A280013 -id:A280013 - 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

A281782 Numbers n such that sum of prime power divisors of n > sum of prime power divisors of m for all m < n.

Original entry on oeis.org

2, 3, 4, 7, 8, 16, 27, 32, 64, 121, 125, 128, 243, 256, 512, 729, 1024, 2048, 4096, 6561, 8192, 15625, 16384, 32761, 32768, 59049, 65536, 117649, 130321, 131072, 177147, 262144, 524287, 524288, 1048576, 1594323, 1953125, 2097152, 4194304, 8388608

Offset: 1

Ilya Gutkovskiy, Apr 14 2017

nonn

Numbers n such that A023889(n) > A023889(m) for all m < n.

Numbers n such that Sum_{p^k|n, p prime, k>=1} p^k > Sum_{p^k|m, p prime, k>=1} p^k for all m < n.

Amiram Eldar, Table of n, a(n) for n = 1..65
Index entries for sequences related to sums of divisors

Cf. A002093, A002182, A023889, A034090, A174572, A246655, A280013.

mx = 0; t = {}; Do[u = DivisorSum[n, # &, PrimePowerQ[#] &]; If[u > mx, mx = u; AppendTo[t, n]], {n, 8500000}]; t

A283416 Number of possible connected steric graphs with n nodes of degree less than 5.

Original entry on oeis.org

1, 1, 2, 8, 43, 405

Offset: 1

Natan Arie Consigli, Mar 07 2017

This gives the number of hypothetical neutrally charged saturated carbon allotropes and hydrocarbons (alkanes, cycloalkanes, alkyl-cycloalkanes) with n carbon atoms satisfying the octet rule counting all isomers, except conformers.
Steric (three-dimensional) graphs can be represented as two-dimensional graphs provided that stereoisomers are taken into account. This means acting the group S_4 on the stereocenters (nodes with 4 different substituents) of each 2D graph and counting all of the resulting non-symmetric molecules.
Many of those "theoretical" molecules are to be discovered and some may not exist due to their strained geometries.
The terms of this sequence (and related ones) were found by drawing and counting the molecules in the PDFs below. If there are any missing molecules you can contact me. I'll add them in the PDF and update the data section.

			a(4) = 8 because there are eight structures with 4 nodes (carbons) following the above rules: butane, isobutane, cyclobutane, methylcyclopropane, (r,r)-bicyclobutane, (s,s)-bicyclobutane, (r,r,r)-tetrahedrane, (s,s,s)-tetrahedrane.

Natan Arie' Consigli, Illustration of a(5) and a(6)

Row sums of A280013.

A290490 Numbers k such that (sum of proper unitary divisors of k) > (sum of proper unitary divisors of m) for all m < k.

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 22, 26, 30, 42, 60, 66, 78, 102, 114, 138, 150, 174, 186, 210, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1590, 1722, 1770, 1830, 2010, 2130, 2190, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6510, 7410, 7590, 7770

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

Numbers k such that A034460(k) > A034460(m) for all m < k.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for sequences related to sums of divisors

Cf. A002182, A002093, A034090, A034448, A034460, A063919, A280013, A281782, A285614.

mx = -1; t = {}; Do[u = DivisorSum[n, # &, GCD[#, n/#] == 1 &] - n; If[u > mx, mx = u; AppendTo[t, n]], {n, 8000}]; t

sumud(n) = sumdiv(n, d, if (gcd(d, n/d)==1, d)) - n;
lista(nn) = {lasts = -1; for (n=1, nn, if ((news = sumud(n)) > lasts, print1(n, ", "); lasts = news););} \\ Michel Marcus, Aug 04 2017

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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A281782 Numbers n such that sum of prime power divisors of n > sum of prime power divisors of m for all m < n.

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A283416 Number of possible connected steric graphs with n nodes of degree less than 5.

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A290490 Numbers k such that (sum of proper unitary divisors of k) > (sum of proper unitary divisors of m) for all m < k.

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