A046837 Internal digits of k^2 include digits of k as substring, k does not end in 0.

3792, 14651, 495475, 505025, 971582, 1713526, 4115964, 5133355, 6933808, 10050125, 48588526, 50050025, 66952741, 88027284, 88819024, 105124922, 146509717, 177656344, 200500625, 250050005, 370156212, 753348181, 758872344, 1000500125, 3199268655, 4005003125, 4935640724, 4999504975

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base

Comments

From David A. Corneth, Oct 10 2023, Oct 12 2023: (Start)

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.

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.

If k has q digits then k^2 has 2*q or 2*q-1 digits.

We now can deploy two stategies to find a term k.

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.

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.

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.

Alternatively 3792 is the only solution with 4 digits such that k*(k - 100) mod 10^6 is < 10^2.

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)

Examples

			3792^2 is in the sequence as 3792^2 = 14379264 contains 3792 in its decimal expansion. - _David A. Corneth_, Oct 10 2023
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..89 (terms <= 10^15)
• David A. Corneth, PARI program

Crossrefs

Cf. A046835.

Programs

PARI

\\ See PARI link David A. Corneth, Oct 10 2023

Extensions

More terms from David A. Corneth, Aug 29 2023, Oct 10 2023