A302934 -id:A302934 - cp's OEIS frontend

A335542 Numbers with a record number of deficient divisors.

1, 2, 4, 8, 16, 30, 60, 90, 150, 210, 315, 630, 990, 1575, 1890, 2310, 3465, 4620, 6930, 11550, 13860, 17325, 20790, 30030, 39270, 45045, 60060, 78540, 90090, 117810, 131670, 180180, 196350, 219450, 225225, 255255, 270270, 353430, 395010, 450450, 510510, 746130

Offset: 1

Amiram Eldar, Jun 13 2020

nonn

The corresponding numbers of deficient divisors are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 16, 17, 18, 22, ...

			2 is in the sequence since it is the least number with 2 deficient divisors, 1 and 2. The next number with more than 2 deficient divisors is 4, which has 3 deficient divisors, 1, 2, and 4.

Amiram Eldar, Table of n, a(n) for n = 1..120

Cf. A000203, A005100, A080226, A302934, A335540.

s[n_] := Count[Divisors[n], _?(DivisorSigma[1, #] < 2*# &)]; sm = -1; seq = {}; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^6}]; seq
Module[{nn=800000,lst},lst=Table[{n,Count[Divisors[n],?(DivisorSigma[1,#]<2#&)]},{n,nn}];DeleteDuplicates[lst,GreaterEqual[#1[[2]],#2[[2]]]&]][[;;,1]] (* _Harvey P. Dale, May 06 2023 *)

Numbers m such that A080226(m) > A080226(k) for all k < m.

A302935 Unitary highly composite deficient numbers: unitary deficient numbers k whose number of unitary divisors ud(k) > ud(m) for all unitary deficient numbers m < k.

Original entry on oeis.org

1, 2, 10, 84, 1155, 25740, 471240, 14549535, 535422888

Offset: 1

Amiram Eldar, Apr 16 2018

The record numbers of unitary divisors are 1, 2, 4, 8, 16, 32, 64, 128, 256, ...

The unitary version of A302934.

Cf. A034444, A034448, A129487, A302934.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; udiv[n_] := 2^PrimeNu[n]; dm = 0; Do[sig = usigma[n]; If[sig >= 2 n, Continue[]]; d = udiv[n]; If[d > dm, Print[n]; dm = d], {n, 1, 1000000000}]

nbud(n) = 1< maxd), print1(n, ", "); maxd = nbud(n);););} \\ Michel Marcus, Apr 17 2018

A302936 Bi-unitary highly composite deficient numbers: bi-unitary deficient numbers k whose number of bi-unitary divisors bd(k) > bd(m) for all bi-unitary deficient numbers m < k.

Original entry on oeis.org

1, 2, 8, 32, 84, 512, 972, 1155, 13365, 25740, 318087, 612612, 11223927, 14549535, 440374077, 746503065, 19013596875

Offset: 1

Amiram Eldar, Apr 16 2018

The record numbers of bi-unitary divisors are 1, 2, 4, 6, 8, 10, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, ...

The bi-unitary version of A302934.

Cf. A188999, A286324, A292982, A293185, A302934.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdivnum[n_] := DivisorSum[n, 1 &, Last@Intersection[f@#, f[n/#]] == 1 &]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; dm = 0; Do[sig = bsigma[n]; If[sig >= 2 n, Continue[]]; d = bdivnum[n]; If[d > dm, Print[n]; dm = d], {n, 1, 1000000000}] (* after Michael De Vlieger at A188999 and A286324 *)

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
lista(nn) = {my(maxd = 0); for(n=1, nn, vbiudiv = biudivs(n); if ((vecsum(vbiudiv) < 2*n) && (#vbiudiv > maxd), print1(n, ", "); maxd = #vbiudiv;););} \\ Michel Marcus, Apr 17 2018

a(15)-a(17) from Amiram Eldar, Jan 26 2019

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A335542 Numbers with a record number of deficient divisors.

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A302935 Unitary highly composite deficient numbers: unitary deficient numbers k whose number of unitary divisors ud(k) > ud(m) for all unitary deficient numbers m < k.

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A302936 Bi-unitary highly composite deficient numbers: bi-unitary deficient numbers k whose number of bi-unitary divisors bd(k) > bd(m) for all bi-unitary deficient numbers m < k.

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