# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a246282 Showing 1-1 of 1 %I A246282 #49 Feb 22 2021 21:19:44 %S A246282 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 A246282 48,49,50,52,54,56,57,60,63,64,66,68,69,70,72,75,76,78,80,81,84,88,90, %U A246282 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 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). %C A246282 Numbers n such that A003961(n) > 2*n. %C A246282 Numbers n such that A048673(n) > n. %C A246282 The sequence grows as: %C A246282 a(10) = 18 %C A246282 a(100) = 192 %C A246282 a(1000) = 1830 %C A246282 a(10000) = 18636 %C A246282 a(100000) = 187350 %C A246282 a(1000000) = 1865226 %C A246282 a(10000000) = 18654333 %C A246282 and the powers of 10 occur at: %C A246282 a(5) = 10 %C A246282 a(53) = 100 %C A246282 a(536) = 1000 %C A246282 a(5423) = 10000 %C A246282 a(53290) = 100000 %C A246282 a(535797) = 1000000 %C A246282 a(5361886) = 10000000 %C A246282 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 A246282 From _Antti Karttunen_, Aug 27 2020: (Start) %C A246282 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 A246282 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 A246282 (End) %H A246282 Antti Karttunen, Table of n, a(n) for n = 1..10000 %H A246282 Index entries for sequences where any odd perfect numbers must occur %H A246282 Index entries for sequences computed from indices in prime factorization %e A246282 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 A246282 4 = 2*2 = p_1 * p_1 is a member, as p_2 * p_2 = 3*3 = 9, and 9/4 > 2. %e A246282 33 = 3*11 = p_2 * p_5 is not a member, as p_3 * p_6 = 5*13 = 65, and 65/33 < 2. %e A246282 35 = 5*7 = p_3 * p_4 is a member, as p_4 * p_5 = 7*11 = 77, and 77/35 > 2. %t A246282 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 A246282 (PARI) %o A246282 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961 %o A246282 isA246282(n) = (A003961(n) > (n+n)); %o A246282 n = 0; i = 0; while(i < 10000, n++; if(isA246282(n), i++; write("b246282.txt", i, " ", n))); %o A246282 (Scheme, with _Antti Karttunen_'s IntSeq-library, two alternative implementations) %o A246282 (define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A003961 n) (* 2 n))))) %o A246282 (define A246282 (MATCHING-POS 1 1 (lambda (n) (> (A048673 n) n)))) %Y A246282 Complement: A246281. %Y A246282 Setwise difference of A246352 and A048674. %Y A246282 Cf. A000040, A003961, A048673, A246362, A252742 (characteristic function), A286385, A326042, A337345. %Y A246282 Positions of positive terms in A252748 and in A337345. %Y A246282 Union of A337372 (primitive terms), A341610 (non-primitive terms). %Y A246282 Subsequences: A000396, A005101, A023196, A326134, A337373, A337374, A337378, A337381, A337384, A337386, A337543, A341611, A341614, A341615. %Y A246282 Cf. also A275717, A275718. %K A246282 nonn %O A246282 1,1 %A A246282 _Antti Karttunen_, Aug 24 2014 %E A246282 A new shorter version of name prepended by _Antti Karttunen_, Aug 27 2020 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE