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

Author

Antti Karttunen, Aug 24 2014

Keywords

• nonn

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)

Examples

			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.
		

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

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.

Programs

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 *)

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.
(define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A003961 n) (* 2 n)))))

Scheme

;; With Antti Karttunen's IntSeq-library.
(define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A048673 n) n))))

Extensions

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