This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
8 qualifies because 8 = 4*2 and 4 is in A019505, but 8 can't be term after 4 in A019505 because smallest number with 4 divisors is 6.
a140635[n_] := NestWhile[#+1&, 1, DivisorSigma[0, n]!=DivisorSigma[0, #]&]
a140635[{m_, n_}] := Map[a140635, Range[m, n]]
a140635[{1, 89}] (* Hartmut F. W. Hoft, Jun 13 2023 *)
A140635(n) = { my(nd = numdiv(n)); for (i=1, n, if (numdiv(i) == nd, return (i))); }; \\ After A139770, Antti Karttunen, May 27 2017
from sympy import divisor_count as d
def a(n):
x=d(n)
m=1
while True:
if d(m)==x: return m
else: m+=1 # Indranil Ghosh, May 27 2017
As k increases, the positive integer k=6 sets or ties the existing records for smallest first, second and third-smallest divisors (1, 2 and 3), as well as for its fourth-smallest (6). Since no smaller integer has more than three divisors, 6 is a term of this sequence.
ge(va, vb) = {for(i=1, min(#va, #vb), if (va[i] > vb[i], return(0));); return(-1);}
isok(k) = {my(dk = divisors(k)); for (m=1, k-1, my(dm = divisors(m)); if (! ge(dk, dm), return(0));); return(1);} \\ Michel Marcus, Mar 16 2022
6 is not in the sequence because 6 is not a multiple of 4, the previous term.
sublist_of_first_proper_multiple_terms_of(v) = { my(u=v[1], lista=List(u)); for(i=2,#v,if((v[i]>u)&&!(v[i]%u), u = v[i]; listput(lista,u))); Vec(lista); };
v133411 = sublist_of_first_proper_multiple_terms_of(v002182); \\ v002182 contains the terms of A002182.
A133411(n) = v133411[n]; \\ Antti Karttunen, Jan 10 2020
For n=8, n*a(n-1) = 8*5040 = 40320 has 96 divisors, but the smallest number with 96 divisors is 27720, so a(8)=27720.
A138113(n)={ local(an1,t) ; if(n<=2, return(n) ) ; an1 = A138113(n-1) ; t=length(divisors(n*an1)) ; return(A005179(t)) ; } {for (n=1,40, print1(A138113(n)", ") ; ) } \\ R. J. Mathar, Mar 20 2010
9*3=27 has 4 divisors, but smallest odd number with 4 divisors is 15.
a(nn) = {ia = 1; print1(ia, ", "); for (n = 1, nn - 1, nd = numdiv(3*ia); forstep(i = 1, 3*ia, 2, if (numdiv(i) == nd, ia = i; break;);); print1(ia, ", "););} \\ Michel Marcus, Jun 14 2013
{/*prints b-file for A140864 - add more for loops for more terms*/ print("#A140864"); print(1" "1); print(2" "3); n = 3; for(p=3,56,tau = numdiv(3*n); exp3n=factor(n)[1,2];delta = bigomega(exp3n+2) - bigomega(exp3n+1); delta = max(delta+1,2); var = exp3n+delta; num = 10^1000; for( n1=1, var, for (n2=0, n1, for( n3=0, n2, for( n4=0, n3, for( n5=0, n4, for( n6=0, n5, for( n7=0, n6, for( n8=0, n7, for( n9=0, n8, for( n10=0, n9, for( n11=0, n10, for( n12=0, n11, for( n13=0, n12, for( n14=0, n13, for( n15=0, n14, if( (n1+1) * (n2+1) * (n3+1) * (n4+1) * (n5+1) * (n6+1) * (n7+1) * (n8+1) * (n9+1) * (n10+1) * (n11+1) * (n12+1) * (n13+1) * (n14+1) * (n15+1) == tau, numtemp = prime(2)^n1 * prime(3)^n2 * prime(4)^n3 * prime(5)^n4 * prime(6)^n5 * prime(7)^n6 * prime(8)^n7 * prime(9)^n8 * prime(10)^n9 * prime(11)^n10 * prime(12)^n11 * prime(13)^n12 * prime(14)^n13 * prime(15)^n14 * prime(16)^n15; if(numtemp < num, num = numtemp); ));););););););) ;);););) ;););); print(p" "num); n=num;)} \\ Dimitri Papadopoulos, May 08 2019
17297280*2 = 34594560; smallest highly composite number <= 34594560 is 32432400.
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