

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 kth prime, A000040(k).


36



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
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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.
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



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 IntSeqlibrary, two alternative implementations)
(define A246282 (MATCHINGPOS 1 1 (lambda (n) (> (A003961 n) (* 2 n)))))


CROSSREFS

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


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



