1,1

Numbers k whose only divisor in A246282 is k itself, i.e., A003961(k) > 2k, but for none of the proper divisors d|k, d<k it holds that A003961(d) > 2d.

Claim 1: The intersection with A005101 is empty, i.e., this is a subsequence of A263837, nonabundant numbers. In other words, any abundant number k (A005101) has A337345(k) > 1 and thus is a term of A341610. This implies the claims made in the sequences A378662, A378664, from which further follows that there are no 1's present in any of these sequences: A378658, A378736, A378740. See the proof given by Jianing Song below.

Claim 2: Furthermore, the intersection with A023196 is equal to {6}. - Added Sep 07 2025

From Antti Karttunen, Dec 06 2024: (Start)

Additional observation: The thirteen initial terms (4, 6, 9, ..., 69, 91) are only semiprimes in A246282, all other semiprimes being in A246281 (but none in A341610), and there seems to be only 678 terms m with A001222(m) = 3, from a(14) = 125 to the last one of them, a(2691) = 519963. There are more than 150000 terms m with A001222(m) = 4. In general, there should be only a finite number of terms m for any given k = A001222(m). Compare for example with A287728.

(End)

Proof of claim 1 by Jianing Song, Dec 11 2024, (slightly extended to cover also claim 2 by Antti Karttunen, Sep 06 2025): (Start)

Let's write p' for the next prime after the prime p. Also, write Q(n) = A003961(n)/sigma(n) which is multiplicative.

Proposition: For n > 1 not being a prime nor twice a prime (e.g., 6 = 2*3), n has a factor p such that Q(n) > p'/p.

This implies that if n is nondeficient [A023196, including any primitively nondeficient n in A006039, and any n in A000396\{6}], then n has a factor p such that A003961(n/p)/(n/p) = (A003961(n)/n)/(p'/p) > sigma(n)/n [which is >= 2 because n is nondeficient], so also n/p is in A246282, meaning that n cannot be in this sequence.

Proof. We see that 1 <= Q(p) <= Q(p^2) <= ..., which implies that if n verifies the proposition, then every multiple of n also verifies it. Since n = p^2 > 4 and n = 8 verify the proposition, it suffices to consider the case where n = pq is the product of two distinct odd primes. Suppose WLOG that p < q, so q >= p', then using q/(q+1) >= p'/(p'+1) we have

Q(n) = p'q'/((p+1)(q+1)) >= p'^2*q'/(q(p+1)(p'+1)) > (p'^2-1)*q'/(q(p+1)(p'+1)) = (p'-1)/(p+1) * q'/q >= q'/q.

(End)

Numbers whose proper multiples are all primeshift-abundant (A246282), and whose proper divisors are all primeshift-deficient (A246281). Compare to A006039. - Antti Karttunen, Aug 31 2025

Antti Karttunen, Table of n, a(n) for n = 1..20000

David A. Corneth, PARI programs

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

Index entries for primes, gaps between.

{k: 1==A337345(k)}.

14 = 2*7 is in the sequence as setting every prime to the next larger prime gives 3*11 = 33 > 28 = 2*14. Doing so for any proper divisor d of 14 gives a number < 2 * d. - David A. Corneth, Dec 07 2024

Block[{a = {}, b = {}}, Do[If[2 i < Times @@ Map[#1^#2 & @@ # &, FactorInteger[i] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[i == 1], AppendTo[a, i]; If[IntersectingQ[Most@ Divisors[i], a], AppendTo[b, i]]], {i, 1400}]; Complement[a, b]] (* Michael De Vlieger, Feb 22 2021 *)

(PARI)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

A252742(n) = (A003961(n) > (2*n));

A337346(n) = sumdiv(n, d, (d<n)&&A252742(d));

isA337372(n) = ((1==A252742(n))&&(0==A337346(n)));

(PARI) is_A337372 = A341609;

(PARI) \\ See Corneth link

Setwise difference A246282 \ A341610.

Cf. A000101, A003961, A246281, A252742, A337346, A378745, A378746.

Cf. A378658, A378662, A378664, A378736, A378740.

Positions of ones in A337345 and in A341609 (characteristic function).

Subsequence of A263837 and thus also of A341614.

Cf. also A005101, A006039, A023196, A091191, A326134.

Cf. also A337543, A387712.

Sequence in context: A036326 A078972 A115652 * A317299 A236026 A193305

Adjacent sequences: A337369 A337370 A337371 * A337373 A337374 A337375

Antti Karttunen, Aug 27 2020

Incorrect comment removed and the comment section revised by Antti Karttunen, Sep 07 2025

approved