cp's OEIS Frontend

A373321 Numbers which are their own imbalance when written in French (see the definition of the imbalance in the Comments section).

%I A373321 #9 Jun 03 2024 18:24:35
%S A373321 4,16,19,22,24,25,34,41,70,97,119,128,138,152,155,170,174,222,232,258,
%T A373321 285,296
%N A373321 Numbers which are their own imbalance when written in French (see the definition of the imbalance in the Comments section).
%C A373321 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.
%C A373321 This sequence is conjectured to be finite and full.
%e A373321 4 has an imbalance of 4 units when written in French and cut between A and T:
%e A373321 Q+U+A = 39 and T+R+E = 43. We have indeed 43 - 39 = 4.
%e A373321 16 has an imbalance of 16 units when written in French and cut between E and I:
%e A373321 S+E = 24 and I+Z+E = 40. We have indeed 40 - 24 = 16.
%e A373321 The cuts in the integers below are made using this sign ——:
%e A373321 QUA——TRE (43——39 = 4)
%e A373321 SE——IZE (24——40 = 16)
%e A373321 DIXN——EUF (51——32 = 19)
%e A373321 VING——TDEUX (52——74 = 22)
%e A373321 VINGTQ——UATRE (89——65 = 24)
%e A373321 VIN——GTCINQ (45——70 = 25)
%e A373321 TRENTEQ——UATRE (99——65 = 34)
%e A373321 QUARA——NTEETUN (58——99 = 41)
%e A373321 SOIXANTE——DIX (107——37 = 70)
%e A373321 QUATR——EVINGTDIXSEPT (77——174 = 97)
%e A373321 C——ENTDIXNEUF (3——122 = 119)
%e A373321 CEN——TVINGTHUIT (22——150 = 128)
%e A373321 CEN——TTRENTEHUIT (22——160 = 138)
%e A373321 CENTCINQUANTEDEU——X (176——24 = 152)
%e A373321 CENTCINQUANTECIN——Q (172——17 = 155)
%e A373321 CE——NTSOIXANTEDIX (8——178 = 170)
%e A373321 CENTSOIXANTEQUATO——RZE (223——49 = 174)
%e A373321 DEUXCENTVINGTDEUX—— (222——0 = 222)
%e A373321 DEUXCENTTRENTEDEUX—— (232——0 = 232)
%e A373321 DEUXCENTCINQUANTEHUIT—— (258——0 = 258)
%e A373321 D——EUXCENTQUATREVINGTCINQ (4——289 = 285)
%e A373321 DE——UXCENTQUATREVINGTSEIZE (9——305 = 296).
%e A373321 No integer has been found that could be cut in more than one way.
%Y A373321 Cf. A104059, A373317 (English version), A083967.
%K A373321 base,nonn,word
%O A373321 1,1
%A A373321 _Eric Angelini_ and _Jean-Marc Falcoz_, Jun 01 2024