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Numbers with a record number of deficient divisors.
+10
2
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
COMMENTS
The corresponding numbers of deficient divisors are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 16, 17, 18, 22, ...
EXAMPLE
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.
MATHEMATICA
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 *)
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.
+10
0
1, 2, 10, 84, 1155, 25740, 471240, 14549535, 535422888
COMMENTS
The record numbers of unitary divisors are 1, 2, 4, 8, 16, 32, 64, 128, 256, ...
MATHEMATICA
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}]
PROG
(PARI) nbud(n) = 1<<omega(n);
usigma(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d));
lista(nn) = {my(maxd = 0); for(n=1, nn, if ((usigma(n) < 2*n) && (nbud(n) > maxd), print1(n, ", "); maxd = nbud(n); ); ); } \\ Michel Marcus, Apr 17 2018
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.
+10
0
1, 2, 8, 32, 84, 512, 972, 1155, 13365, 25740, 318087, 612612, 11223927, 14549535, 440374077, 746503065, 19013596875
COMMENTS
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, ...
MATHEMATICA
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 *)
PROG
(PARI) 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
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