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.
%I A091538 #35 Jul 27 2025 10:32:17
%S A091538 1,0,2,0,3,4,0,5,6,8,0,7,9,12,16,0,11,10,18,24,32,0,13,14,20,36,48,64,
%T A091538 0,17,15,27,40,72,96,128,0,19,21,28,54,80,144,192,256,0,23,22,30,56,
%U A091538 108,160,288,384,512,0,29,25,42,60,112,216,320,576,768,1024
%N A091538 Triangle built from m-primes as columns.
%C A091538 m-primes (also called m-almost primes) are the numbers which have precisely m prime factors counting multiple factors. 1 is included as 0-prime.
%C A091538 The number N>=1 appears in column no. m = A001222(N).
%H A091538 Chai Wah Wu, <a href="/A091538/b091538.txt">Table of n, a(n) for n = 0..10000</a>
%H A091538 Wolfdieter Lang, <a href="/A091538/a091538.txt">First 11 rows</a>.
%F A091538 For n>=m>=1: a(n, m)= (n-m+1)-th member in the strictly monotonically increasing sequence of numbers N satisfying: N=product(p(k)^(e_k), k=1..) with p(k) := A000040(k) (k-th prime) such that sum(e_k, k=1..) = m, where the e_k are nonnegative. if m=0 : a(n, 0)=1 if n=0 else 0. If n<m then a(n, m)=0.
%e A091538 From _Michael De Vlieger_, May 24 2017: (Start)
%e A091538 Chart a(n,m) read by antidiagonals:
%e A091538 n | m ->
%e A091538 ------------------------------------------------
%e A091538 0 | 1 0 0 0 0 0 0 ... (A000007)
%e A091538 1 | 2 3 5 7 11 13 17 (A000040)
%e A091538 2 | 4 6 9 10 14 15 21 (A001358)
%e A091538 3 | 8 12 18 20 27 28 30 (A014612)
%e A091538 4 | 16 24 36 40 54 56 60 (A014613)
%e A091538 5 | 32 48 72 80 108 112 120 (A014614)
%e A091538 6 | 64 96 144 160 216 224 240 (A046306)
%e A091538 7 | 128 192 288 320 432 448 480 (A046308)
%e A091538 8 | 256 384 576 640 864 896 960 (A046310)
%e A091538 ...
%e A091538 Triangle begins:
%e A091538 0 | 1
%e A091538 1 | 0 2
%e A091538 2 | 0 3 4
%e A091538 3 | 0 5 6 8
%e A091538 4 | 0 7 9 12 16
%e A091538 5 | 0 11 10 18 24 32
%e A091538 6 | 0 13 14 20 36 48 64
%e A091538 7 | 0 17 15 27 40 72 96 128
%e A091538 8 | 0 19 21 28 54 80 144 192 256
%e A091538 ...
%e A091538 (End)
%t A091538 With[{nn = 11}, Function[s, Function[t, Table[Function[m, If[m == 1, Boole[k == 1], t[[m, k]]]][n - k + 1], {n, nn}, {k, n, 1, -1}]]@ Map[Position[s, #][[All, 1]] &, Range[0, nn]]]@ PrimeOmega@ Range[2^nn]] (* or *)
%t A091538 a = {1}; Do[Block[{r = {Prime@ n}}, Do[AppendTo[r, SelectFirst[ Range[a[[-(n - i)]] + 1, 2^n], PrimeOmega@ # == i &]], {i, 2, n - 1}]; a = Join[a, {0}, If[n == 1, {}, r], {2^n}]], {n, 11}]; a (* _Michael De Vlieger_, May 24 2017 *)
%o A091538 (Python)
%o A091538 from math import isqrt, comb, prod
%o A091538 from sympy import prime, primerange, integer_nthroot, primepi
%o A091538 def A091538(n):
%o A091538 a = (m:=isqrt(k:=n+1<<1))+(k>m*(m+1))
%o A091538 r = n-comb(a,2)
%o A091538 w = a-r
%o A091538 if r==0: return int(w==1)
%o A091538 if r==1: return prime(w)
%o A091538 def bisection(f,kmin=0,kmax=1):
%o A091538 while f(kmax) > kmax: kmax <<= 1
%o A091538 kmin = kmax >> 1
%o A091538 while kmax-kmin > 1:
%o A091538 kmid = kmax+kmin>>1
%o A091538 if f(kmid) <= kmid:
%o A091538 kmax = kmid
%o A091538 else:
%o A091538 kmin = kmid
%o A091538 return kmax
%o A091538 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)))
%o A091538 def f(x): return int(w+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,r)))
%o A091538 return bisection(f,w,w) # _Chai Wah Wu_, Jun 11 2025
%Y A091538 The column sequences (without leading zeros) are: A000007, A000040 (primes), A001358, A014612-4, A046306, A046308, A046310, A046312, A046314, A069272-A069281 for m=0..20, respectively.
%Y A091538 A078840 is this table with the zeros omitted.
%K A091538 nonn,easy,tabl
%O A091538 0,3
%A A091538 _Wolfdieter Lang_, Feb 13 2004