A397475 Odd exponential abundant numbers that are not exponential unitary abundant.

38101088025, 111419102025, 173851472025, 647718496425, 723920672475, 876325024575, 1104931552725, 1181133728775, 1409740256925, 1448448326325, 1562144609025, 1638346785075, 1790751137175, 2019357665325, 2116962938475, 2247964193475, 2260069136325, 2324166369525

Offset

1,1

Comments

The odd terms in A391085.

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

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.

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)).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Prog

(PARI) fun1(p, e) = sumdiv(e, d, p^d);
fun2(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
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; }
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); }
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); }
list(2.5*10^12)

Crossrefs

Intersection of A005408 and A391085.

Subsequence of A130897 and A321147.

A397476 is a subsequence.

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

Sequence in context: A178221 A038832 A038821 * A397476 A139780 A234072

Adjacent sequences: A397472 A397473 A397474 * A397476 A397477 A397478

Keyword

nonn

Author

Amiram Eldar, Jun 27 2026

Status

approved