A348266
k-digit numbers whose digit(s) are the number of distinct prime factors in each of the preceding k integers.
Original entry on oeis.org
22, 313, 2232, 2323, 2333, 32215, 432152, 2434332, 4222423, 43332543, 332325334, 2535332433, 4532543535234, 5435433351423
Offset: 1
22 is a term because omega(20) = 2 and omega(21) = 2, whose concatenation is 22.
313 is a term because preceding it omega(310) = 3, omega(311) = 1 and omega(312) = 3, and their concatenation is 313.
32215 is a term because, the number of distinct prime divisors of 32210, 32211, 32212, 32213 and 32214 are 3, 2, 2, 1, 5 and their ordered concatenation gives the next number 32215.
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Select[Range[33000], FromDigits[PrimeNu /@ (# - Range[IntegerLength[#], 1, -1])] == # &] (* Amiram Eldar, Oct 09 2021 *)
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isok(m) = {my(s="", k=m, i=1); while(1, s = concat(s, Str(omega(k))); if (eval(s) == m+i, return (i)); if (eval(s) > m+i, return(0)); k++; i++;);}
lista(nn) = my(nb); for(n=1, nn, if (nb=isok(n), print1(n+nb, ", "))); \\ Michel Marcus, Oct 09 2021
A323084
k-digit numbers whose digit(s) are the number of distinct prime factors (with multiplicity) in each of the following k integers.
Original entry on oeis.org
0, 1, 12, 21, 4224, 153426, 442451, 471614, 523291, 4336232, 474335342, 3624263478, 36443455382, 244936365228, 452527642826, 593326437534, 4372566243537
Offset: 1
4224 is a term because 4224 is a 4-digit number whose digits (4,2,2,4) are the number of prime factors (with multiplicity) in each of the following 4 integers; i.e., 4225 = 5^2*13^2 (four prime factors), 4226 = 2*2113 (two prime factors), 4227 = 3*1409 (two prime factors), and 4228 = 2^2*7*151 (four prime factors), therefore (4,2,2,4) -> 4224.
A348187
Integers m with k digits (for some k) that lie in an interval of k integers and the digits of m are the total number of distinct prime factors of all the integers in that interval.
Original entry on oeis.org
12, 22, 31, 122, 132, 312, 322, 2323, 3312, 14421, 23222, 24243, 33333, 51243, 333424, 342332, 432241, 523233, 1333232, 1432243, 2424341, 2442253, 5134334, 15232343, 24243232, 24424243, 25514234, 26134354, 32334533, 33252335, 33341415, 33343412, 34332425, 43523432, 53224343
Offset: 1
12 is a term because omega([11, 12]) gives [1, 2], the digits of 12.
33333 is a term because omega([33332, 33333, 33334, 33335, 33336]) but also omega([33333, 33334, 33335, 33336, 33337]) both give [3, 3, 3, 3, 3].
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vecn(n) = {my(list = List()); for (k=10^(n-1), 10^n-1-n, my(w = apply(omega, vector(n, i, k+i-1))); my(m = fromdigits(w)); if ((m>=k) && (m<=k+n-1), listput(list, m));); Set(list);}
lista(nn) = {my(list = List()); for (n=1, nn, my(w=vecn(n)); for (k=1, #w, listput(list, w[k]));); Set(list);}
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