A129499 -id:A129499 - cp's OEIS frontend

A285614 Unitary highly abundant numbers: numbers n such that usigma(n) > usigma(m) for all m < n, where usigma(n) = sum of unitary divisors of n (A034448).

1, 2, 3, 4, 5, 6, 10, 12, 14, 18, 21, 22, 26, 30, 42, 60, 66, 78, 90, 102, 114, 130, 138, 150, 170, 174, 186, 210, 294, 318, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1554, 1590, 1722, 1770, 1830, 1974

Offset: 1

Amiram Eldar, Apr 22 2017

nonn

Corresponds to A002093 (Highly abundant numbers), with usigma(n) = sum of unitary divisors of n (divisors d such that gcd(d, n/d)=1, A034448) instead of sigma(n) (sum of divisors, A000203).

Contains many terms of A280013 (sum of squarefree divisors instead of unitary divisors), but not all of them - the first terms of A280013 that are not in this sequence are 16530, 26070, 8734110, 8757210,...

			The first 9 values of usigma(n) for n=1..9 are: 1, 3, 4, 5, 6, 12, 8, 9, 10. usigma(10)=18 is higher than these 9 values, thus 10 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A034448, A002093, A034683, A129499, A280013.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; a = {}; k = 0; Do[s = usigma[n]; If[s > k, AppendTo[a, n]; k = s], {n, 1000}]; a

A129498 Unitary abundancy of n-th unitary abundant number: usigma(k)-2k if this is > 0.

Original entry on oeis.org

12, 12, 12, 4, 12, 12, 12, 12, 12, 12, 12, 156, 12, 12, 12, 12, 12, 12, 204, 12, 12, 228, 12, 120, 12, 12, 228, 12, 12, 276, 12, 252, 300, 12, 12, 12, 180, 12, 12, 120, 12, 348, 300, 12, 12, 12, 188, 120, 12, 324, 12, 12, 48, 300, 420, 12, 12, 196, 72, 444, 12, 372

Offset: 1

Ant King, Apr 20 2007

The transforms of this sequence are discussed in A129499.

			The fourth unitary abundant number is 70. As the unitary divisors of 70 are 1, 2, 5, 7, 10, 14, 35 and 70, we have a(4) = 1+2+5+7+10+14+35+70-2 * 70 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A034448, A034460, A129468, A034683, A129499.

uab[1]=-1; uab[n_] := Times @@ (1 + Power @@@ FactorInteger[n]) - 2n; seq={}; Do[u=uab[n]; If[u>0, AppendTo[seq, u]], {n, 1, 1000}]; seq (* Amiram Eldar, Jun 18 2019 *)

A034448(k)-2k = A034460(k)-k, whenever these are positive.

a(n) = A129468(A034683(n)). - Amiram Eldar, Jun 18 2019

cp's OEIS Frontend

A285614 Unitary highly abundant numbers: numbers n such that usigma(n) > usigma(m) for all m < n, where usigma(n) = sum of unitary divisors of n (A034448).

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