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.
The list begins as follows: 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 ... 4 9 25 49 ... 6 10 14 15 21 22 26 33 34 35 38 39 46 51 ... 8 27 ... 12 18 20 28 44 45 50 52 ... 16 ... Note: the above array, without the initial 1, is given by A095904 (and its transpose A179216). - _Antti Karttunen_, May 15 2017
p:= proc() 0 end:
a:= proc(n) option remember; local t; a(n-1);
t:= (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
sort(map(i-> i[2], ifactors(n)[2]), `>`));
p(t):= p(t)+1
end: a(0):=0:
seq(a(n), n=1..100); # Alois P. Heinz, May 31 2020
prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Count[Array[prisig,n],prisig[n]],{n,30}] (* Gus Wiseman, Jul 08 2019 *)
a(10)=3 since 6,8,10 each have four divisors. a(11)=5 since 2,3,5,7,11 each have two divisors.
N:= 1000: # to get a(1) to a(N) R:= Vector(N): for n from 1 to N do v:= numtheory:-tau(n); R[v]:= R[v]+1; A[n]:= R[v]; od: seq(A[n],n=1..N); # Robert Israel, May 04 2015
b[_] = 0;
a[n_] := a[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)
a(n)=my(d=numdiv(n)); sum(k=1,n,numdiv(k)==d) \\ Charles R Greathouse IV, Sep 02 2015
a(11)=8 since 2,3,4,5,7,8,9,11 each have one distinct prime factor. a(12)=3 since 6,10,12 each have two distinct prime factors.
From _Gus Wiseman_, Dec 28 2018: (Start)
Column n lists the a(n) positive integers less than or equal to n with the same number of distinct prime factors as n:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
---------------------------------------------------------------------
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 3 4 5 7 8 6 9 10 11 12 14 13 16 15 17 18
2 3 4 5 7 8 6 9 10 12 11 13 14 16 15
2 3 4 5 7 8 6 10 9 11 12 13 14
2 3 4 5 7 6 8 9 10 11 12
2 3 4 5 7 8 6 9 10
2 3 4 5 7 8 6
2 3 4 5 7
2 3 4 5
2 3 4
2 3
2
(End)
Table[Length[Select[Range[n],PrimeNu[#]==PrimeNu[n]&]],{n,100}] (* Gus Wiseman, Dec 28 2018 *)
a(n) = my(nb = #factor(n)~); sum(k=1, n, #factor(k)~ == nb); \\ Michel Marcus, Jul 13 2019
a(10) = 3 because bigomega(10) = 2 and also bigomega(4) = bigomega(6) = bigomega(9) = 2.
A:= NULL:
for n from 1 to 100 do
t:= numtheory:-bigomega(n);
if not assigned(R[t]) then
A:= A,0;
R[t]:= 1;
else
A:= A, R[t];
R[t]:= R[t]+1;
fi
od:
A; # Robert Israel, Oct 24 2021
Table[Length[Select[Range[n - 1], PrimeOmega[#] == PrimeOmega[n] &]], {n, 80}]
a(n)={my(t=bigomega(n)); sum(k=1, n-1, bigomega(k)==t)} \\ Andrew Howroyd, Oct 31 2020
from math import prod, isqrt from sympy import isprime, primepi, primerange, integer_nthroot, primeomega def A335097(n): if n==1: return 0 if isprime(n): return primepi(n)-1 def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1))) return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,primeomega(n)))-1) # Chai Wah Wu, Aug 28 2024
Column n lists the a(n) positive integers less than n with more prime factors than n:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
---------------------------------------------------------------------
4 6 8 8 10 12 12 12 16 16 18 16
4 9 10 8 8 15 16
8 9 14 15
6 8 12 14
4 6 10 12
4 9 10
8 9
6 8
4 6
4
Table[Length[Select[Range[n],PrimeOmega[#]>PrimeOmega[n]&]],{n,100}]
Column n lists the a(n) positive integers less than n with more distinct prime factors than n:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
---------------------------------------------------------------------
6 6 6 10 12 15 15 18
6 10 14 14 15
6 12 12 14
10 10 12
6 6 10
6
b:= proc(n) option remember; nops(numtheory[factorset](n)) end:
a:= proc(n) option remember;
(t-> add(`if`(b(i)>t, 1, 0), i=1..n-1))(b(n))
end:
seq(a(n), n=1..100); # Alois P. Heinz, Dec 28 2018
Table[Length[Select[Range[n],PrimeNu[#]>PrimeNu[n]&]],{n,100}]
a(n) = my(omegan=omega(n)); sum(k=1, n-1, omega(k) > omegan); \\ Michel Marcus, Dec 29 2018
first(n) = {my(t = 1, pp = 1, res = vector(n)); forprime(p = 2, oo, pp*=p; if(pp > n, v = vector(t); break); t++); for(i = 1, n, o = omega(i); res[i] = v[o+1]; for(j = 1, o, v[j]++)); res} \\ David A. Corneth, Dec 29 2018
{x|x <= 6, A072047(x) = A072047(6) = 1} = {2,3,4,6}, therefore a(6) = 4.
{x|x <= 28, A072047(x) = A072047(28) = 3} = {19,28}, therefore a(28) = 2.
a340313 n = a340313_list !! (n-1)
a340313_list = repetitions a072047_list
where
repetitions [] = []
repetitions (a:as) = 1 : h a as (repetitions as)
h [] = []
h b (c:cs) (r:rs) = (if c == b then succ else id) r : h b cs rs
with(numtheory):
b:= proc(n) option remember; local k; if n=1 then 1 else
for k from 1+b(n-1) while not issqrfree(k) do od; k fi
end:
p:= proc() 0 end:
a:= proc(n) option remember; local h; a(n-1);
h:= bigomega(b(n)); p(h):= p(h)+1;
end: a(0):=0:
seq(a(n), n=1..100); # Alois P. Heinz, Jan 06 2021
b[n_] := b[n] = Module[{k}, If[n == 1, 1,
For[k = 1 + b[n - 1], !SquareFreeQ[k], k++]; k]];
p[_] = 0;
a[n_] := a[n] = Module[{h}, a[n - 1];
h = PrimeOmega[b[n]]; p[h] = p[h]+1];
a[0] = 0;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 28 2022, after Alois P. Heinz *)
first(n) = {v = vector(5); n--; res = vector(n); t = 0; for(i = 2, oo, f = factor(i)[,2]; if(vecmax(f) == 1, if(#f > #v, v = concat(v, vector(#f - #v)) ); t++; v[#f]++; res[t] = v[#f]; if(t >= n, return(concat(1, res)) ) ) ) } \\ David A. Corneth, Jan 07 2021
from math import isqrt, prod from sympy import primerange, integer_nthroot, mobius, primenu, primepi def A340313(n): if n == 1: return 1 def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1))) def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)) def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax kmax = bisection(f) return int(sum(primepi(kmax//prod(c[1] for c in a))-a[-1][0] for a in g(kmax,0,1,1,m)) if (m:=primenu(kmax)) > 1 else primepi(kmax)) # Chai Wah Wu, Aug 31 2024
for k = 1, 2, 3, 4, 5, 6, ...: p_k(3) = 5, 9, 18, 36, 72, 144, ... (the 3rd k-almost prime); p_k(3)^2 = 25, 81, 324, 1296, 5184, 20736, ...; b_3(k) = 9, 8, 8, 8, 8, 8, ... (index in the 2k-almost primes); so since b_3(3) = b_3(2) = 8, a(3) = 8.
get_p(m,k) = {local(i,n);i=0;n=1;while(iA209934(n) = {local(m,k,k_old);m=3;k_old=get_k(2,get_p(1,n)^2);k=get_k(4,get_p(2,n)^2);while(kMichael B. Porter, Mar 20 2012
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