A397475 - OEIS

%I #10 Jun 29 2026 15:19:46

%S 38101088025,111419102025,173851472025,647718496425,723920672475,

%T 876325024575,1104931552725,1181133728775,1409740256925,1448448326325,

%U 1562144609025,1638346785075,1790751137175,2019357665325,2116962938475,2247964193475,2260069136325,2324166369525

%N Odd exponential abundant numbers that are not exponential unitary abundant.

%C The odd terms in A391085.

%C Odd numbers k such that A322857(k) <= 2*k < A051377(k).

%C If k is a term and m is an odd squarefree number coprime to k, then k*m is also a term. The primitive terms in this sequence (A397476) are the powerful (A001694) terms. All the terms are of the form k*m where k is primitive and m is an odd squarefree number coprime to k.

%C The asymptotic density of this sequence is Sum_{n>=1} f(A397476(n)) = 7.9...*10^(-12), where f(n) = (4/(Pi^2*n)) * Product_{prime p|n}(p/(p+1)).

%H Amiram Eldar, <a href="/A397475/b397475.txt">Table of n, a(n) for n = 1..10000</a>

%o (PARI) fun1(p, e) = sumdiv(e, d, p^d);

%o fun2(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));

%o is1(k) = {my(f = factor(k)); prod(i = 1, #f~, fun1(f[i, 1], f[i, 2])) > 2*k && prod(i = 1, #f~, fun2(f[i, 1], f[i, 2])) <= 2*k;}

%o listprim(lim) = {my(s = List(), m); forstep(j = 1, sqrtnint(lim, 3), 2, forstep(i = 1, sqrtint(lim\j^3), 2, m = i^2*j^3; if(is1(m), listput(s, m)))); s = Set(s);}

%o list(lim) = {my(p = listprim(lim), s = List()); for(i = 1, #p, forsquarefree(j = 1, lim\p[i], if(j[1]%2 && gcd(j[1], p[i]) == 1, listput(s, j[1] * p[i])))); Set(s);}

%o list(2.5*10^12)

%Y Intersection of A005408 and A391085.

%Y Subsequence of A130897 and A321147.

%Y A397476 is a subsequence.

%Y Cf. A001694, A051377, A056911, A209061, A322857.

%K nonn

%O 1,1

%A _Amiram Eldar_, Jun 27 2026