A375669 -id:A375669 - cp's OEIS frontend

A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Gus Wiseman, Mar 02 2025

nonn

First differs from A335740 in lacking 72, which has prime indices {1,1,1,2,2} and section-sum partition (3,3,1).
Also numbers whose section-sum partition of prime indices (A381436) ends with a number > 1.
Includes all squarefree numbers (A005117) except 2.

			The terms together with their prime indices begin:
     3: {2}        25: {3,3}        45: {2,2,3}
     5: {3}        26: {1,6}        46: {1,9}
     6: {1,2}      27: {2,2,2}      47: {15}
     7: {4}        29: {10}         49: {4,4}
     9: {2,2}      30: {1,2,3}      50: {1,3,3}
    10: {1,3}      31: {11}         51: {2,7}
    11: {5}        33: {2,5}        53: {16}
    13: {6}        34: {1,7}        54: {1,2,2,2}
    14: {1,4}      35: {3,4}        55: {3,5}
    15: {2,3}      36: {1,1,2,2}    57: {2,8}
    17: {7}        37: {12}         58: {1,10}
    18: {1,2,2}    38: {1,8}        59: {17}
    19: {8}        39: {2,6}        61: {18}
    21: {2,4}      41: {13}         62: {1,11}
    22: {1,5}      42: {1,2,4}      63: {2,2,4}
    23: {9}        43: {14}         65: {3,6}

The LHS (exponent of 2) is A007814.
The complement is A360013 = 2*A360015 (if we prepend 1), counted by A241131 (shifted right and starting with 1 instead of 0).
The case of equality is A360014, inclusive A360015.
The RHS (greatest exponent of an odd prime factor) is A375669.
These are positions of terms > 1 in A381437.
Partitions of this type are counted by A381544.
A000040 lists the primes, differences A001223.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A381436 gives section-sum partition of prime indices, Heinz number A381431.
A381438 counts partitions by last part part of section-sum partition.
Cf. A000720, A001221, A001222, A001694, A003557, A005117, A066328, A130091, A181819, A212166, A380955.

Select[Range[100],FactorInteger[2*#][[1,2]]-1<=Max@@Last/@Rest[FactorInteger[2*#]]&]

Positive integers n such that A007814(n) <= A375669(n).

A375670 The maximum exponent in the prime factorization of the largest 5-rough divisor of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 23 2024

First differs from A106799 at n = 35.

The largest exponent among the exponents of the primes that are larger than 3 in the prime factorization of n.

Cf. A003586, A007310 (5-rough numbers), A051903, A065330, A106799, A375669.

a[n_] := Module[{m = n / Times@@({2,3}^IntegerExponent[n,{2,3}])}, If[m == 1, 0, Max[FactorInteger[m][[;; , 2]]]]]; Array[a, 100]

a(n) = {my(m = n >> valuation(n, 2)/3^valuation(n, 3)); if(m == 1, 0,vecmax(factor(m)[,2]));}

a(n) = A051903(A065330(n)).
a(n) = 0 if and only if n is a 3-smooth number (A003586).
a(n) = 1 if and only if n is a product of a squarefree 5-rough number larger than 1 and a 3-smooth number.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * d(k) = 1.1034178389191320571029... , where d(k) is the asymptotic density of the occurrences of k in this sequence: d(1) = 3/(2*zeta(2)), and d(k) = (1/zeta(k+1)) / ((1-1/2^(k+1))*(1-1/3^(k+1))) - (1/zeta(k)) / ((1-1/2^k)*(1-1/3^k)) for k >= 2.
In general, the asymptotic mean of the maximum exponent in the prime factorization of the largest p-rough divisor of n is Sum_{k>=1} k * d(k), where d(1) = 1/(zeta(2) * f(p, 2)), d(k) = 1/(zeta(k+1) * f(p, k+1)) - 1/(zeta(k) * f(p, k)) for k >= 2, and f(p, m) = Product_{q prime < p} (1-1/q^m).

A376645 The maximum exponent in the factorization of n into powers of Gaussian primes.

Original entry on oeis.org

0, 2, 1, 4, 1, 2, 1, 6, 2, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 6, 2, 2, 3, 4, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 4, 2, 2, 1, 8, 2, 2, 1, 4, 1, 3, 1, 6, 1, 2, 1, 4, 1, 2, 2, 12, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 2, 4, 1, 2, 1, 8, 4, 2, 1, 4, 1, 2, 1

Offset: 1

Amiram Eldar, Oct 01 2024

a(n) = 0 only for n = 1. a(n) = k occurs infinitely many times for k >= 1. The numbers n = 2^e * m = 2^A007814(n) * A000265(n) for which a(n) = k and their asymptotic density are as follows:
  1. k = 1: n is an odd squarefree number (A056911) and the density is d(1) = 2/(3*zeta(2)) = 0.405284... (A185199).
  2. k >= 3 is odd: e < (k+1)/2 and m is a (k+1)-free number that is not a k-free number: d(k) = (1 - 1/2^((k+1)/2)) * (f(k+1)/zeta(k+1) - f(k)/zeta(k)), where f(k) = 1 - 1/2^k.
  3. k >= 2 is even: e = k/2 and m is a (k+1)-free number, or e < k/2 and m is a (k+1)-free number that is not a k-free number: d(k) = (1/2^(k/2+1)) * f(k+1)/zeta(k+1) + (1-1/2^(k/2)) * (f(k+1)/zeta(k+1) - f(k)/zeta(k)), where f(k) is defined above.
The asymptotic mean of this sequence is Sum_{k>=1} k * d(k) = 2.64836785173193409440576... .

			a(2) = 2 because 2 = -i * (1+i)^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gaussian Prime.
Wikipedia, Gaussian integer: Unique factorization.

Cf. A000265, A051903, A007814, A056911, A078458, A086275, A185199, A335852, A375669.

a[n_] := Max[FactorInteger[n, GaussianIntegers -> True][[;; , 2]]]; a[1] = 0; Array[a, 100]
(* or *)
a[n_] := Module[{e = IntegerExponent[n, 2], od, em}, odd = n / 2^e; Max[2*e, If[odd == 1, 0, Max[FactorInteger[odd][[;;, 2]]]]]]; Array[a, 100]

a(n) = if(n == 1, 0, vecmax(factor(n*I)[, 2]));

a(n) = my(e = valuation(n, 2), es = factor(n >> e)[, 2]); max(2*e, if(#es, vecmax(es), 0));

a(n) = max(2*A007814(n), A051903(A000265(n))) = max(2*A007814(n), A375669(n)).

cp's OEIS Frontend

A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.

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A375670 The maximum exponent in the prime factorization of the largest 5-rough divisor of n.

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A376645 The maximum exponent in the factorization of n into powers of Gaussian primes.

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