This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A046837 #31 Aug 15 2025 21:04:53 %S A046837 3792,14651,495475,505025,971582,1713526,4115964,5133355,6933808, %T A046837 10050125,48588526,50050025,66952741,88027284,88819024,105124922, %U A046837 146509717,177656344,200500625,250050005,370156212,753348181,758872344,1000500125,3199268655,4005003125,4935640724,4999504975 %N A046837 Internal digits of k^2 include digits of k as substring, k does not end in 0. %C A046837 From _David A. Corneth_, Oct 10 2023, Oct 12 2023: (Start) %C A046837 If k is a term and has q digits then k * (k - 10^t) has a string of q consective 0 for some t > 0 such that there are t digits after 'k' in k^2. This might ease the search for terms. %C A046837 For example 3792^2 = 14379264 so after 3792 there are 2 digits so t = 2. Then looking at 3792 * (3792 - 10^2) = 14000064 we see 4 consecutive zeros. %C A046837 If k has q digits then k^2 has 2*q or 2*q-1 digits. %C A046837 We now can deploy two stategies to find a term k. %C A046837 1. Solve k * (k - 10^t) >= m*10^(q + t) for the smallest such integer k where the number of digits of m is 2*q - (q+t) = q-t or 2*q - 1 - (q+t) = q-t-1. %C A046837 2. Find k such that k * (k - 10^t) mod 10^(q + t) < 10^t if they exist. Depending on q and t one might prefer to use one method over the other. %C A046837 Applied to the case k = 3792 we find that it is an example for smallest k such that k * (k - 10^2) >= 14*10^(4 + 2) that k is such that k * (k - 10^2) mod 10^(4+2) < 10^2. %C A046837 Alternatively 3792 is the only solution with 4 digits such that k*(k - 100) mod 10^6 is < 10^2. %C A046837 This sequence is infinite as it contains, among other families, 10^(2*n + 3) + 5 * 10^(n + 2) + 5^3, i.e., 10050125, 1000500125, 100005000125,... (End) %H A046837 David A. Corneth, <a href="/A046837/b046837.txt">Table of n, a(n) for n = 1..89</a> (terms <= 10^15) %H A046837 David A. Corneth, <a href="/A046837/a046837.gp.txt">PARI program</a> %e A046837 3792^2 is in the sequence as 3792^2 = 14379264 contains 3792 in its decimal expansion. - _David A. Corneth_, Oct 10 2023 %o A046837 (PARI) \\ See PARI link _David A. Corneth_, Oct 10 2023 %Y A046837 Cf. A046835. %K A046837 nonn,base %O A046837 1,1 %A A046837 _David W. Wilson_ %E A046837 More terms from _David A. Corneth_, Aug 29 2023, Oct 10 2023