|
%I #49 Feb 22 2021 21:19:44
%S 4,6,8,9,10,12,14,15,16,18,20,21,24,27,28,30,32,35,36,39,40,42,44,45,
%T 48,49,50,52,54,56,57,60,63,64,66,68,69,70,72,75,76,78,80,81,84,88,90,
%U 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
%N 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).
%C Numbers n such that A003961(n) > 2*n.
%C Numbers n such that A048673(n) > n.
%C The sequence grows as:
%C a(10) = 18
%C a(100) = 192
%C a(1000) = 1830
%C a(10000) = 18636
%C a(100000) = 187350
%C a(1000000) = 1865226
%C a(10000000) = 18654333
%C and the powers of 10 occur at:
%C a(5) = 10
%C a(53) = 100
%C a(536) = 1000
%C a(5423) = 10000
%C a(53290) = 100000
%C a(535797) = 1000000
%C a(5361886) = 10000000
%C 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.
%C From _Antti Karttunen_, Aug 27 2020: (Start)
%C 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.
%C 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.
%C (End)
%H Antti Karttunen, <a href="/A246282/b246282.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e 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.
%e 4 = 2*2 = p_1 * p_1 is a member, as p_2 * p_2 = 3*3 = 9, and 9/4 > 2.
%e 33 = 3*11 = p_2 * p_5 is not a member, as p_3 * p_6 = 5*13 = 65, and 65/33 < 2.
%e 35 = 5*7 = p_3 * p_4 is a member, as p_4 * p_5 = 7*11 = 77, and 77/35 > 2.
%t 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 *)
%o (PARI)
%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o isA246282(n) = (A003961(n) > (n+n));
%o n = 0; i = 0; while(i < 10000, n++; if(isA246282(n), i++; write("b246282.txt", i, " ", n)));
%o (Scheme, with _Antti Karttunen_'s IntSeq-library, two alternative implementations)
%o (define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A003961 n) (* 2 n)))))
%o (define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A048673 n) n))))
%Y Complement: A246281.
%Y Setwise difference of A246352 and A048674.
%Y Cf. A000040, A003961, A048673, A246362, A252742 (characteristic function), A286385, A326042, A337345.
%Y Positions of positive terms in A252748 and in A337345.
%Y Union of A337372 (primitive terms), A341610 (non-primitive terms).
%Y Subsequences: A000396, A005101, A023196, A326134, A337373, A337374, A337378, A337381, A337384, A337386, A337543, A341611, A341614, A341615.
%Y Cf. also A275717, A275718.
%K nonn
%O 1,1
%A _Antti Karttunen_, Aug 24 2014
%E A new shorter version of name prepended by _Antti Karttunen_, Aug 27 2020
|
