136, 112, 171, 144, 153, 170, 253, 211, 300, 240, 351, 300, 406, 237, 465, 496, 384, 416, 445, 539, 666, 522, 558, 633, 820, 444, 903, 780, 772, 770, 1081, 887, 1176, 912, 951, 1104

Offset: 1

R. J. Mathar, Nov 21 2011

Previous name was "Bi-unitary Euler function of n".

a(n) is the sum over all entries equal to 1 in row A165430(n,), weighted by the column index.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function J. Int. Seq. 12 (2009), Article 09.5.2.

Cf. A116550, A165430, A188999

a200723 = sum . zipWith (*) [1..] . map a063524 . a165430_row
-- Reinhard Zumkeller, Mar 04 2013

A200723 := proc(n)
        local a,k ;
        a := 0 ;
        for k from 1 to n do
                if A165430(k,n) = 1 then
                        a := a+ k ;
                end if;
        end do;
        a ;
end proc:
seq(A200723(n),n=1..80) ;

T[n_, k_] := Module[{d = Divisors[GCD[n, k]]}, Max[Select[d, CoprimeQ[#, k/#] && CoprimeQ[#, n/#] &]]]; a[n_] := Sum[k * Boole[T[n, k] == 1], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, May 23 2025 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
a(n) = sum(k=1, n, if (vecmax(setintersect(udivs(n), udivs(k))) == 1, k)); \\ Michel Marcus, Jun 28 2023

a(6) = 1*1 + 4*1 +5*1 = 10 corresponding to the three 1's in row 6 of A165430.

New name from Amiram Eldar, May 23 2025

A307160 Decimal expansion of the constant c in the asymptotic formula for the partial sums of the bi-unitary divisors sum function, A307159(k) ~ c*k^2.

Original entry on oeis.org

7, 5, 2, 8, 3, 8, 7, 4, 1, 0, 0, 2, 2, 9, 4, 3, 1, 1, 5, 4, 3, 3, 3, 0, 9, 5, 1, 5, 5, 3, 0, 4, 1, 2, 7, 6, 5, 1, 9, 5, 2, 5, 4, 6, 7, 5, 6, 5, 2, 2, 1, 0, 8, 5, 8, 7, 7, 9, 0, 3, 2, 8, 7, 8, 6, 8, 1, 2, 5, 2, 2, 6, 0, 5, 5, 8, 1, 4, 8, 7, 8, 4, 7, 7, 4, 1, 8, 6, 0, 4, 7, 8, 2, 5, 8, 0, 7, 0, 0, 1, 1, 9, 9, 4, 1, 3

Offset: 0

Amiram Eldar, Mar 27 2019

The asymptotic mean of the bi-unitary abundancy index lim_{n->oo} (1/n) * Sum_{k=1..n} A188999(k)/k = 2*c = 1.505677... - Amiram Eldar, Jun 10 2020

			0.75283874100229431154333095155304127651952546756522...

D. Suryanarayana and M. V. Subbarao, Arithmetical functions associated with the biunitary k-ary divisors of an integer, Indian J. Math., Vol. 22 (1980), pp. 281-298.

László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1, section 4.13.

Cf. A188999, A306071, A306072, A307159.

$MaxExtraPrecision = 1000; nm=1000; c = Rest[CoefficientList[Series[Log[1 - 2*x^3 + x^4 + x^5 - x^6],{x,0,nm}],x] * Range[0, nm]]; RealDigits[(Zeta[2]*Zeta[3]/2) * Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]

Equals (zeta(2)*zeta(3)/2)* Product_{p}(1 - 2/p^3 + 1/p^4 + 1/p^5 - 1/p^6).

More terms from Vaclav Kotesovec, May 29 2020

A331110 The sum of dual-Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the dual Zeckendorf expansion (A104326) of each s(i) contains only terms that are in the dual Zeckendorf expansion of r(i).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 27, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 45, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 108, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 135, 84, 144

Offset: 1

Amiram Eldar, Jan 09 2020

First differs from A188999 at n = 32.

			a(32) = 45 since 32 = 2^5 and the dual Zeckendorf expansion of 5 is 110, i.e., its dual Zeckendorf representation is a set with 2 terms: {2, 3}. There are 4 possible exponents of 2: 0, 2, 3 and 5, corresponding to the subsets {}, {2}, {3} and {2, 3}. Thus 32 has 4 dual-Zeckendorf-infinitary divisors: 2^0 = 1, 2^2 = 4, 2^3 = 8, and 2^5 = 32, and their sum is 1 + 4 + 8 + 32 = 45.

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

The number of dual-Zeckendorf-infinitary divisors of n is in A331109.

Cf. A049417, A049418, A074847, A097863, A104326, A188999, A331107.

fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr];
dualZeck[n_] := Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, {}, v[[i[[1, 1]] ;; -1]]]];
f[p_, e_] := p^Fibonacci[1 + Position[Reverse@dualZeck[e], _?(# == 1 &)]];
a[1] = 1; a[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); Array[a, 100]

Multiplicative with a(p^e) = Product_{i} (p^s(i) + 1), where s(i) are the terms in the dual Zeckendorf representation of e (A104326).

A331971 a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 1, 3, 2, 3, 1, 3, 1, 3, 2, 4, 1, 6, 1, 4, 2, 4, 2, 5, 0, 3, 1, 4, 2, 5, 1, 4, 2, 4, 1, 6, 2, 5, 2, 5, 2, 8, 1, 6, 1, 4, 2, 7, 1, 5, 3, 5, 2, 8, 0, 5, 1, 6, 1, 8, 2, 5, 3, 6, 3, 9, 0, 6, 2, 5, 1, 9, 1, 7, 1

Offset: 2

Amiram Eldar, Feb 03 2020

nonn

The bi-unitary version of A048138.

The offset is 2 as in A048138 since there are infinitely many numbers k (the primes and squares of primes) for which A331970(k) = 1.

			a(8) = 2 since 8 is the sum of the proper bi-unitary divisors of 2 numbers: 10 (1 + 2 + 5) and 12 (1 + 3 + 4).

Amiram Eldar, Table of n, a(n) for n = 2..30000

Cf. A048138, A188999, A222266, A324938, A331970, A331973.

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]); bs[n_] := bsigma[n] - n; m = 300; v = Table[0, {m}]; Do[b = bs[k]; If[2 <= b <= m, v[[b]]++], {k, 1, m^2}]; Rest @ v

A331972 Bi-unitary highly touchable numbers: numbers m > 1 such that a record number of numbers k have m as the sum of the proper bi-unitary divisors of k.

Original entry on oeis.org

2, 6, 8, 17, 29, 31, 55, 79, 91, 115, 121, 175, 181, 211, 295, 301, 361, 391, 421, 481, 511, 571, 631, 781, 841, 991, 1051, 1231, 1261, 1471, 1561, 1651, 1681, 1891, 2101, 2311, 2731, 3151, 3361, 3571, 3991, 4201, 4291, 4411, 4621, 5251, 5461, 6091, 6511, 6931

Offset: 1

Amiram Eldar, Feb 03 2020

nonn

The corresponding record values are 0, 1, 2, 3, 4, 6, 8, 9, 10, 11, 14, 15, ...

The bi-unitary version of A238895.

			a(1) = 2 since it is the first number which is not the sum of proper bi-unitary divisors of any number.
a(2) = 6 since it is the least number which is the sum of proper bi-unitary divisors of one number: 6 = A331970(6).
a(3) = 8 since it is the least number which is the sum of proper bi-unitary divisors of 2 numbers: 8 = A331970(10) = A331970(12).

Amiram Eldar, Table of n, a(n) for n = 1..73 (terms below 30000)

Cf. A188999, A238895, A324276, A325177, A331970, A331974.

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]); bs[n_] := bsigma[n] - n; m = 300; v = Table[0, {m}]; Do[b = bs[k]; If[2 <= b <= m, v[[b]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 2, m}]; s

A335053 Odd bi-unitary abundant numbers whose bi-unitary abundancy is closer to 2 than that of any smaller odd bi-unitary abundant number.

Original entry on oeis.org

945, 25515, 46035, 49875, 83265, 354585, 359205, 361515, 366135, 382305, 389235, 396165, 400785, 403095, 407715, 414645, 416955, 423885, 430815, 437745, 442365, 13351635, 132335385, 159030135, 1756753845, 6561644355, 10394173335, 13455037365, 37456183215

Offset: 1

Amiram Eldar, May 21 2020

nonn

The bi-unitary abundancy of a number k is bsigma(k)/k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

			The bi-unitary abundancies of the first terms are 2.031..., 2.005..., 2.0019..., 2.0018..., 2.0015..., ...

Cf. A188999, A188263, A293186, A302571, A335052, A335055.

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]); seq = {}; r = 3; Do[s = bsigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10^6, 2}]; seq

A293618 Numbers n that equal the sum of their first k consecutive aliquot bi-unitary divisors, but not all of them (i.e k < A286324(n)-1).

Original entry on oeis.org

24, 360, 432, 1344, 2016, 19440, 45360, 68544, 714240, 864000, 1468800, 1571328, 1900800, 2391120, 2888704, 3057600, 4586400, 5241600, 103194000

Offset: 1

Amiram Eldar, Oct 13 2017

The bi-unitary version of Erdős-Nicolas numbers (A194472).

If all the aliquot bi-unitary divisors are permitted (i.e. k <= A286324(n)-1), then the 3 bi-unitary perfect numbers, 6, 60 and 90, are included.

			24 is in the sequence since its aliquot bi-unitary divisors are 1, 2, 3, 4, 6, 8, 12 and 24 and 1 + 2 + 3 + 4 + 6 + 8 = 24.

Cf. A188999, A194472, A222266, A286324.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Drop[bdiv[n], -2]]; a = {}; Do[ If[selDivs[n] == 0, AppendTo[a, n]; Print[n]], {n, 2, 10^6}]; a (* after Alonso del Arte at A194472 *)

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A331970 The sum of the proper bi-unitary divisors of n.

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A332037 Indices of records in A332036.

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A200723 The sum of integers k from 1 to n such that the greatest common unitary divisor of k and n is 1.

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A307160 Decimal expansion of the constant c in the asymptotic formula for the partial sums of the bi-unitary divisors sum function, A307159(k) ~ c*k^2.

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A331110 The sum of dual-Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the dual Zeckendorf expansion (A104326) of each s(i) contains only terms that are in the dual Zeckendorf expansion of r(i).

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A331971 a(n) is the number of values of m such that the sum of proper bi-unitary divisors of m (A331970) is n.

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