A341358 -id:A341358 - cp's OEIS frontend

A381742 Numbers k such that k^2 is abundant but d*k is nonabundant for any proper divisor d of k.

14, 124, 585, 1016, 16748, 32085, 33892, 37882, 39962, 41925, 46665, 121605, 134589, 181305, 212175, 388455, 495465, 522488, 524224, 544065, 839865, 1061565, 1152921, 1165515, 1243275, 1247103, 1335411, 1676829, 1943638, 2151075, 2290869, 2478075, 2625514, 2673998

Offset: 1

Amiram Eldar, Mar 06 2025

nonn

Numbers k such that k^2 is primitive abundant number (A091191).

If p is an odd Mersenne exponent (A174265), then 2^((p-1)/2) * (2^p-1) is a term.

Amiram Eldar, Table of n, a(n) for n = 1..775

Subsequence of A381738.
A379950 is a subsequence.
Cf. A005101, A091191, A174265, A263837, A341358, A379949.

q[k_] := DivisorSigma[-1, k^2] > 2 &&  AllTrue[Divisors[k], DivisorSigma[-1, #*k] <= 2 || # == k &]; Select[Range[10^6], q]

isok(k) = fordiv(k, d, if(d < k && sigma(d*k, -1) > 2, return(0))); sigma(k^2, -1) > 2;

A381743 The number of divisors d of n such that d*n is abundant.

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 6, 0, 1, 0, 0, 0, 6, 0, 6, 0, 0, 0, 8, 0, 0, 0, 5, 0, 8, 0, 0, 0, 0, 0, 9, 0, 0, 0, 8, 0, 8, 0, 4, 0, 0, 0, 10, 0, 3, 0, 4, 0, 8, 0, 8, 0, 0, 0, 12, 0, 0, 0, 0, 0, 8, 0, 2, 0, 8, 0, 12, 0, 0, 0, 2, 0, 8, 0, 10, 0, 0, 0, 12, 0, 0

Offset: 1

Amiram Eldar, Mar 06 2025

The indices of records of this sequence are the highly composite numbers (A002182) without 2 and 4.

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

Cf. A000005, A000396, A002182, A005101, A341358, A381738, A381742.

a[n_] := DivisorSum[n, 1 &, DivisorSigma[-1, #*n] > 2 &]; Array[a, 100]

a(n) = sumdiv(n, d, sigma(d*n, -1) > 2);

a(n) = A000005(n) if and only if n is abundant (A005101).
a(n) = A000005(n) - 1 if and only if n is either 1, a perfect number (A000396), or in A341358.
a(n) >= 1 if and only if n is in A381738.
a(n) = 1 if and only if n is in A381742.

cp's OEIS Frontend

A381742 Numbers k such that k^2 is abundant but d*k is nonabundant for any proper divisor d of k.

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