A378746 -id:A378746 - 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.

4, 6, 9, 10, 14, 15, 21, 35, 39, 49, 57, 69, 91, 125, 242, 275, 286, 325, 338, 363, 418, 425, 442, 475, 494, 506, 561, 575, 598, 646, 682, 715, 722, 725, 754, 775, 782, 806, 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

Offset: 1

Antti Karttunen, Aug 27 2020

nonn

Numbers k whose only divisor in A246282 is k itself, i.e., A003961(k) > 2k, but for none of the proper divisors d|k, dA003961(d) > 2d.
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).
From Antti Karttunen, Dec 06 2024: (Start)
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.
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.
(End)
Proof of the latter observation by Jianing Song, Dec 11 2024: (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, n has a factor p such that Q(n) > p'/p.
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.
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)

			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

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.

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, A091191, A326134.
Cf. also A337543.

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 *)

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,(dA252742(d));
isA337372(n) = ((1==A252742(n))&&(0==A337346(n)));

PARI
```
is_A337372 = A341609;
```
PARI
```
\\ See Corneth link
```

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

A063124 a(n) = # { primes p | prime(n) <= p < 2*prime(n) } where prime(n) is the n-th prime.

Original entry on oeis.org

2, 2, 2, 3, 4, 4, 5, 5, 6, 7, 8, 10, 10, 10, 10, 12, 14, 13, 14, 15, 14, 16, 16, 17, 20, 21, 20, 20, 19, 19, 24, 24, 26, 26, 28, 27, 29, 29, 29, 29, 31, 31, 33, 33, 33, 33, 36, 39, 39, 39, 40, 40, 40, 42, 43, 44, 43, 43, 43, 43, 43, 45, 50, 51, 50, 50, 55, 55, 57, 56, 56, 56, 58

Offset: 1

Reinhard Zumkeller, Aug 08 2001

nonn

a(n) is the number of primes between prime(n) and 2*prime(n) inclusive. - Sean A. Irvine, Apr 18 2023

Also for x = Product_{i=n..n+k} A000040(i), the least k such that A003961(x) > 2*x. - Antti Karttunen, Dec 08 2024

			a(10) = 7 as there are 7 primes between prime(10) = 29 and 58 = 29*2: 29, 31, 37, 41, 43, 47, 53.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 2000 terms from Harry J. Smith]

Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.
Cf. A003961, A108227, A331677, A334051, A378746.

A062134 := proc(n) numtheory:-pi(2*ithprime(n))-n+1; end; # N. J. A. Sloane, Oct 19 2024
[seq(A062134(n),n=1..100)];

Table[PrimePi[2*Prime[n]] - n + 1, {n, 100}] (* Paolo Xausa, Oct 22 2024 *)

a(n)={1 + primepi(2*prime(n)) - n} \\ Harry J. Smith, Aug 19 2009

a(n) = A035250(prime(n)).
a(n) = A070046(n) + 1. - Sean A. Irvine, Apr 18 2023
From Antti Karttunen, Dec 08 2024: (Start)
a(n) = n-A331677(n) = 1+n-A334051(n).
a(n) = 1+A000720(2*A000040(n))-n. [After Harry J. Smith's PARI-program]
a(n) < A108227(n). [Assuming M. F. Hasler's interpretation in May 08 2017 comment in the latter]
a(n) = A001222(A378746(n)).
(End)

Definition clarified by N. J. A. Sloane, Oct 04 2024

A378745 a(n) = prime(n)^A378744(n).

Original entry on oeis.org

4, 9, 125, 49, 161051, 2197, 410338673, 130321, 12167, 12200509765705829, 923521, 94931877133, 1555098314991537910888601, 11688200277601, 10779215329, 1174711139837, 15413179794770734626518662321719325259, 191707312997281, 8182718904632857144561, 19118715823042429491729074582041753821507871751, 58871586708267913

Offset: 1

Antti Karttunen, Dec 08 2024

nonn

Terms are all present in A337372, i.e., are primitively prime-shift abundant.

Antti Karttunen, Table of n, a(n) for n = 1..170

Cf. A000040, A003961, A337372, A378744.

Cf. also A378746.

A378745(n) = { my(p=prime(n), q=prime(1+n)); for(k=1,oo,if(q^k > 2*(p^k), return(p^k))); };

a(n) = A000040(n)^A378744(n).

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.

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A063124 a(n) = # { primes p | prime(n) <= p < 2*prime(n) } where prime(n) is the n-th prime.

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A378745 a(n) = prime(n)^A378744(n).

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