A391909 Numbers k such that -1 <= A195017(k) <= 1.

1, 2, 3, 5, 6, 7, 11, 12, 13, 14, 15, 17, 18, 19, 23, 26, 28, 29, 30, 31, 33, 35, 36, 37, 38, 41, 42, 43, 45, 47, 51, 52, 53, 58, 59, 61, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 83, 84, 86, 89, 90, 93, 95, 97, 98, 99, 101, 102, 103, 105, 106, 107, 108, 109, 113, 114, 116, 119, 122, 123, 127, 130

Offset

1,2

Comments

These are the numbers such that if we count their odd-indexed prime factors and even-indexed prime factors separately (and count repeated factors) the 2 totals (see A257991 and A257992) differ by no more than 1. In other words, each multiset of prime factors is partitioned as evenly as possible by the parity of index of the primes.

Notably, we can view the terms as representing polynomials (encoded using prime factors as described in A206284). So considered, the set of terms represents a submonoid of N[x] under the operation A297845(.,.), which represents multiplication. Stern polynomials (see A260443) are a subset of the polynomials in this submonoid.

Precisely, the polynomials in the submonoid are those where the sums of the coefficients of odd, respectively even, powers of x differ by no more than 1. If we further restrict the sums to be equal, the submonoid we get is an ideal of N[x], the ideal generated by x+1. This ideal is represented likewise by A325698.

The sequence includes the products of any number of consecutive primes (A073485). A perfect power, m^k, k >= 2, is present if and only if m is in A325698.

Presumably, this sequence has zero asymptotic density.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Wikipedia, Submonoids

Index entries for sequences related to prime indices in the factorization of n.

Formula

{a(n)} = {k >= 1 : |A257991(k) - A257992(k)| <= 1}.

{a(n)} = A325698 U {A325698(i) * A000040(j) : i, j >= 1}.

A391746(a(n)) = n.

Example

10 = 2 * 5. 2 is the 1st prime and 5 is the 3rd prime, so 10 has a total of 2 odd-indexed prime factors but 0 even-indexed prime factors; these totals differ by more than 1, so 10 is not in the sequence. 10 represents the polynomial x^2 + 1.
90 = 2 * 3 * 3 * 5. 2 is the 1st prime, 3 is the 2nd prime and 5 is the 3rd prime, so 90 has a total of 2 odd-indexed prime factors and also 2 even-indexed prime factors (counting repetitions); these totals do not differ by more than 1, so 90 is in the sequence. 90 represents the polynomial x^2 + 2x + 1.

Prog

(PARI)
A195017(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * (-1)^(1+primepi(f[i, 1])))); };
A391747(n) = (abs(A195017(n)) <= 1);
is_A391909 = A391747;

Crossrefs

Cf. A195017, A206284, A257991, A257992, A260443, A297845, A391746 (a left inverse), A391747 (characteristic function).

Subsequences (see also those that they list): A000040, A073485, A260442, A325698.

Sequence in context: A059041 A129128 A355647 * A367585 A164922 A205523

Adjacent sequences: A391906 A391907 A391908 * A391910 A391911 A391912

Keyword

nonn,easy

Author

Antti Karttunen and Peter Munn, Dec 23 2025

Status

approved