A165411
Primes p such that each of p's digits d appears consecutively exactly d times and p contains each nonzero digit up to its maximum digit.
Original entry on oeis.org
223331, 122555554444333, 224444333555551, 224444555553331, 225555544441333, 333555554444221, 555552233344441, 555552244441333, 555554444221333, 122444455555666666333, 122555554444666666333, 144446666662255555333
Offset: 1
1333444455555226666667777777 is a term because it is a prime meeting the criteria: It contains all digits 1 through 7, its maximum, each appearing in a single run of length equal to the value of the digit.
A247700
Numbers which have d digits "d", whenever one of their digits is "d", ordered by largest digit, then by size of the number.
Original entry on oeis.org
1, 22, 122, 212, 221, 333, 1333, 3133, 3313, 3331, 22333, 23233, 23323, 23332, 32233, 32323, 32332, 33223, 33232, 33322, 122333, 123233, 123323, 123332, 132233, 132323, 132332, 133223, 133232, 133322, 212333, 213233, 213323, 213332
Offset: 1
In base 2, the only number with this property is a(1) = 1.
In base 3, this property is again satisfied by 1, but also by the 4 additional terms 22, 122, 212 and 221. They are listed "as such" (without conversion from base 3 to base 10) as a(2),...,a(5).
In base 4, there are 75 more terms (involving three digits "3"), listed as a(6),...,a(80).
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a=[];for(d=1,3,n=[10^d\9*d]; for(i=1,#a,t=vector(d+#s=digits(a[i]),j,10^j)~\10;forvec(v=vector(d,j,[1,#t]),c=0;n=concat(n,vector(#t,j,if(setsearch(v,j),d,s[c++]))*t),2));a=concat(a,vecsort(n)));a \\ M. F. Hasler, Sep 25 2014
A356369
Numbers such that each digit "d" occurs d times, for every digit from 1 to the largest digit.
Original entry on oeis.org
1, 122, 212, 221, 122333, 123233, 123323, 123332, 132233, 132323, 132332, 133223, 133232, 133322, 212333, 213233, 213323, 213332, 221333, 223133, 223313, 223331, 231233, 231323, 231332, 232133, 232313, 232331, 233123, 233132, 233213, 233231, 233312, 233321, 312233, 312323
Offset: 1
213323 is a term because the digit 1 occurs once, the digit 2 twice and 3 three times. Every digit from 1 to 3 is present.
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from itertools import islice
from sympy.utilities.iterables import multiset_permutations
def agen():
for m in range(1, 10):
s = "".join(str(k)*k for k in range(1, m+1))
yield from (int("".join(p)) for p in multiset_permutations(s))
print(list(islice(agen(), 65))) # Michael S. Branicky, Oct 17 2022
A247701
Numbers whose digits are nondecreasing and which have exactly d digits "d" whenever there is at least one digit "d".
Original entry on oeis.org
1, 22, 122, 333, 1333, 4444, 14444, 22333, 55555, 122333, 155555, 224444, 666666, 1224444, 1666666, 2255555, 3334444, 7777777, 12255555, 13334444, 17777777, 22666666, 33355555, 88888888, 122666666, 133355555, 188888888, 223334444, 227777777, 333666666, 999999999
Offset: 1
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a=[]; N=9; for(m=1,min(N,9), a=concat(a,n=10^m\9*m); for(i=1,#a-1, #Str(a[i])>N-m && break; a=concat(a,a[i]*10^m+n))); Set(a)
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