A353897 -id:A353897 - cp's OEIS frontend

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

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).

A356194 a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 25, 26, 81, 28, 29, 30, 31, 256, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 162, 55, 112, 57, 58, 59, 60, 61, 62, 63, 256, 65, 66, 67

Offset: 1

Amiram Eldar, Jul 29 2022

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

Cf. A138302, A353897.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193.

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

s(n) = {my(e=logint(n,2)); if(n == 2^e, n, 2^(e+1))};
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))).

a(n) = n iff n is in A138302.

A368781 The maximal exponent in the unique factorization of n in terms of distinct "Fermi-Dirac primes".

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 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, Jan 05 2024

First differs from A335428 at n = 36. Differs from A050377, A344417 and A347437 at n = 1 and then at n = 36.

In the unique factorization of n in terms of distinct "Fermi-Dirac primes", n is represented as a product of prime powers (A246655) whose exponents are powers of 2 (A000079). a(n) is the maximal exponent of these prime powers (and not the maximal exponent of the exponents that are powers of 2). Thus, a(n) is a power of 2 for n >= 2.

			For n = 972 = 2^2 * 3^5, the unique factorization of 972 in terms of distinct "Fermi-Dirac primes" is 2^(2^1) * 3^(2^0) * 3^(2^2). Therefore, a(972) = 2^2 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Fermi-Dirac prime.

Cf. A000079, A050376, A051903, A053644, A182979, A246655, A299090, A353897.

Cf. A050377, A335428, A344417, A347437.

a[n_] := 2^Floor[Log2[Max[FactorInteger[n][[;; , 2]]]]]; a[1] = 0; Array[a, 100]

a(n) = if(n > 1, 2^exponent(vecmax(factor(n)[, 2])), 0);

from sympy import factorint
def A368781(n): return 1<1 else 0 # Chai Wah Wu, Apr 11 2025

a(n) = A053644(A051903(n)).
a(n) = 2^(A299090(n)-1) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} (2^(k-1) * (1 - 1/zeta(2^k))) = 1.56056154773294953123... .
a(n) = A051903(A353897(n)). - Amiram Eldar, May 07 2024

A366906 The largest exponentially evil divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 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, 8, 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, 8, 27, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Oct 27 2023

The largest divisor of n that is an exponentially evil number (A262675).

The number of exponentially evil divisors of n is A366902(n) and their sum is A366904(n).

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

Cf. A004709, A262675, A366902, A366904.

Similar sequences: A353897, A365683, A366905.

maxEvil[e_] := Module[{k = e}, While[OddQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^maxEvil[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = {my(k = n); while(hammingweight(k)%2, k--); k;}
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}

Multiplicative with a(p^e) = p^max{k=1..e, k evil}.
a(n) <= n, with equality if and only if n is exponentially evil number (A262675).
a(n) >= 1, with equality if and only if n is a cubefree number (A004709).

A366905 The largest exponentially odious divisor of n.

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, Oct 27 2023

First differs from A353897 at n = 128.
The largest divisor of n that is an exponentially odious number (A270428).
The number of exponentially odious divisors of n is A366901(n) and their sum is A366903(n).

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

Cf. A270428, A366901,A366903.

Similar sequences: A353897, A365683, A366906.

maxOdious[e_] := Module[{k = e}, While[EvenQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^maxOdious[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = {my(k = n); while(!(hammingweight(k)%2), k--); k;}
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}

Multiplicative with a(p^e) = p^max{k=1..e, k odious}.
a(n) <= n, with equality if and only if n is exponentially odious number (A270428).
Sum_{k=1..n} a(k) ~ c*n^2, where c = (1/2) * Product_{p prime} (1 + Sum_{e>=1} (p^f(e) - p^(f(e-1)+1))/p^(2*e)) = 0.4636829525..., f(e) = max{k=1..e, k odious} for e >= 1, and f(0) = 0.

A372379 The largest divisor of n whose number of divisors is a power of 2.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 8, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 8, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 24, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70

Offset: 1

Amiram Eldar, Apr 29 2024

First differs from A350390 at n = 32.
The largest term of A036537 dividing n.
The largest divisor of n whose exponents in its prime factorization are all of the form 2^k-1 (A000225).

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

Cf. A000225, A000523, A036537, A138302, A162643, A282940, A350390, A353897, A372380, A372381.

f[p_, e_] := p^(2^Floor[Log2[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~, f[i, 1]^(2^exponent(f[i, 2]+1)-1));}

from math import prod
from sympy import factorint
def A372379(n): return prod(p**((1<<(e+1).bit_length()-1)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Apr 30 2024

Multiplicative with a(p^e) = p^(2^floor(log_2(e+1)) - 1).
a(n) = n if and only if n is in A036537.
a(A162643(n)) = A282940(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.7907361848... = 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))-1 for k >= 1, and f(0) = 0.

A369760 The number of divisors of the smallest multiple of n whose prime factorization exponents are all powers of 2.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 9, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 9, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12

Offset: 1

Amiram Eldar, Jan 31 2024

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

Cf. A000005, A138302, A353897, A356194, A369761, A369762.

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

s(n) = {my(e=logint(n, 2)); if(n == 2^e, n, 2^(e+1))};
a(n) = vecprod(apply(x -> s(x) + 1, factor(n)[, 2]));

a(n) = A000005(A356194(n)).
Multiplicative with a(p^e) = 2^ceiling(log_2(e)) + 1.
a(n) >= A000005(n), with equality if and only if n is in A138302.

A369761 The sum of divisors of the smallest multiple of n whose prime factorization exponents are all powers of 2.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 31, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 124, 31, 42, 121, 56, 30, 72, 32, 511, 48, 54, 48, 91, 38, 60, 56, 186, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 363, 72, 248, 80, 90, 60, 168, 62, 96, 104, 511, 84

Offset: 1

Amiram Eldar, Jan 31 2024

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

Cf. A000203, A138302, A353897, A356194, A369760, A369762.

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

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

a(n) = A000203(A356194(n)).
Multiplicative with a(p^e) = (p^(2^ceiling(log_2(e))+1)-1)/(p-1).
a(n) >= A000203(n), with equality if and only if n is in A138302.

A369762 The sum of unitary divisors of the smallest multiple of n whose prime factorization exponents are all powers of 2.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 17, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 68, 26, 42, 82, 40, 30, 72, 32, 257, 48, 54, 48, 50, 38, 60, 56, 102, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 246, 72, 136, 80, 90, 60, 120, 62, 96, 80, 257, 84, 144

Offset: 1

Amiram Eldar, Jan 31 2024

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

Cf. A034448, A138302, A353897, A356194, A369760, A369761.

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

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

a(n) = A034448(A356194(n)).
Multiplicative with a(p^e) = p^(2^ceiling(log_2(e))) + 1.
a(n) >= A034448(n), with equality if and only if n is in A138302.

A369890 The number of divisors of the largest divisor of n whose exponents in its prime factorization are all powers of 2.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 15 2024

First differs from A369015 at n = 32.

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

Cf. A000005, A000079, A053644, A138302, A353897, A369015.

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

a(n) = vecprod(apply(x -> 2^logint(x, 2) + 1, factor(n)[, 2]));

a(n) = A000005(A353897(n)).
Multiplicative with a(p^e) = A053644(e) + 1.
a(n) = 2 if and only if n is prime.
a(n) <= A000005(n), with equality if and only if n is in A138302.

cp's OEIS Frontend

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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A356194 a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.

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A368781 The maximal exponent in the unique factorization of n in terms of distinct "Fermi-Dirac primes".

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A366906 The largest exponentially evil divisor of n.

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A366905 The largest exponentially odious divisor of n.

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A372379 The largest divisor of n whose number of divisors is a power of 2.

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A369760 The number of divisors of the smallest multiple of n whose prime factorization exponents are all powers of 2.

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A369761 The sum of divisors of the smallest multiple of n whose prime factorization exponents are all powers of 2.