A138302 -id:A138302 - cp's OEIS frontend

A367170 The number of divisors of the largest unitary divisor of n that is a term of A138302.

1, 2, 2, 3, 2, 4, 2, 1, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 2, 3, 4, 1, 6, 2, 8, 2, 1, 4, 4, 4, 9, 2, 4, 4, 2, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 2, 4, 2, 4, 4, 2, 12, 2, 4, 6, 1, 4, 8, 2, 6, 4, 8, 2, 3, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Nov 07 2023

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

Cf. A000005, A048298, A138302, A367168, A367169, A367171.

Similar sequences: A365401, A365402.

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

Multiplicative with a(p^e) = A048298(e) + 1.
a(n) = A000005(A367168(n)).
a(n) <= A000005(n), with equality if and only if n is in A138302.

A367171 The sum of divisors of the largest unitary divisor of n that is a term of A138302.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 1, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 4, 31, 42, 1, 56, 30, 72, 32, 1, 48, 54, 48, 91, 38, 60, 56, 6, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 3, 72, 8, 80, 90, 60, 168, 62, 96, 104, 1, 84, 144, 68, 126

Offset: 1

Amiram Eldar, Nov 07 2023

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

Cf. A000203, A048298, A138302, A306633, A367168, A367169, A367170.

Similar sequences: A351568, A351569.

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

Multiplicative with a(p^e) = (p^(A048298(e)+1)-1)/(p-1).
a(n) = A000203(A367168(n)).
a(n) <= A000203(n), with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)/zeta(3) = 1.368432... (A306633).

A369933 The maximal exponent in the prime factorization of the exponentially 2^n numbers (A138302).

Original entry on oeis.org

0, 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, 2, 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, Feb 06 2024

Differs from A368473 at n = 1, 32, 89, 126, 159, ... .

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

Cf. A013661, A051903, A271727, A138302, A368473, A369934.

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[AllTrue[e, pow2Q], Max @@ e, Nothing]]; f[1] = 0; Array[f, 150]

ispow2(n) = n >> valuation(n, 2) == 1;
lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = factor(k)[, 2]; if(ispow2(vecprod(e)), print1(vecmax(e), ", "))); }

a(n) = A051903(A138302(n)).
a(n) = 2^A369934(n), for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/zeta(2) + Sum_{k>=1} (2^k * (d(k) - d(k-1)))) / A271727 = 1.40540547368932408503..., where d(k) = Product_{p prime} (1 - 1/p^3 + Sum_{i=2..k} (1/p^(2^i)-1/p^(2^i+1))) for k >= 1, and d(0) = 1/zeta(2).

A370077 The product of exponents of the prime factorization of the largest unitary divisor of n that is a term of A138302.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 09 2024

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

Cf. A005361, A006519, A138302, A268335, A367168, A370078.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

ispow2(n) = n >> valuation(n, 2) == 1;
a(n) = vecprod(apply(x -> if(ispow2(x), x, 1), factor(n)[, 2]));

a(n) = A005361(A367168(n)).
a(n) = A006519(A005361(n)).
a(n) = 2^A370078(n).
a(n) = 1 if and only if n is an exponentially odd number (A268335).
a(n) <= A005361(n), with equality if and only if n is an exponentially 2^n-number (A138302).
Multiplicative with a(p^e) = e if e is a power of 2, and 1 otherwise.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (2^k-1)*(1/p^(2^k) - 1/p^(2^k+1))) = 1.47219167074464124662... .

A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A367515 at n = 128.

The sum of these divisors is A385043(n), and the largest of them is A367168(n).

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

The unitary analog of A353898.
Cf. A005117, A034444, A138302, A209229, A367168, A367515, A385043.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), this sequence (exponentially 2^n), A385044 (5-rough).

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

PARI
```
a(n) = vecprod(apply(x -> (x == 1<
				
```

Multiplicative with a(p^e) = A209229(e) + 1.
a(n) <= A034444(n), with equality if and only if n is in A138302.
a(n) <= A353898(n), with equality if and only if n is squarefree (A005117).

A365297 a(n) is the smallest number k such that k*n is a number whose prime factorization exponents are all powers of 2 (A138302).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 31 2023

First differs from A270419 at n = 128.

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

Cf. A138302, A270419, A356194.

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

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

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e)) - e).
a(n) = A356194(n)/n.
a(n) = 1 if and only if n is in A138302.

A376471 Lexicographically earliest strictly increasing sequence of numbers whose partial products are all exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 11, 13, 17, 19, 20, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 208, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Amiram Eldar, Sep 24 2024

nonn

All the primes are terms.

			1 * 2 = 2^1 and 1 = 2^0.
1 * 2 * 3 = 6 = 2^1 * 3^1 and 1 = 2^0.
1 * 2 * 3 * 5 * 6 = 180 = 2^2 * 3^2 * 5^1, 1 = 2^0 and 2 = 2^1.

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

Disjoint union of A000040 and A376472.
Similar sequences:
  Sequence      | Partial products are in | Exponents are in
  --------------+-------------------------+------------------------
  A050376       | A037992                 | A000225 \ {0} (2^n-1)
  A089237       | A268335                 | A005408 (odd numbers)
  {1} U A246551 | A246551                 | A000290 \ {0} (squares)
  this sequence | A138302                 | A000079 (powers of 2)

expPow2Q[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 2^IntegerExponent[#, 2] &]; a[1] = 1; a[n_] := a[n] = Module[{prod = Times @@ Array[a, n - 1], k = a[n - 1] + 1}, While[! expPow2Q[prod*k], k++]; k]; Array[a, 100]

ispow2(n) = if(n == 0, 1, n >> valuation(n, 2) == 1);
lista(pindmax) = {my(pmax = prime(pindmax), v = vector(pindmax), f, pind, prd); print1(1, ", "); for(k = 2, pmax, f = factor(k); pind = apply(x -> primepi(x), f[,1]); for(i = 1, #pind, v[pind[i]] += f[i, 2]); if(vecprod(apply(x -> ispow2(x), v)) > 0, print1(k, ", "), for(i = 1, #pind, v[pind[i]] -= f[i, 2])));}

A386537 Exponent of the highest power of 2 dividing the n-th number whose prime factorization exponents are all powers of 2 (A138302).

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1

Offset: 1

Amiram Eldar, Jul 25 2025

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

Cf. A007404, A007814, A138302.

Similar sequences: A383004, A383005, A383006, A383007, A383029, A383032, A386536.

exp2nQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 2^IntegerExponent[#, 2] &];
IntegerExponent[Select[Range[200], exp2nQ], 2]

isexp2n(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] >> valuation(f[i, 2], 2) > 1, return (0))); 1;}
list(lim) = for(k = 1, lim, if(isexp2n(k), print1(valuation(k, 2), ", ")));

a(n) = A007814(A138302(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1 + Sum_{k>=0} (2^k + 1)/2^(2^k)) / (1 + Sum_{k>=0} 1/2^(2^k)) - 1 = 0.70550483007968767769... .

A368540 The smallest unitary divisor d of n such that n/d is a term of A138302.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 125, 1, 1

Offset: 1

Amiram Eldar, Dec 29 2023

First differs from A368167 at n = 64 and from A367513 at n = 128.

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

Cf. A048298, A077610, A138302, A367168, A367513, A368167.

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

from math import prod
from sympy import factorint
def A368540(n): return prod(p**e for p, e in factorint(n).items() if not e or (e&-e)^e) # Chai Wah Wu, Dec 30 2023

a(n) = n / A367168(n).
Multiplicative with a(p^e) = p^(e-A048298(e)).
a(n) >= 1, with equality if and only if n is in A138302.

A268335 Exponentially odd numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Vladimir Shevelev, Feb 01 2016

nonn

The sequence is formed by 1 and the numbers whose prime power factorization contains only odd exponents.
The density of the sequence is the constant given by A065463.
Except for the first term the same as A002035. - R. J. Mathar, Feb 07 2016
Also numbers k all of whose divisors are bi-unitary divisors (i.e., A286324(k) = A000005(k)). - Amiram Eldar, Dec 19 2018
The term "exponentially odd integers" was apparently coined by Cohen (1960). These numbers were also called "unitarily 2-free", or "2-skew", by Cohen (1961). - Amiram Eldar, Jan 22 2024

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.
Eckford Cohen, Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, Distribution of unitarily k-free integers, Journal of the Australian Mathematical Society, Vol. 20 , No. 2 (1975), pp. 129-141.

Cf. A002035, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A124010.

Select[Range@ 100, AllTrue[Last /@ FactorInteger@ #, OddQ] &] (* Version 10, or *)
Select[Range@ 100, Times @@ Boole[OddQ /@ Last /@ FactorInteger@ #] == 1 &] (* Michael De Vlieger, Feb 02 2016 *)

isok(n)=my(f = factor(n)); for (k=1, #f~, if (!(f[k,2] % 2), return (0))); 1; \\ Michel Marcus, Feb 02 2016

from itertools import count, islice
from sympy import factorint
def A268335_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(e&1 for e in factorint(n).values()),count(max(startvalue,1)))
A268335_list = list(islice(A268335_gen(),20)) # Chai Wah Wu, Jun 22 2023

Sum_{a(n)<=x} 1 = C*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c = 4*sqrt(2.4/log 2) = 7.44308... and C = Product_{prime p} (1 - 1/p*(p + 1)) = 0.7044422009991... (A065463).

Sum_{n>=1} 1/a(n)^s = zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)), s>1. - Amiram Eldar, Sep 26 2023

cp's OEIS Frontend

A367170 The number of divisors of the largest unitary divisor of n that is a term of A138302.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A367171 The sum of divisors of the largest unitary divisor of n that is a term of A138302.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A369933 The maximal exponent in the prime factorization of the exponentially 2^n numbers (A138302).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A370077 The product of exponents of the prime factorization of the largest unitary divisor of n that is a term of A138302.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A365297 a(n) is the smallest number k such that k*n is a number whose prime factorization exponents are all powers of 2 (A138302).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A376471 Lexicographically earliest strictly increasing sequence of numbers whose partial products are all exponentially 2^n-numbers (A138302).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A386537 Exponent of the highest power of 2 dividing the n-th number whose prime factorization exponents are all powers of 2 (A138302).

Views

Author

Keywords

Links