A268335 -id:A268335 - cp's OEIS frontend

A367695 Numbers k such that k and k+1 are both exponentially odd numbers (A268335).

1, 2, 5, 6, 7, 10, 13, 14, 21, 22, 23, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 46, 53, 54, 55, 56, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 87, 88, 93, 94, 95, 96, 101, 102, 103, 104, 105, 106, 109, 110, 113, 114, 118, 119, 122, 127, 128

Offset: 1

Amiram Eldar, Nov 27 2023

nonn

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 48, 478, 4734, 47195, 471707, 4716892, 47168363, 471681183, 4716806520, ... . Apparently, the asymptotic density of this sequence exists and equals Product_{p prime} (1 - 2/(p*(p+1))) = 0.47168... (A307868).

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

Subsequence of A268335.
Cf. A307868.
Subsequences: A007674, A325058.
Similar sequences: A071318, A121495, A340152, A367696.

expOddQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; Select[Range[128], And @@ expOddQ /@ {#, # + 1} &]

isexpodd(n) = {my(f = factor(n)); for(i=1, #f~, if (!(f[i, 2] % 2), return (0))); 1;}
is(n) = isexpodd(n) && isexpodd(n+1)

A368977 The number of bi-unitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 3, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 4, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 6, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 3, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 6, 3, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 11 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A005117, A062503, A222266, A268335, A286324, A368978.

Similar sequences: A055076, A322483, A363825, A368979.

f[p_, e_] := If[OddQ[e], (e+3)/2, 2*Floor[e/4]+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, (x+3)/2, 2*(x\4)+1), factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2 + 2*X^3 - X^4)/(1 - X - X^4 + X^5))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024

Multiplicative with a(p^e) = (e+3)/2 if e is odd, and 2*floor(e/4)+1 if e is even.
a(n) >= 1, with equality if and only if n is in A062503.
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
From Vaclav Kotesovec, Jan 11 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*s)))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (1 - p + 2*p^2) / (p*(1 + p)*(1 + p^2))) = 0.5715031234451924252215041182933420817059774181158824297150124265420835...,
f'(1) = f(1) * Sum_{p prime} (4*p^5 - p^4 + 2*p^3 + 2*p + 1) * log(p) / (p^7 + 2*p^6 + p^5 + 3*p^4 + p^3 + p - 1) = f(1) * 1.1422556395248477875508983912036578244050011522937179465478688905880430...
and gamma is the Euler-Mascheroni constant A001620. (End)

A374460 Indices of the nonsquarefree terms in the sequence of exponentially odd numbers (A268335).

Original entry on oeis.org

7, 18, 20, 24, 31, 39, 41, 63, 69, 74, 86, 89, 91, 97, 98, 109, 115, 121, 131, 135, 154, 161, 167, 174, 177, 179, 189, 194, 200, 211, 212, 223, 234, 243, 244, 249, 250, 265, 266, 268, 273, 290, 296, 302, 314, 325, 328, 338, 343, 348, 350, 366, 367, 373, 382, 388, 393

Offset: 1

Amiram Eldar, Jul 09 2024

The asymptotic density of this sequence is 1 - A059956 / A065463 = 0.13700925215474602945... .

			The first 7 exponentially odd numbers are 1, 2, 3, 5, 6, 7, and 8. A268335(7) = 8 = 3^3 is the least nonsquarefree term. Therefore a(1) = 7.

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

Cf. A005117, A059956, A065463, A268335.

Similar sequences: A361936, A363189, A371186, A371188.

Position[Select[Range[120], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &], _?(!SquareFreeQ[#] &), Heads -> False] // Flatten

isexpodd(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] % 2), return(0))); 1;}
lista(kmax) = {my(f, c = 0); for(k = 1, kmax, if(isexpodd(k), c++; if(!issquarefree(k), print1(c, ", "))));}

A268335(a(n)) = A374459(n).

A384559 The sum of the exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 10, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 30, 5, 26, 30, 14, 29, 30, 31, 34, 33, 34, 35, 6, 37, 38, 39, 50, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 60, 55, 70, 57, 58, 59, 30, 61, 62, 21, 10, 65, 66, 67, 34, 69

Offset: 1

Amiram Eldar, Jun 03 2025

First differs from A384558 at n = 512: a(512) = 514, while A384558(512) = 522.

The number of these divisors is A384557(n), and the largest of them is A331737(n).

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

Cf. A005117, A051377, A072587, A268335, A331737, A368979, A374459, A384557, A384558.

f[p_, e_] := DivisorSum[e, p^# &, OddQ[#] && CoprimeQ[#, e/#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, sumdiv(f[i,2], d, (d % 2) * (gcd(d, f[i,2]/d) == 1) * f[i,1]^d));}

Multiplicative with a(p^e) = Sum_{d|e, d odd, gcd(d, e/d) = 1} p^d.
a(n) = n if and only if n is squarefree (A005117).
a(n) < n if and only if n is in A072587.
a(n) > n if and only if n is in A374459.

A334899 Bi-unitary practical numbers (A334898) that are not exponentially odd numbers (A268335).

Original entry on oeis.org

48, 72, 192, 240, 288, 320, 336, 360, 432, 448, 504, 528, 600, 624, 648, 768, 792, 800, 810, 816, 912, 936, 960, 1050, 1104, 1134, 1152, 1176, 1200, 1224, 1280, 1296, 1344, 1350, 1368, 1392, 1400, 1440, 1470, 1488, 1568, 1650, 1656, 1680, 1728, 1776, 1782, 1792

Offset: 1

Amiram Eldar, May 16 2020

nonn

Practical numbers (A005153) that are exponentially odd (A268335) are also bi-unitary practical numbers (A334898), since all of their divisors are bi-unitary.

Of the first 2500 bi-unitary practical numbers, only 847 are in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..850

Cf. A005153, A268335, A287173, A334898, A334902.

biunitaryDivisorQ[div_, n_] := If[Mod[#2, #1] == 0, Last @ Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; bdivs[n_] := Module[{d = Divisors[n]}, Select[d, biunitaryDivisorQ[#, n] &]]; bPracQ[n_] := Module[{d = bdivs[n], sd, x}, sd = Plus @@ d; Min @ CoefficientList[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, sd}], x] >  0]; expOddQ[n_] := AllTrue[Last /@ FactorInteger[n], OddQ]; Select[Range[1000], !expOddQ[#] && bPracQ[#] &]

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[#] &]

A335989 Terms of A301517 that are not exponentially odd numbers (A268335).

Original entry on oeis.org

12500, 18252, 21600, 37500, 50000, 67228, 84500, 87500, 91260, 127764, 137500, 146016, 150000, 151200, 162500, 200000, 200772, 201684, 212500, 231868, 237500, 237600, 253500, 262500, 268912, 274400, 280800, 287500, 310284, 336140, 337500, 346788, 350000, 362500

Offset: 1

Amiram Eldar, Jul 03 2020

nonn

If k = Product p^e, then A162296(k) / A048250(k) = -1 + Product (p^(e+1) - 1)/(p^2 - 1). If k is exponentially odd, then e = 2*m - 1 is odd for all the prime factors p of k and p^(e+1) - 1 = (p^2)^m - 1 is divisible by p^2 - 1. Therefore, A162296(k) / A048250(k) is an integer for all exponentially odd numbers, and it is a positive integer for all the nonsquarefree (A013929) exponentially odd numbers.
It seems that most of the terms of A301517 are exponentially odd numbers. For example, the first 10^4 terms of A301517 include only 9 terms that are not exponentially odd numbers. Up to 10^8 there are 9660732 terms of A301517, and only 9107 of them are not exponentially odd numbers.
The number of terms of this sequence that do not exceed 10^k, for k = 5, 6, ... are 9, 92, 916, 9107, 91172, 911187, .... Apparently, this sequence has an asymptotic density c = 0.000091... If this is true, then the asymptotic density of A301517 is c + A065463 - A059956 = 0.096606... (A065463 is the density of the exponentially odd numbers, and A059956 is the density of the squarefree numbers which are a subset of the exponentially odd numbers).

			12500 = 2^2 * 5^5 is a term since the exponent of its prime factor 2 is 2 which even, and therefore it is not an exponentially odd number, and the sum of its squarefree divisors, A048250(12500) = 18 divides the sum of its nonsquarefree divisors, A162296(12500) = 27324 = 18 * 1518.

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

Cf. A048250, A059956, A065463, A162296, A268335, A301517.

f[p_, e_] := (p^(e + 1) - 1)/(p^2 - 1); Select[Range[2, 4*10^5], Max[Last /@ (fct = FactorInteger[#])] > 1 && ! AllTrue[Last /@ fct, OddQ] && (r =  Times @@ (f @@@ fct)) > 1 && IntegerQ[r] &]

A368470 a(n) is the number of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 4, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Dec 26 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A000290, A005117, A033634, A268335, A350389, A368471.

f[p_, e_] := If[OddQ[e], (e + 3)/2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, (f[i,2]+3)/2, 1));}

a(n) = A033634(A350389(n)).
Multiplicative with a(p^e) = (e+3)/2 if e is odd and 1 otherwise.
a(n) >= 1, with equality if and only if n is a square (A000290).
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(2*s)^2 * Product_{p prime} (1 + 2/p^s - 1/p^(2*s) - 1/p^(3*s)).
From Vaclav Kotesovec, Dec 26 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + p^s/(1 + p^s)^2).
Let f(s) = Product_{p prime} (1 - 1/p^s + p^s/(1 + p^s)^2).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (2*p+1) / (p*(p+1)^2)) = 0.528940778823659679133966695786017426052491935740673837882972347697...,
f'(1) = f(1) * Sum_{p prime} (4*p^2 + 3*p + 1) * log(p) / (p^4 + 3*p^3 + p^2 - 2*p - 1) = f(1) * 1.36109933267802415215189866467122940932493907539386280428818...
and gamma is the Euler-Mascheroni constant A001620. (End)

A368471 a(n) is the sum of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 11, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 44, 1, 42, 31, 8, 30, 72, 32, 43, 48, 54, 48, 1, 38, 60, 56, 66, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 93, 72, 88, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144

Offset: 1

Amiram Eldar, Dec 26 2023

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

Cf. A000203, A000290, A005117, A033634, A268335, A350389, A368470.

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

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

a(n) = A033634(A350389(n)).
Multiplicative with a(p^e) = (p^(e+2) - p)/(p^2 - 1) + 1 if e is odd and 1 otherwise.
a(n) >= 1, with equality if and only if n is a square (A000290).
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 = (Pi^6/1080) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.51287686448947428073... .

A374457 The Dedekind psi function values of the exponentially odd numbers (A268335).

Original entry on oeis.org

1, 3, 4, 6, 12, 8, 12, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 48, 42, 36, 30, 72, 32, 48, 48, 54, 48, 38, 60, 56, 72, 42, 96, 44, 72, 48, 72, 54, 108, 72, 96, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 144, 90

Offset: 1

Amiram Eldar, Jul 09 2024

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

Cf. A001615, A065463, A268335.
Similar sequences related to psi: A000082, A033196, A323332, A371413, A371415.
Similar sequences related to exponentially odd numbers: A366438, A366439, A366534, A366535, A367417, A368711, A374456.

f[p_, e_] := If[OddQ[e], (p+1) * p^(e-1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] % 2, (f[i, 1]+1) * f[i, 1]^(f[i, 2] - 1), 0));}
lista(kmax) = {my(s1); for(k = 1, kmax, s1 = s(k); if(s1 > 0, print1(s1, ", ")));}

a(n) = A001615(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 / A065463^2 = 2.01515877170903249510... .

cp's OEIS Frontend

A367695 Numbers k such that k and k+1 are both exponentially odd numbers (A268335).

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A368977 The number of bi-unitary divisors of n that are exponentially odd numbers (A268335).

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A374460 Indices of the nonsquarefree terms in the sequence of exponentially odd numbers (A268335).

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A384559 The sum of the exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

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A334899 Bi-unitary practical numbers (A334898) that are not exponentially odd numbers (A268335).

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

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A335989 Terms of A301517 that are not exponentially odd numbers (A268335).

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A368470 a(n) is the number of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

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