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Roots of self-stuffable numbers A322323.
2

%I #52 Mar 12 2019 03:58:32

%S 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,

%T 75,88,96,108,125,126,128,138,143,144,165,225,231,275,288,297,333,375,

%U 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

%N Roots of self-stuffable numbers A322323.

%C Numbers m, not divisible by 10, such that for some k, m*10^k is in A322323. - _David A. Corneth_, Dec 09 2018, corrected by _M. F. Hasler_, Dec 19 2018

%C 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.

%C 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]

%C 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

%H Ray Chandler, <a href="/A322002/b322002.txt">Table of n, a(n) for n = 1..413</a>

%H Ray Chandler, <a href="/A322002/a322002.txt">Some known self-stuffable primitive roots</a>

%F { 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.

%e 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.

%e Similarly, a(2) = 3 has E(3) = 3 - 1 = 2 != 3, so (300, 300300300, ...) is a subsequence of A322323.

%e 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.

%e 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.

%o (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}

%Y Cf. A322323, A007953, A010879, A055642.

%K nonn,base

%O 1,1

%A _M. F. Hasler_, Dec 09 2018