A337691
a(n) is the least positive integer divisible by exactly n primitive nondeficient numbers (A006039).
Original entry on oeis.org
1, 6, 60, 140, 420, 3780, 17160, 28600, 40040, 138600, 120120, 180180, 300300, 360360, 600600, 1351350, 900900, 4144140, 1801800, 3063060, 5405400, 6126120, 8558550, 7657650, 19399380, 20720700, 17117100, 15315300, 29099070, 30630600, 45945900, 70450380, 91891800, 87297210
Offset: 0
The least nondeficient number, therefore the least primitive nondeficient number is 6. So a(1) = 6, as the smallest number divisible by exactly 1 primitive nondeficient number.
Table of n, a(n) and the relevant divisors starts:
n a(n) divisors in A006039
0 1 (none);
1 6 6;
2 60 6, 20;
3 140 20, 28, 70;
4 420 6, 20, 28, 70;
5 3780 6, 20, 28, 70, 945;
6 17160 6, 20, 88, 104, 572, 1430;
7 28600 20, 88, 104, 550, 572, 650, 1430;
8 40040 20, 28, 70, 88, 104, 572, 1430, 2002; ...
Note that a(6), a(7), a(8) are 3*5720, 5*5720, 7*5720.
See
A000203 and
A023196 for definitions of deficient and nondeficient.
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\\ Code for A337690 given under that entry.
A337691list(search_up_to_n) = { my(m=Map(),lista=List([]),t); for(n=1,search_up_to_n,if(!(n%(2^24)),print1("(",n,")")); t=A337690(n); if(!mapisdefined(m,t), mapput(m,t,n))); for(n=0,oo,if(mapisdefined(m,n,&t), listput(lista,t), return(Vec(lista)))); };
v337691 = A337691list(2^27);
A337691(n) = v337691[1+n];
A335541
Numbers with a record value of the ratio of the number of abundant divisors to the total number of divisors.
Original entry on oeis.org
1, 12, 24, 36, 72, 120, 144, 216, 360, 432, 720, 1440, 2160, 2880, 4320, 8640, 12960, 17280, 20160, 25920, 30240, 40320, 51840, 60480, 80640, 120960, 181440, 241920, 362880, 483840, 604800, 725760, 967680, 1209600, 1451520, 1814400, 2177280, 2419200, 2903040, 3628800
Offset: 1
36 has 9 divisors, {1, 2, 3, 4, 6, 9, 12, 18, 36}, 3 of which are abundant, {12, 18, 36}. The ratio 3/9 = 1/3 is larger than the ratios for all the numbers below 36. Hence 36 is a term.
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f[n_] := Count[(d = Divisors[n]), _?(DivisorSigma[1, #] > 2# &)]/Length[d]; fm = -1; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^4}]; s
A335542
Numbers with a record number of deficient divisors.
Original entry on oeis.org
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
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.
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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 *)
Showing 1-3 of 3 results.
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