[seqfan] Re: Concatenate the sums of the neighboring digits

[Hans Havermann]

HH: "Roughly (I just iterated numbers 50 times and noted large final terms): ..."

The 109 terms I provided may be said to increase without limit on the basis of the existence in each number's repeated application of the map defined in A053392 a term containing a strictly internal substring of more than two adjacent 9's. Here is a table:

1496: a(41)  <9089>   999  (7845,7847), (8163,8165)
1497: a(32)  <1019>  9999  (720,723)
1498: a(32)  <1032>  9999  (720,723)
1499: a(31)   <771>   999  (496,498)
1587: a(36)  <2412>   999  (753,755)
1588: a(36)  <2412>   999  (753,755)
1589: a(36)  <2412>   999  (753,755)
1691: a(34)  <1032>  9999  (720,723)
1692: a(27)    <96>   999  (86,88)
1693: a(27)    <96>   999  (86,88)
1694: a(41)  <7745>   999  (6206,6208)
1695: a(41)  <7773>   999  (6206,6208)
1696: a(38)  <3682>   999  (3607,3609)
1697: a(35)  <1757>   999  (1684,1686)
1698: a(32)   <880>   999  (797,799)
1699: a(32)   <876>   999  (797,799)
1719: a(36)  <1032>  9999  (720,723)
1728: a(31)    <96>   999  (86,88)
1729: a(31)    <96>   999  (86,88)
1783: a(37)   <175>   999  (35,37)
1784: a(39)   <520>  9999  (83,86);  999  (409,411)
1785: a(28)   <118>   999  (73,75)
1786: a(28)   <159>   999  (73,75)
1787: a(28)   <161>   999  (73,75)
1788: a(39)  <3138>   999  (1223,1225)
1789: a(36)  <1436>   999  (1430,1432)
1791: a(36)  <2744>  9999  (720,723);  999  (2389,2391)
1792: a(36)  <3056>  9999  (720,723);  999  (2084,2086)
1793: a(32)   <876>   999  (797,799)
1794: a(32)   <894>   999  (797,799)
1795: a(32)   <894>   999  (797,799)
1796: a(40)  <8067>   999  (6261,6263), (6474,6476)
1797: a(36)  <3726>   999  (3706,3708)
1798: a(37)  <4948>  9999  (4792,4795)
1799: a(39)  <7721>   999  (6206,6208)
1819: a(56)  <2835>   999  (1034,1036)
1867: a(48)   <853>   999  (203,205)
1868: a(34)   <880>  9999  (227,230)
1869: a(34)   <880>  9999  (227,230)
1874: a(38)  <1373>   999  (496,498)
1875: a(37)  <1722>   999  (639,641)
1876: a(33)  <1166>   999  (1146,1148)
1877: a(34)  <1618>  9999  (1462,1465)
1878: a(36)  <2259>   999  (744,746)
1879: a(36)  <2283>   999  (744,746)
1880: a(31)   <118>   999  (73,75)
1881: a(31)   <159>   999  (73,75)
1882: a(26)   <183>   999  (104,106)
1883: a(25)   <165>   999  (105,107)
1884: a(26)   <220>   999  (149,151)
1885: a(21)    <64>   999  (55,57)
1886: a(21)    <64>   999  (55,57)
1887: a(30)   <711>   999  (514,516)
1888: a(31)   <829>   999  (667,669)
1889: a(31)   <829>   999  (667,669)
1890: a(30)   <810>   999  (496,498)
1891: a(37)  <9089>   999  (7845,7847), (8163,8165)
1892: a(28)  <1019>  9999  (720,723)
1893: a(28)  <1032>  9999  (720,723)
1894: a(27)   <771>   999  (496,498)
1895: a(27)   <772>   999  (496,498)
1896: a(27)   <772>   999  (496,498)
1897: a(27)   <810>   999  (496,498)
1898: a(27)   <854>   999  (496,498)
1899: a(27)   <857>   999  (496,498)
1955: a(41)  <2412>   999  (753,755)
1956: a(41)  <2412>   999  (753,755)
1957: a(41)  <2412>   999  (753,755)
1958: a(40)  <1513>   999  (414,416), (1305,1307)
1959: a(27)    <83>   999  (12,14)
1964: a(40)  <2510>   999  (1923,1925)
1965: a(32)  <1029>   999  (926,928)
1966: a(37)  <3664>   999  (3266,3268)
1967: a(36)  <2844>   999  (1248,1250)
1968: a(39)  <7292>   999  (431,433)
1969: a(35)  <2777>  9999  (2478,2481)
1970: a(27)   <123>   999  (56,58)
1971: a(27)   <125>   999  (56,58)
1972: a(27)   <128>   999  (56,58)
1973: a(28)   <323>   999  (212,214)
1974: a(28)   <377>   999  (212,214)
1975: a(24)   <123>   999  (56,58)
1976: a(24)   <125>   999  (56,58)
1977: a(24)   <128>   999  (56,58)
1978: a(33)  <1335>   999  (696,698), (900,902)
1979: a(33)  <1417>   999  (696,698)
1980: a(35)  <2412>   999  (753,755)
1981: a(35)  <2412>   999  (753,755)
1982: a(27)   <314>   999  (145,147)
1983: a(28)   <549>   999  (470,472)
1984: a(27)   <451>   999  (391,393)
1985: a(28)   <577>   999  (506,508)
1986: a(23)   <165>   999  (156,158)
1987: a(23)   <165>   999  (156,158)
1988: a(32)  <1736>   999  (417,419), (1539,1541)
1989: a(32)  <1681>   999  (417,419)
1991: a(27)   <628>   999  (202,204)
1992: a(27)   <628>   999  (202,204)
1993: a(27)   <628>   999  (202,204)
1994: a(27)   <628>   999  (202,204)
1995: a(27)   <628>   999  (202,204)
1996: a(27)   <628>   999  (202,204)
1997: a(27)   <628>   999  (202,204)
1998: a(27)   <628>   999  (202,204)
1999: a(27)   <628>   999  (202,204)
2496: a(36)  <2412>   999  (753,755)
2497: a(36)  <2412>   999  (753,755)
2498: a(36)  <2412>   999  (753,755)
2499: a(36)  <2412>   999  (753,755)

Each number is followed by the first a(n) containing more than two adjacent 9's. This is followed by that term's number of decimal digits. Finally, the adjacent-9's substring(s) and its/their position(s) counting from the left. Hopefully there's no mistakes.