A222266 -id:A222266 - cp's OEIS frontend

A319072 a(n) is the sum of the non-bi-unitary divisors of n.

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 4, 0, 9, 0, 12, 0, 0, 0, 0, 5, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 24, 18, 0, 0, 16, 7, 15, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 24, 8, 0, 0, 0, 36, 0, 0, 0, 45, 0, 0, 20, 40, 0, 0, 0, 24, 9, 0, 0, 64, 0, 0, 0, 0, 0, 54, 0, 48, 0, 0, 0, 0, 0, 21, 36, 87

Offset: 1

Omar E. Pol, Sep 22 2018

nonn

			For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12, and the bi-unitary divisors of 12 are 1, 3, 4, 12, hence the non-bi-unitary divisors of 12 are 2 and 6, and the sum of them is 2 + 6 = 8, so a(12) = 8. Also the sum of the divisors of 12 is 28, and the sum of the bi-unitary divisors of 12 is 20, so a(12) = 28 - 20 = 8.

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

Cf. A000203, A027750, A188999, A222266.

f1[p_, e_] := (p^(e+1) - 1)/(p - 1); f2[p_, e_] := f1[p, e] - If[OddQ[e], 0, p^(e/2)]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100] (* Amiram Eldar, Apr 04 2024 *)

a(n) = A000203(n) - A188999(n).

A335938 Bi-unitary pseudoperfect numbers (A292985) that are not exponentially odd numbers (A268335).

Original entry on oeis.org

48, 60, 72, 80, 90, 150, 162, 192, 240, 288, 294, 320, 336, 360, 420, 432, 448, 504, 528, 540, 560, 576, 600, 624, 630, 648, 660, 704, 720, 726, 756, 768, 780, 792, 800, 810, 816, 832, 880, 912, 924, 936, 960, 990, 1008, 1014, 1020, 1040, 1050, 1092, 1104, 1134

Offset: 1

Amiram Eldar, Jun 30 2020

nonn

Pseudoperfect numbers (A005835) that are exponentially odd (A268335) are also bi-unitary pseudoperfect numbers (A292985), since all of their divisors are bi-unitary.

First differs from A335216 at n = 28.

			48 is a term since it is not exponentially odd number (48 = 2^4 * 3 and 4 is even), so not all of its divisors are bi-unitary, and it is the sum of a subset of its bi-unitary divisors: 8 + 16 + 24 = 48.

Amiram Eldar, Table of n, a(n) for n = 1..3000

Subsequence of A005835 and A292985.

Cf. A222266, A268335, A295830, A335216.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; bPspQ[n_] := Module[{d = Most @ bdiv[n], x}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0]; expOddQ[n_] := AllTrue[Last /@ FactorInteger[n], OddQ]; Select[Range[1000], ! expOddQ[#] && bPspQ[#] &]

A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 2, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 3, 1, 2, 2, 5, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 4, 3, 2, 1, 3, 2, 2, 2, 4, 1, 3, 2, 2, 2, 2, 2, 6, 1, 2, 2, 2, 1, 3, 1, 4, 3

Offset: 1

Amiram Eldar, Nov 15 2021

The total number of prime powers (not including 1) that divide n is A001222(n).

The least number k such that a(k) = m is A122756(m).

			12 has 4 bi-unitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.

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

Cf. A000961, A001222, A122756, A162641, A222266, A246655, A286324, A268335, A350390.
Similar sequences: A001222, A349258, A349281.
Cf. A083342, A179119.

f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> if(x%2, x, x-1), factor(n)[, 2])); \\ Amiram Eldar, Sep 29 2023

Additive with a(p^e) = e if e is odd, and e-1 if e is even.
a(n) <= A001222(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A286324(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
a(n) = A001222(n) - A162641(n). - Amiram Eldar, May 18 2023
From Amiram Eldar, Sep 29 2023: (Start)
a(n) = A001222(A350390(n)) (the number of prime factors of the largest exponentially odd number dividing n, counted with multiplicity).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B_2 - C), where B_2 = A083342 and C = A179119. (End)

A361786 Bi-unitary arithmetic numbers: numbers for which the arithmetic mean of the bi-unitary divisors is an integer.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99

Offset: 1

Amiram Eldar, Mar 24 2023

nonn

First differs from A361386 at n = 35.

Number k such that A286324(k) divides A188999(k).

			6 is a term since the arithmetic mean of its bi-unitary divisors, {1, 2, 3, 6}, is 3 which is an integer.

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

Cf. A188999, A222266, A286324.

Similar sequences: A003601, A103826, A361386.

f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e]; q[1] = True; q[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], q]

is(n) = {my(f = factor(n), p, e); denominator(prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e))) == 1; }

A362854 The sum of the divisors of n that are both bi-unitary and exponential.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 10, 11, 12, 13, 14, 15, 18, 17, 18, 19, 20, 21, 22, 23, 30, 25, 26, 30, 28, 29, 30, 31, 34, 33, 34, 35, 36, 37, 38, 39, 50, 41, 42, 43, 44, 45, 46, 47, 54, 49, 50, 51, 52, 53, 60, 55, 70, 57, 58, 59, 60, 61, 62, 63, 70, 65, 66, 67, 68

Offset: 1

Amiram Eldar, May 05 2023

The number of these divisors is A362852(n).

The indices of records of a(n)/n are the primorials (A002110) cubed, i.e., 1 and the terms of A115964.

			a(8) = 10 since 8 has 2 divisors that are both bi-unitary and exponential, 2 and 8, and 2 + 8 = 10.

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

Cf. A002110, A004709, A051377, A082020, A115964, A188999, A222266, A322791, A361810, A362852.

f[p_, e_] := DivisorSum[e, p^# &] - If[OddQ[e], 0, p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(p, e) = sumdiv(e, d, p^d*(2*d != e));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2]));}

Multiplicative with a(p^e) = Sum_{d|e} p^d if e is odd, and (Sum_{d|e} p^d) - p^(e/2) if e is even.
a(n) >= n, with equality if and only if n is cubefree (A004709).
limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p)*(1 + Sum_{e>=1} Sum_{d|e, d != e/2}, p^(d-2*e))) = 0.5124353304539905... .

A363334 a(n) is the sum of divisors of n that are both coreful and bi-unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 14, 9, 10, 11, 12, 13, 14, 15, 26, 17, 18, 19, 20, 21, 22, 23, 42, 25, 26, 39, 28, 29, 30, 31, 62, 33, 34, 35, 36, 37, 38, 39, 70, 41, 42, 43, 44, 45, 46, 47, 78, 49, 50, 51, 52, 53, 78, 55, 98, 57, 58, 59, 60, 61, 62, 63, 118, 65, 66, 67

Offset: 1

Amiram Eldar, May 28 2023

First differs from A363331 at n = 16.

The number of these divisors is A363332(n).

			a(8) = 14 since 8 has 3 divisors that are both bi-unitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.

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

Cf. A004709, A057723, A188999, A222266, A362852, A363331, A363332.

f[p_, e_] := (p^(e+1) - 1)/(p - 1) - 1 - If[OddQ[e], 0, p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1 - if(f[i, 2]%2, 0, f[i, 1]^(f[i, 2]/2)));}

Multiplicative with a(p^e) = (p^(e+1) - 1)/(p - 1) - 1, if e is odd, and (p^(e+1) - 1)/(p - 1) - p^(e/2) - 1 if e is even.
a(n) >= n, with equality if and only if n is cubefree (A004709).
a(n) >= A362852(n), with equality if and only if n = 1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/2) * Product_{p prime} (p/(p+1))*(1+1/p-1/p^3+2/p^5) = 0.557782322450569540209... .
Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(3*s-2) + 2/p^(3*s-1) - 2/p^(4*s-2)). - Amiram Eldar, Oct 01 2023

A335939 Bi-unitary weird numbers (A292986) that are not exponentially odd numbers (A268335).

Original entry on oeis.org

5390, 7400, 11830, 20230, 24010, 25270, 37030, 58870, 67270, 95830, 117670, 129430, 154630, 196630, 243670, 260470, 314230, 352870, 373030, 436870, 482230, 491744, 507232, 554470, 658630, 714070, 742630, 801430, 831670, 893830, 1129030, 1201270, 1313830, 1352470

Offset: 1

Amiram Eldar, Jun 30 2020

nonn

Weird numbers (A006037) that are exponentially odd (A268335) are also bi-unitary weird numbers (A292986), since all of their divisors are bi-unitary.

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

Intersection of A072587 and A292986.

Cf. A006037, A222266, A268335, A335936, A335938.

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]); biuabQ[n_] := bsigma[n] > 2*n; f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; bweirdQ[n_] := biuabQ[n] && Module[{d = Most @ bdiv[n], x}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 0]; expOddQ[n_] := AllTrue[Last /@ FactorInteger[n], OddQ]; Select[Range[1000], !expOddQ[#] && bweirdQ[#] &]

More terms from Amiram Eldar, Mar 25 2023

A361787 Bi-unitary arithmetic numbers k whose mean bi-unitary divisor is a bi-unitary divisor of k.

Original entry on oeis.org

1, 6, 60, 270, 420, 630, 672, 2970, 5460, 8190, 10080, 22848, 30240, 99792, 136500, 172900, 204750, 208656, 245700, 249480, 312480, 332640, 342720, 385560, 491400, 695520, 708288, 791700, 819000, 861840, 1028160, 1037400, 1187550, 1228500, 1421280, 1528800, 1571328

Offset: 1

Amiram Eldar, Mar 24 2023

nonn

Also, bi-unitary harmonic numbers k whose harmonic mean of the bi-unitary divisors of k is a bi-unitary divisor of k.

			6 is a term since the arithmetic mean of its bi-unitary divisors, {1, 2, 3, 6}, is 3, and 3 is also a bi-unitary divisor of 6.
60 is a term since the arithmetic mean of its bi-unitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, is 15, and 15 is also a bi-unitary divisor of 60.

Amiram Eldar, Table of n, a(n) for n = 1..236

Subsequence of A286325 and A361786.
Similar sequence: A007340, A353039, A361387.
Cf. A188999, A222266, A286324.

biudivQ[f_, d_] := AllTrue[f, OddQ[Last[#]] || IntegerExponent[d, First[#]] != Last[#]/2 &]; biuDivs[n_] := Module[{d = Divisors[n], f = FactorInteger[n]}, Select[d, biudivQ[f, #] &]]; Select[Range[10^5], IntegerQ[(r = Mean[(i = biuDivs[#])])] && MemberQ[i, r] &]

isbdiv(f, d) = {for (i=1, #f~, if(f[i, 2]%2 == 0 && valuation(d, f[i, 1]) == f[i, 2]/2, return(0))); 1;}
is(n) = {my(f = factor(n), r, p, e); r = prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, (p^(e+1)-1)/((e + 1)*(p-1)), ((p^(e+1)-1)/(p-1)-p^(e/2))/e)); denominator(r) == 1 && n%r==0 && isbdiv(f, r); }

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A319072 a(n) is the sum of the non-bi-unitary divisors of n.

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A335938 Bi-unitary pseudoperfect numbers (A292985) that are not exponentially odd numbers (A268335).

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A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

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A361786 Bi-unitary arithmetic numbers: numbers for which the arithmetic mean of the bi-unitary divisors is an integer.

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A362854 The sum of the divisors of n that are both bi-unitary and exponential.

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A363334 a(n) is the sum of divisors of n that are both coreful and bi-unitary.

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A335939 Bi-unitary weird numbers (A292986) that are not exponentially odd numbers (A268335).

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A361787 Bi-unitary arithmetic numbers k whose mean bi-unitary divisor is a bi-unitary divisor of k.

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