A246282 Numbers k for which A003961(k) > 2*k; numbers n such that if n = Product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) > 2*n, where p_k indicates the k-th prime, A000040(k).

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 91, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144

Offset

1,1

Comments

Numbers n such that A003961(n) > 2*n.

Numbers n such that A048673(n) > n.

The sequence grows as:

a(10) = 18

a(100) = 192

a(1000) = 1830

a(10000) = 18636

a(100000) = 187350

a(1000000) = 1865226

a(10000000) = 18654333

and the powers of 10 occur at:

a(5) = 10

a(53) = 100

a(536) = 1000

a(5423) = 10000

a(53290) = 100000

a(535797) = 1000000

a(5361886) = 10000000

suggesting that the ratio a(n)/n is converging to an constant and an arbitrary natural number is slightly more likely to be in this sequence than in the complement A246281. See also comments at A246351 and compare to quite a different ratio present in the "inverse" case A246362.

From Antti Karttunen, Aug 27 2020: (Start)

Any perfect number, including all odd perfect numbers (if such numbers exist), must occur in this sequence. See A286385 and A326042 for the reason why.

Like abundancy index (ratio A000203(n)/n), also ratio A003961(n)/n is multiplicative and always > 1 for all n > 1. Thus if the number has a proper divisor that is in this sequence, then the number itself also is. See A337372 for terms included here, but with no proper divisor in this sequence.

(End)

Links

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

Index entries for sequences where any odd perfect numbers must occur

Index entries for sequences computed from indices in prime factorization

Example

3 = p_2 (3 is the second prime, A000040(2)) is not a member, because p_3 = 5 (5 is the next prime after 3, A000040(3)) and 5/3 < 2.
4 = 2*2 = p_1 * p_1 is a member, as p_2 * p_2 = 3*3 = 9, and 9/4 > 2.
33 = 3*11 = p_2 * p_5 is not a member, as p_3 * p_6 = 5*13 = 65, and 65/33 < 2.
35 = 5*7 = p_3 * p_4 is a member, as p_4 * p_5 = 7*11 = 77, and 77/35 > 2.

Mathematica

Select[Range[144], 2 # < Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &] (* Michael De Vlieger, Feb 22 2021 *)

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
isA246282(n) = (A003961(n) > (n+n));
n = 0; i = 0; while(i < 10000, n++; if(isA246282(n), i++; write("b246282.txt", i, " ", n)));
(Scheme, with Antti Karttunen's IntSeq-library, two alternative implementations)
(define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A003961 n) (* 2 n)))))
(define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A048673 n) n))))

Crossrefs

Complement: A246281.

Setwise difference of A246352 and A048674.

Cf. A000040, A003961, A048673, A246362, A252742 (characteristic function), A286385, A326042, A337345.

Positions of positive terms in A252748 and in A337345.

Union of A337372 (primitive terms), A341610 (non-primitive terms).

Subsequences: A000396, A005101, A023196, A326134, A337373, A337374, A337378, A337381, A337384, A337386, A337543, A341611, A341614, A341615.

Cf. also A275717, A275718.

Sequence in context: A343597 A199009 A070807 * A046352 A046355 A335419

Adjacent sequences: A246279 A246280 A246281 * A246283 A246284 A246285

Keyword

nonn

Author

Antti Karttunen, Aug 24 2014

Extensions

A new shorter version of name prepended by Antti Karttunen, Aug 27 2020

Status

approved