cp's OEIS Frontend

A373317 Numbers which are their own imbalance when written in English (see the definition of the imbalance in the Comments section).

%I A373317 #32 Jun 03 2024 18:24:23
%S A373317 17,25,100,115,116,144,157,182,184,186,191,193,199,223,236
%N A373317 Numbers which are their own imbalance when written in English (see the definition of the imbalance in the Comments section).
%C A373317 Take a number. Write this number in capital letters. Remove any spaces or hyphens to form a single block of contiguous letters. Cut this block into two parts using a vertical line which is tangent to at least one of the letters. The word ONE, for example, admits four cuts, the first before the O, the second between the O and the N, the third between the N and the E, the fourth after the E. These respective cuts produce the four pairs of subsets of following letters: —/ONE, O/NE, ON/E, ONE/—. A weight is now assigned to each subset. To do this, we replace each letter in the subset by its rank in the alphabet. Then we add up these ranks within the subset. The weights of the four successive pairs above are therefore {0;34}, {15;19}, {29;5}, {34;0}. The absolute difference of the two numbers that form such a pair is called imbalance. The successive imbalances in the example here are 34, 4, 24 and 34. None is equal to 1 ("ONE"). Thus 1 is not in the sequence.
%C A373317 This sequence is conjectured to be finite and full.
%C A373317 It is true that there are no more terms. Computationally, there are none < 10^8. Let v(n) be the sum of the valuations of all the letters in the name of n.  Terms in the sequence require n < v(n). Now note that if 1000^(e-1) <= n < 1000^e, then v(n) <= (314+216)*e, where the loose upper bound comes from "sevenhundredseventyseven" and "quattuordecillion" considering all terms up to 10^66 using English names of large numbers (see Wikipedia link; similar bounds can be derived for extended naming schemes). Thus, (i) v(n) <= 530*e <= 1000^(e-1) for e >= 3; and (ii) for e < 3, v(n) <= 314 + (314+102) + (314+84) = 1128, where 102 and 84 come from "thousand" and "million", respectively. Bound (i) shows there can be no terms >= 10^6; (ii) that there are no terms >= 1128; indeed, (ii) can be extended down to show there are no terms >= 314). - _Michael S. Branicky_, Jun 03 2024
%H A373317 Wikipedia, <a href="https://en.wikipedia.org/wiki/Names_of_large_numbers">Names of Large Numbers</a>
%e A373317 17 has an imbalance of 17 units when cut between V and E:
%e A373317 S+E+V = 46 and E+N+T+E+E+N = 63. We have indeed 63 - 46 = 17.
%e A373317 25 has an imbalance of 25 units when cut between N and T:
%e A373317 T+W+E+N = 62 and T+Y+F+I+V+E = 87. We have indeed 87 - 62 = 25.
%e A373317 The cuts in the integers below are made using the sign ——:
%e A373317 SEV——ENTEEN (46——63 = 17)
%e A373317 TWEN——TYFIVE (87——62 = 25)
%e A373317 ONEHUNDRE——D (104——4 = 100)
%e A373317 ON——EHUNDREDFIFTEEN (29——144 = 115)
%e A373317 ONEHUNDREDSIX——TEEN (160——44 = 116)
%e A373317 ONEHUNDREDFORTYF——OUR (198——54 = 144)
%e A373317 ONEHUNDREDFIFTYSE——VEN (198——41 = 157)
%e A373317 ON——EHUNDREDEIGHTYTWO (29——211 = 182)
%e A373317 ON——EHUNDREDEIGHTYFOUR (29——213 = 184)
%e A373317 ONEHUNDREDEIGHTYSI——X (210——24 = 186)
%e A373317 ONEHUNDREDNINETYO——NE (210——19 = 191)
%e A373317 ON——EHUNDREDNINETYTHREE (29——222 = 193)
%e A373317 ONEHUNDREDNINETYNI——NE (218——19 = 199)
%e A373317 TWOHUNDREDTWENTYT——HREE (259——36 = 223)
%e A373317 TWOHUNDREDTHIRTYSI——X (260——24 = 236).
%e A373317 No integer has been found that could be cut in more than one way.
%o A373317 (Python)
%o A373317 from num2words import num2words
%o A373317 def n2w(n): return "".join(c for c in num2words(n).replace(" and", "") if c.isalpha())
%o A373317 def ok(n):
%o A373317     d = [ord(c) - ord('A') + 1 for c in n2w(n).upper()]
%o A373317     if sum(d) < n: return False
%o A373317     return any(abs(sum(d[:i])-sum(d[i:])) == n for i in range(1, len(d)))
%o A373317 print([k for k in range(1005) if ok(k)]) # _Michael S. Branicky_, Jun 03 2024
%Y A373317 Cf. A104059, A373321 (French version), A083967, A372222.
%K A373317 base,nonn,word,fini,full
%O A373317 1,1
%A A373317 _Eric Angelini_ and _Jean-Marc Falcoz_, Jun 01 2024