cp's OEIS Frontend

A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

%I A348738 #29 Jul 12 2024 21:16:48
%S A348738 2,3,5,6,7,8,10,11,13,14,15,17,19,20,21,22,23,24,26,27,28,29,30,31,33,
%T A348738 34,35,37,38,39,40,41,42,43,45,46,47,50,51,52,53,54,55,56,57,58,59,60,
%U A348738 61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,82,83,84,85,86,87,88,89,90,91,92,93,94,95,97
%N A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).
%C A348738 Claim: If there is an odd term y of A336702 larger than one, and it is the least one of such terms, then it should satisfy condition that for all nontrivial unitary divisor pairs d and x/d of x = A064989(y) [with gcd(d,x/d) = 1, 1 < d < x], the other divisor should reside in this sequence, and the other divisor in A348739. Proof: Applying A064989 to the odd terms of A336702 gives the fixed points of A326042. Suppose there are other odd terms in A336702 in addition to its initial 1, and let y be the least of these odd terms > 1 and x = A064989(y). Because A326042 (from here on indicated with f) is multiplicative, it follows that if we take any two nontrivial unitary divisors a and b of x, with x = a*b, gcd(a,b) = 1, 1 < a,b < x, then f(a)*f(b) = f(x) = x. Because f(x)/x = 1, we must have f(a)/a * f(b)/b = 1, as also the ratio f(n)/n is multiplicative. But f(a)/a and f(b)/b cannot be equal to 1, because then a and b would also be fixed by f, which contradicts our assumption that x were the least such fixed point larger than one. Therefore f(a) < a and f(b) > b, or vice versa. See also the comments in A348930, A348933.
%C A348738 Moreover, all odd perfect numbers (a subsequence of A336702), if such numbers exist, should also satisfy the same condition, regardless of whether they are the least of such numbers or not, because having a non-deficient proper divisor will push the abundancy index (ratio sigma(n)/n) of any number over 2. That is, for any such pair of nontrivial unitary divisors d and x/d, both A003961(d) and A003961(x/d) should be deficient, i.e., neither one should be in A337386. See also the condition given in A347383.
%C A348738 Terms that occur also in A337386 are: 120, 240, 360, 420, 480, 504, 540, 600, 630, ...
%H A348738 Antti Karttunen, <a href="/A348738/b348738.txt">Table of n, a(n) for n = 1..20000</a>
%H A348738 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A348738 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%t A348738 f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; Select[Range[100], s1[DivisorSigma[1, s2[#]]] < # &] (* _Amiram Eldar_, Nov 04 2021 *)
%o A348738 (PARI)
%o A348738 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A348738 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A348738 A326042(n) = A064989(sigma(A003961(n)));
%o A348738 isA348738(n) = (A326042(n)<n);
%Y A348738 Positions of positive terms in A348736, positions of 1's in A348737 (characteristic function).
%Y A348738 Almost complement of A348739.
%Y A348738 Cf. A000203, A003961, A003973, A006039, A027687, A064989, A326042, A336702, A337386, A347383, A348741, A348748 (corresponding odd numbers).
%Y A348738 Subsequences: A000040, A374464 (after its initial 1).
%Y A348738 Cf. also A348930, A348933.
%K A348738 nonn
%O A348738 1,1
%A A348738 _Antti Karttunen_, Nov 02 2021