A138302 -id:A138302 - cp's OEIS frontend

A367168 The largest unitary divisor of n that is a term of A138302.

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Nov 07 2023

First differs from A056192 at n = 32 and from A270418 at n = 128.

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

Cf. A048298, A065465, A138302, A367169, A367170, A367171.
Cf. A056192, A270418.
Similar sequences: A350388, A350389, A360539.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], p^e, 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] == 1 << valuation(f[i, 2], 2), f[i, 1]^f[i, 2], 1));}

from math import prod
from sympy import factorint
def A367168(n): return prod(p**e for p,e in factorint(n).items() if not(e&-e)^e) # Chai Wah Wu, Nov 10 2023

Multiplicative with a(p^e) = p^A048298(e).
a(n) <= n, with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.881513... (A065465).

A353897 a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 16, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 16, 65, 66, 67, 68

Offset: 1

Amiram Eldar, May 10 2022

			a(27) = 9 since 9 = 3^2 is the largest divisor of 27 with an exponent in its prime factorization, 2, that is a power of 2.

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

Cf. A138302, A353898.

Similar sequences: A000265, A007947, A008834, A055071, A350390.

f[p_, e_] := p^(2^Floor[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Multiplicative with a(p^e) = p^(2^floor(log_2(e))).
a(n) = n if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.4616988732... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = 2^floor(log_2(k)) and f(0) = 0.

A271727 Decimal expansion of the density of exponentially 2^n-numbers (A138302).

Original entry on oeis.org

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

Offset: 0

Juan Arias-de-Reyna, Apr 13 2016

			0.87249717935391281355800771433253186691958393977733373765412...

Juan Arias-de-Reyna, Table of n, a(n) for n = 0..1000
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015-2016.
Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.

Density of A138302.

Cf. A271726 (Expansion of log(f(x))).

$MaxExtraPrecision = m = 500; em = 10; f[x_] := Log[1 - x^3 + Sum[x^(2^e) - x^(1 + 2^e), {e, 2, em}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]] (* Amiram Eldar, Sep 09 2022 *)

Equals Product_{prime p} f(1/p), where f(x) = 1-x^3+Sum_{n>=2}(x^(2^n)-x^(1+2^n)).

A353898 a(n) is the number of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, May 10 2022

First differs from A049599 and A282446 at n=32.

			The divisors of 8 are 1, 2 = 2^1, 4 = 2^2 and 8 = 2^3. 3 of these divisors, 1, 2 and 4, are in A138302. Therefore, a(8) = 3.

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

Cf. A000005, A004709, A049599, A138302, A282446, A353897.

Similar sequences: A034444, A048105, A286324, A049419.

f[p_, e_] := Floor[Log2[e]] + 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Multiplicative with a(p^e) = floor(log_2(e)) + 2.
a(n) > 1 for n > 1 and a(n) = 2 if and only if n is a prime.
a(n) = A000005(n) if and only if n is cubefree (A004709).

A353900 a(n) is the sum of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 23, 18, 39, 20, 42, 32, 36, 24, 28, 31, 42, 13, 56, 30, 72, 32, 23, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 39, 72, 56, 80, 90, 60, 168, 62, 96, 104, 23, 84, 144

Offset: 1

Amiram Eldar, May 10 2022

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

Cf. A000203, A004709, A138302, A353898.

Similar sequences: A034448, A048146, A051377, A188999.

f[p_, e_] := 1 + Sum[p^(2^k), {k, 0, Floor[Log2[e]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 0, logint(f[i,2], 2), f[i,1]^(2^k)));} \\ Amiram Eldar, Nov 19 2022

Multiplicative with a(p^e) = 1 + Sum_{k=0..floor(log_2(e))} p^(2^k).
a(n) = A000203(n) if and only if n is cubefree (A004709).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1-1/p)*(1 + Sum_{k>=1} (Sum_{j=0..floor(log_2(k))} p^(2^j)/p^(2*k)))) = 0.7176001667... . - Amiram Eldar, Nov 19 2022

A368473 Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2

Offset: 1

Amiram Eldar, Dec 26 2023

The terms of A005361 that are powers of 2 (A000079).

The first position of 2^k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.

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

Cf. A000079, A005361, A085629, A138302, A271727.
Cf. A366538, A367169.
Similar sequences: A322327, A368472, A368474.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[p == 2^IntegerExponent[p, 2], p, Nothing]]; Array[f, 150]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p >> valuation(p,2) == 1, print1(p, ", ")));}

a(n) = A005361(A138302(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=0} 2^k/p^(2^k)) = 1.504710204899266020302..., where d = A271727 is the asymptotic density of the exponentially 2^n-numbers.

A385043 The sum of the unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A385042(n), and the largest of them is A367168(n).

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

The unitary analog of A353900.
Cf. A005117, A034448, A138302, A209229, A385042.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), this sequence (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], p^e + 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] == 1<

Multiplicative with a(p^e) = p^(A209229(e)) + 1.
a(n) <= A034448(n), with equality if and only if n is in A138302.
a(n) <= A353900(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1/(p*(p+1)) + Sum_{k>=2} (1/p^(2^k)-1/p^(2^k-1))) = 1.21427559551509410114... .

A385139 The sum of divisors d of n such that n/d has exponents in its prime factorization that are all powers of 2 (A138302).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 14, 13, 18, 12, 28, 14, 24, 24, 29, 18, 39, 20, 42, 32, 36, 24, 56, 31, 42, 39, 56, 30, 72, 32, 58, 48, 54, 48, 91, 38, 60, 56, 84, 42, 96, 44, 84, 78, 72, 48, 116, 57, 93, 72, 98, 54, 117, 72, 112, 80, 90, 60, 168, 62, 96, 104, 116, 84, 144

Offset: 1

Amiram Eldar, Jun 19 2025

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

Cf. A004709, A138302, A353900.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), this sequence (exponentially 2^n).

f[p_, e_] := p^e + Sum[p^(e - 2^k), {k, 0, Floor[Log2[e]]}]; 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] + sum(k = 0, logint(f[i, 2], 2), f[i, 1]^(f[i, 2]-2^k)));}

Multiplicative with a(p^e) = p^e + Sum_{k=0..floor(log_2(e))} p^(e-2^k).		
a(n) <= A000203(n), with equality if and only if n is cubefree (A004709).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + (1-1/p)*(Sum_{k>=1} (Sum_{j=0..floor(log_2(k))} 1/p^(k+2^j)))) = 1.62194750148969761827... .

A366538 The number of unitary divisors of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 4, 4, 2, 8, 2, 4, 4, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 8, 2, 4, 4, 8, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4, 2, 8, 4, 4, 4, 4, 4, 2, 4

Offset: 1

Amiram Eldar, Oct 12 2023

Also, the number of infinitary divisors of the terms of A138302, since A138302 is also the sequence of numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide.

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

Cf. A034444, A037445, A077609, A077610, A138302, A366539.

Similar sequences: A366534, A366536.

f[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, If[AllTrue[e, # == 2^IntegerExponent[#, 2] &], 2^Length[e], Nothing]]; f[1] = 1; Array[f, 150]

lista(max) = for(k = 1, max, my(e = factor(k)[, 2], is = 1); for(i = 1, #e, if(e[i] >> valuation(e[i], 2) > 1, is = 0; break)); if(is, print1(2^#e, ", ")));

a(n) = A034444(A138302(n)).

a(n) = A037445(A138302(n)).

A366539 The sum of unitary divisors of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 26, 42, 40, 30, 72, 32, 48, 54, 48, 50, 38, 60, 56, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 72, 80, 90, 60, 120, 62, 96, 80, 84, 144, 68, 90, 96, 144, 72, 74, 114, 104

Offset: 1

Amiram Eldar, Oct 12 2023

Also the sum of infinitary divisors of the terms of A138302, since A138302 is also the sequence of numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide.

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

Cf. A034448, A049417, A077609, A077610, A138302, A271727, A366538.

Similar sequences: A034676, A366535, A366537.

s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, # == 2^IntegerExponent[#, 2] &], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], is = 1); for(i = 1, #e, if(e[i] >> valuation(e[i], 2) > 1, is = 0; break)); if(is, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));

a(n) = A034448(A138302(n)).
a(n) = A049417(A138302(n)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1/d^2) * Product_{p prime} f(1/p) = 1.58107339851877782285..., d = A271727 is the asymptotic density of A138302, and f(x) = 1 + x^2 + 2 * Sum_{k>=2} (x^(2^k)-x^(2^k+1)).
The asymptotic mean of the unitary abundancy index of A138302: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A138302(k) = c * d = 1.37948208055913856387... .

cp's OEIS Frontend

A367168 The largest unitary divisor of n that is a term of A138302.

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A353897 a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).

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A271727 Decimal expansion of the density of exponentially 2^n-numbers (A138302).

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A353898 a(n) is the number of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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A353900 a(n) is the sum of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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A368473 Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).

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A385043 The sum of the unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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A385139 The sum of divisors d of n such that n/d has exponents in its prime factorization that are all powers of 2 (A138302).

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