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A339144
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a(n) is the smallest positive integer such that n*a(n) contains n+a(n) as a substring. If no such number exists then a(n) = -1.
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9
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-1, 2, -1, 68, -1, -1, -1, 44, -1, 890, 110, 60, 44, 35, 30, 27, 25, 23, 22, 20, 929, 19, 18, 88, 17, -1, 16, 16, 68, 15, 15, 60, 58, 56, 14, 14, 14, 371, 48, 360, 336, 562, 9104, 8, 13, 13, 283, 39, 269, 450, 37, 452, 245, 18, 679, 34, 225, 33, 2053, 12, 12, 12, 12, 12, 30, 369, 889, 4, 16961
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OFFSET
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1,2
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COMMENTS
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For n = 1, 3, 5, 6, 7, 9, 26 no value has been found for which n*a(n) contains n + a(n) as a substring (obviously true for n = 1) for a(n) up to 5x10^10. It is likely, although unproven, that this is the complete list of values for which a(n) = -1.
The sequence values display erratic behavior. Most of the term values appear random but there are ranges of n values with the same value. The largest such range for the first one million terms is a(501000) to a(501499), 500 terms, all of which have a(n) = 1002. Patterns also appear for n value corresponding to multiples of powers-of-ten. For example if n=10^k then a(n) = 89*10^k. The largest value in the first one million terms is a(554635) = 879948670.
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LINKS
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EXAMPLE
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a(2) = 2 as 2*2 = 4 which contains 2 + 2 = 4 as a substring.
a(4) = 68 as 4*68 = 272 which contains 4+68 = 72 as a substring.
a(69) = 16961 as 69*16961 = 1170309 which contains 69+16961 = 17030 as a substring.
a(501000) = 1002 as 501000*1002 = 502002000 which contains 501000+1002 = 502002 as a substring. This is the first of 500 consecutive terms with a(n) = 1002.
a(554635) = 879948670 as 554635*879948670 = 488050330585450 which contain 554635+879948670 = 880503305 as a substring. This is the largest value of a(n) for the first one million terms.
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PROG
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(PARI) isok(n, k) = #strsplit(Str(n*k), Str(n+k)) > 1;
a(n) = {if (vecsearch([1, 3, 5, 6, 7, 9, 26], n), return (-1)); my(k=1); while (! isok(k, n), k++); k; } \\ Michel Marcus, Dec 02 2020 and Jan 23 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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