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A322002
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Roots of self-stuffable numbers A322323.
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2
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2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 16, 18, 22, 24, 25, 36, 37, 44, 45, 48, 63, 64, 65, 66, 72, 75, 88, 96, 108, 125, 126, 128, 138, 143, 144, 165, 225, 231, 275, 288, 297, 333, 375, 404, 444, 549, 576, 625, 666, 765, 768, 777, 803, 808, 825, 999, 1026, 1125, 1314, 1408, 1485, 1728, 1875, 2048, 2088, 2178, 2455, 2628, 2688, 2907, 3123
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OFFSET
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1,1
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COMMENTS
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In A322323 it is shown that the set of these self-stuffable numbers is the union of infinite subsequences produced from the roots M listed here, as follows: If E(M) = A007953(M) - A055642(M) = sumdigits(M) - #digits(M) > 0, then N = M*10^E(M) has a number of digits equal to its sum of digits, L = A007953(N) = A055642(N). Then S(N) is well defined, where S(N) is the number obtained from N by inserting d[i] spaces after each nonzero digit d[i] of N except for the last digit, and then filling these spaces by a second copy of the digits of N. (More generally, this S(x) is well defined iff E(x) = A010879(x).) If additionally N | S(N), then N is in A322323, a self-stuffable number. Moreover, all numbers N_m = Sum_{k=0..2m} 10^(k*L)*N, m >= 0, have the same property. If 10*M leads to such an N, then M leads to the same N. Therefore we call roots and list here only those M leading to self-stuffable N which are not divisible by 10.
a(13) = 22 and a(31) = 126 are the only known cases of primitive roots M with the additional property that E(M) = A010879(M), the last digit of M, in which case all M_m = Sum_{k=0..2m} 10^(k*L)*M, m >= 0, are also self-stuffable (if M | S(M)). A322323 is the union of the sets {N_m, m >= 0}, and {M_m, m >= 0} if E(M) = A010879(M), where M runs over all terms listed here. [Edited by M. F. Hasler, Dec 19 2018]
Lars Blomberg found another primitive (see A322323 for definition) root 21021021021021021021 that is self-stuffable to join the previously known examples 22 and 126. - Ray Chandler, Jan 02 2019
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LINKS
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FORMULA
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{ M > 0 | M != 0 (mod 10), E(M) > 0 and M*10^E(M) | S(M*10^E(M)) } with E(M) = A007953(M) - A055642(M) and S(x) defined in COMMENTS.
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EXAMPLE
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The root a(1) = 2 has E(2) = 2 - 1 = 1 != 2 so it is not a self-stuffable number itself, but any odd number of concatenations of 2*10^E(2) = 20, 202020, 2020202020, ... is in A322323.
Similarly, a(2) = 3 has E(3) = 3 - 1 = 2 != 3, so (300, 300300300, ...) is a subsequence of A322323.
a(13) = 22 has E(22) = 4 - 2 = 2 = last digit of 22 and M = 22 | 2222 = S(M), so not only (2200, 220022002200, ...) but also (22, 2200220022, 220022002200220022, ...) is a subsequence of A322323.
a(31) = 126 has E(126) = 9 - 3 = 6 = last digit of 126 and M = 126 | 112266 = S(M), so not only (126000000, 126000000126000000126000000, ...) but also (126, 126000000126000000126, ...) is a subsequence of A322323.
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PROG
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(PARI) is_A322002(n)={my(c=0, d=digits(n), e=vecsum(d)-#d); d[#d] && e>0 && fromdigits(concat(vector(#d, i, vector(d[i]+1, k, if(k==1, d[i], c<#d, d[c+=1])))))%n==0}
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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