cp's OEIS Frontend

A337372 Primitively primeshift-abundant numbers: Numbers that are included in A246282 (k with A003961(k) > 2k), but none of whose proper divisors are.

%I A337372 #72 Dec 13 2024 09:37:55
%S A337372 4,6,9,10,14,15,21,35,39,49,57,69,91,125,242,275,286,325,338,363,418,
%T A337372 425,442,475,494,506,561,575,598,646,682,715,722,725,754,775,782,806,
%U A337372 845,847,867,874,925,957,962,1023,1025,1045,1054,1058,1066,1075,1105,1118,1175,1178,1221,1222,1235,1265,1309,1325,1334,1353
%N A337372 Primitively primeshift-abundant numbers: Numbers that are included in A246282 (k with A003961(k) > 2k), but none of whose proper divisors are.
%C A337372 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.
%C A337372 Question: Do the odd terms in A326134 all occur here? Answer is yes, if the following conjecture holds: This is a subsequence of A263837, nonabundant numbers. In other words, we claim that any abundant number k (A005101) has A337345(k) > 1 and thus is a term of A341610. (The conjecture indeed holds. See the proof below).
%C A337372 From _Antti Karttunen_, Dec 06 2024: (Start)
%C A337372 Observation 1: 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.
%C A337372 Observation 2: The intersection with A005101 (and thus also with A091191) is empty, which then 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.
%C A337372 (End)
%C A337372 Proof of the latter observation by _Jianing Song_, Dec 11 2024: (Start)
%C A337372 Let's write p' for the next prime after the prime p. Also, write Q(n) = A003961(n)/sigma(n) which is multiplicative.
%C A337372 Proposition: For n > 1 not being a prime nor twice a prime, n has a factor p such that Q(n) > p'/p.
%C A337372 This implies that if n is abundant [including any primitively abundant n in A091191], 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 abundant], so n/p is in A246282, meaning that n cannot be in this sequence.
%C A337372 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
%C A337372 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.
%C A337372 (End)
%H A337372 Antti Karttunen, <a href="/A337372/b337372.txt">Table of n, a(n) for n = 1..20000</a>
%H A337372 David A. Corneth, <a href="/A337372/a337372_1.gp.txt">PARI programs</a>
%H A337372 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%H A337372 <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>.
%F A337372 {k: 1==A337345(k)}.
%e A337372 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
%t A337372 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 *)
%o A337372 (PARI)
%o A337372 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A337372 A252742(n) = (A003961(n) > (2*n));
%o A337372 A337346(n) = sumdiv(n,d,(d<n)&&A252742(d));
%o A337372 isA337372(n) = ((1==A252742(n))&&(0==A337346(n)));
%o A337372 (PARI) is_A337372 = A341609;
%o A337372 (PARI) \\ See Corneth link
%Y A337372 Setwise difference A246282 \ A341610.
%Y A337372 Cf. A000101, A003961, A246281, A252742, A337346, A378745, A378746.
%Y A337372 Cf. A378658, A378662, A378664, A378736, A378740.
%Y A337372 Positions of ones in A337345 and in A341609 (characteristic function).
%Y A337372 Subsequence of A263837 and thus also of A341614.
%Y A337372 Cf. also A005101, A091191, A326134.
%Y A337372 Cf. also A337543.
%K A337372 nonn
%O A337372 1,1
%A A337372 _Antti Karttunen_, Aug 27 2020