A357014 -id:A357014 - cp's OEIS frontend

A357017 Decimal expansion of the asymptotic density of odd numbers whose exponents in their prime factorization are squares.

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

Offset: 0

Amiram Eldar, Sep 09 2022

Equivalently, the asymptotic density of numbers whose sum of their exponential divisors (A051377) is odd (A357014).

			0.40979744671331970751092295652844049998230163939067...

Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015-2016.

Cf. A000290, A010052, A051377, A197680, A357014, A357016.

$MaxExtraPrecision = m = 1000; em = 100; f[x_] := Log[1 + Sum[x^(e^2), {e, 2, em}] - Sum[x^(e^2 + 1), {e, 1, em}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[(1/2) * Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]

Equals (1/2) * Product_{p odd prime} (1 + Sum_{k>=2} (c(k)-c(k-1))/p^k), where c(k) is the characteristic function of the squares (A010052).

A376218 Odd exponentially odd numbers.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 149

Offset: 1

Amiram Eldar, Sep 16 2024

First differs from its subsequence A182318 at n = 8318: a(8318) = 19683 = 3^9 = 3^(3^2) is not a term of A182318.
Numbers whose prime factorization contains only odd primes and odd exponents.
Numbers whose sum of coreful divisors (A057723) is odd (a coreful divisor d of a number k is a divisor that is divisible by every prime that divides k, see also A307958).
The even exponentially odd numbers are numbers of the form 2^k * m, where k is odd and m is a term of this sequence.
The asymptotic density of this sequence is (3/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (3/5) * A065463 = 0.42266532... .

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

Intersection of A005408 and A268335.
Cf. A057723, A065463, A182318, A307958.
Other numbers with an odd sum of divisors: A000079 (unitary divisors), A028982 (all divisors), A069562 (non-unitary divisors), A357014 (exponential divisors).

Select[Range[1, 150, 2], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

is(k) = k % 2 && vecprod(factor(k)[,2]) % 2;

A357015 Nonsquarefree numbers whose sum of exponential divisors (A051377) is odd.

Original entry on oeis.org

81, 405, 567, 625, 891, 1053, 1377, 1539, 1863, 1875, 2349, 2401, 2511, 2835, 2997, 3321, 3483, 3807, 4293, 4375, 4455, 4779, 4941, 5265, 5427, 5751, 5913, 6237, 6399, 6723, 6875, 6885, 7203, 7209, 7371, 7695, 7857, 8125, 8181, 8343, 8667, 8829, 9153, 9315, 9639

Offset: 1

Amiram Eldar, Sep 09 2022

nonn

The squarefree numbers are excluded from this sequence since the sum of the exponential divisors of any squarefree number k is A005117(k) = k, so the sum of the exponential divisors of any odd squarefree number (A056911) is odd.
Equivalently, odd nonsquarefree numbers whose exponents in their prime factorization are squares.
The asymptotic density of this sequence is A357017 - 4/Pi^2 = 0.0045127121... .

			81 = 3^4 is a term since it is not squarefree and A051377(81) = 93 is odd.

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

Intersection of A013929 and A357014.

Cf. A005117, A051377, A056911, A185199.

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^4], ! SquareFreeQ[#] && OddQ[esigma[#]] &]

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A357017 Decimal expansion of the asymptotic density of odd numbers whose exponents in their prime factorization are squares.

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A376218 Odd exponentially odd numbers.

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A357015 Nonsquarefree numbers whose sum of exponential divisors (A051377) is odd.

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