A348506 -id:A348506 - cp's OEIS frontend

A348505 a(n) = usigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

1, 1, 1, 5, 1, 1, 1, 3, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 3, 26, 1, 7, 5, 1, 1, 1, 11, 1, 1, 1, 50, 1, 1, 1, 3, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 7, 1, 3, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 6, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1, 1, 5, 1, 1, 1, 3, 1, 10, 1, 5, 1, 1, 1, 11, 1, 50, 10, 130

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 6 != 3*10 = a(8) * a(9).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A005117, A034448, A048146, A063880, A348503, A348504, A348506 (positions of ones).

Cf. also A344697, A348049.

f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := (usigma = Times @@ f1 @@@ (fct = FactorInteger[n])) / GCD[usigma, Times @@ f2 @@@ fct]; Array[a, 100] (* Amiram Eldar, Oct 29 2021 *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348505(n) = { my(u=A034448(n)); (u/gcd(u, sigma(n))); };

a(n) = A034448(n) / A348503(n) = A034448(n) / gcd(A000203(n), A034448(n)).

A367801 Numbers that are both exponentially odd (A268335) and exponentially odious (A270428).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106

Offset: 1

Amiram Eldar, Dec 01 2023

First differs from its subsequence A005117 at n = 79: a(79) = 128 is not a squarefree number.
First differs from A077377 at n = 63, and from A348506 at n = 68.
Numbers whose prime factorization contains only exponents that are odd odious numbers (A092246).
The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.61156148494581943994..., where f(x) = (1-x) * (1 + x/(2*(1-x^2)) + (Product_{k>=0} (1-(-x)^(2^k)) - Product_{k>=0} (1-x^(2^k))))/2.

Amiram Eldar, Table of n, a(n) for n = 1..12232 (terms below 20000)
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

Intersection of A268335 and A270428.
Cf. A367802, A367803, A367804.
Subsequences: A005117, A092759.
Cf. A092246.
Cf. A077377, A348506.

odQ[n_] := OddQ[n] && OddQ[DigitCount[n, 2, 1]]; Select[Range[150], AllTrue[FactorInteger[#][[;;, 2]], odQ] &]

is(n) = {my(f = factor(n)); for (i = 1, #f~, if(!(f[i, 2]%2 && hammingweight(f[i, 2])%2), return (0))); 1;}

cp's OEIS Frontend

A348505 a(n) = usigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

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