A357474 Squarely correct numbers.

1, 4, 9, 11, 14, 16, 19, 25, 36, 41, 44, 49, 64, 81, 91, 94, 99, 100, 111, 114, 116, 119, 121, 125, 136, 141, 144, 149, 161, 164, 169, 181, 191, 194, 196, 199, 225, 251, 254, 256, 259, 289, 324, 361, 364, 369, 400, 411, 414, 416, 419, 425, 436, 441, 444, 449, 464

Offset: 1

Author

Freddy Barrera, Sep 29 2022

Keywords

• nonn
• easy
• base

Comments

A positive integer is a squarely correct number if its base-10 representation consists entirely of one or more adjacent blocks of digits that are positive perfect squares with no leading zeros. (See George Berzsenyi link below.)

Examples

			14401 is not a term, but 14411 is.
		

Links

• George Berzsenyi, Consecutively and Squarely Correct (P439), Math Horizons, Vol. 30, Issue 1, The Playground, August 2022.

Crossrefs

Cf. A000290, A036435, A018851.

Programs

PARI

is(n)=if(n<11, issquare(n), n%10, my(d=digits(n)); for(i=1,#d, if(issquare(n%10^i) && d[#d+1-i] && is(n\10^i), return(1))); 0, my(k=valuation(n,10)); k%2==0 && is(n/10^k)) \\ Charles R Greathouse IV, Oct 04 2022

Python

from sympy.ntheory.primetest import is_square
def ok(n):
    if is_square(n): return True
    s = str(n)
    return any(s[i]!="0" and ok(int(s[:i])) and ok(int(s[i:])) for i in range(1, len(s)))
print([k for k in range(1, 465) if ok(k)]) # Michael S. Branicky, Oct 03 2022

Formula

5n/3 < a(n) << n^k for n > 1, where k = log 10/log 3 = 2.0959.... - Charles R Greathouse IV, Oct 03 2022