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

4, 16, 19, 22, 24, 25, 34, 41, 70, 97, 119, 128, 138, 152, 155, 170, 174, 222, 232, 258, 285, 296

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Jun 01 2024

Keywords

• base
• nonn
• word

Comments

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.

This sequence is conjectured to be finite and full.

Examples

			4 has an imbalance of 4 units when written in French and cut between A and T:
Q+U+A = 39 and T+R+E = 43. We have indeed 43 - 39 = 4.
16 has an imbalance of 16 units when written in French and cut between E and I:
S+E = 24 and I+Z+E = 40. We have indeed 40 - 24 = 16.
The cuts in the integers below are made using this sign ——:
QUA——TRE (43——39 = 4)
SE——IZE (24——40 = 16)
DIXN——EUF (51——32 = 19)
VING——TDEUX (52——74 = 22)
VINGTQ——UATRE (89——65 = 24)
VIN——GTCINQ (45——70 = 25)
TRENTEQ——UATRE (99——65 = 34)
QUARA——NTEETUN (58——99 = 41)
SOIXANTE——DIX (107——37 = 70)
QUATR——EVINGTDIXSEPT (77——174 = 97)
C——ENTDIXNEUF (3——122 = 119)
CEN——TVINGTHUIT (22——150 = 128)
CEN——TTRENTEHUIT (22——160 = 138)
CENTCINQUANTEDEU——X (176——24 = 152)
CENTCINQUANTECIN——Q (172——17 = 155)
CE——NTSOIXANTEDIX (8——178 = 170)
CENTSOIXANTEQUATO——RZE (223——49 = 174)
DEUXCENTVINGTDEUX—— (222——0 = 222)
DEUXCENTTRENTEDEUX—— (232——0 = 232)
DEUXCENTCINQUANTEHUIT—— (258——0 = 258)
D——EUXCENTQUATREVINGTCINQ (4——289 = 285)
DE——UXCENTQUATREVINGTSEIZE (9——305 = 296).
No integer has been found that could be cut in more than one way.
		

Crossrefs

Cf. A104059, A373317 (English version), A083967.