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A378737 Abundant numbers k for which A378665(k) > A378664(k).

%I A378737 #14 Dec 08 2024 17:24:36
%S A378737 66,102,174,186,222,246,258,282,318,354,366,402,426,438,474,498,534,
%T A378737 582,606,618,642,654,678,748,762,786,822,834,894,906,942,978,1002,
%U A378737 1038,1074,1086,1146,1158,1182,1194,1266,1338,1362,1374,1398,1434,1446,1496,1506,1542,1578,1614,1626,1662,1686,1698,1758,1842
%N A378737 Abundant numbers k for which A378665(k) > A378664(k).
%C A378737 For most of these numbers k, A378664(k) = 6. Note that A003961(6) = A003961(2*3) = 3*5 = 15 > 2*6, while sigma(6) = 12, making 6 non-abundant. Note that there seems to be only a finite amount (namely 13, see A337372) of such "stopper semiprimes" that would prevent of A378736 obtaining value 1.
%C A378737 Other possible values that A378664 obtains on these numbers are for example 68, 136, 170, 256, 290, 370, 410, 430, 470, 530, 646, 682, 754, 1276, 1292, 1364, 1508, 1628, 1804, 1892, 2068, 2332, 4756, 6844, 10846, 15334, etc. See A378740, which contains some of these.
%H A378737 Antti Karttunen, <a href="/A378737/b378737.txt">Table of n, a(n) for n = 1..20000</a>
%H A378737 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%H A378737 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A378737 {A005101(i) for such indices i where A378735(i) > A378736(i)}.
%e A378737 66 is a term as A378665(66) = 33, but A378664(66) = 6.
%e A378737 748 is a term as A378665(748) = 374, but A378664(748) = 68.
%e A378737 1866 is a term as A378665(1866) = 933, but A378664(1866) = 6.
%e A378737 1870 is a term as A378665(1870) = 935, but A378664(1870) = 170.
%o A378737 (PARI)
%o A378737 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A378737 A294935(n) = (sigma(n)<=(2*n));
%o A378737 A341612(n) = ((sigma(n)<=(2*n))&&((2*n)<A003961(n)));
%o A378737 A378664(n) = { fordiv(n,d,if(A341612(n/d), return(n/d))); (1); };
%o A378737 A378665(n) = { fordiv(n,d,if(A294935(n/d), return(n/d))); (1); };
%o A378737 is_A378737(n) = (!A294935(n) && (A378664(n)!=A378665(n)));
%Y A378737 Subsequence of A005101.
%Y A378737 Cf. A378738 (subsequence).
%Y A378737 Cf. A337372, A378664, A378665, A378735, A378736, A378740.
%K A378737 nonn
%O A378737 1,1
%A A378737 _Antti Karttunen_, Dec 06 2024