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

1,2

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.)

Links

Table of n, a(n) for n=1..57.

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

Example

14401 is not a term, but 14411 is.

Prog

(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
(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

Crossrefs

Cf. A000290, A036435, A018851.

Sequence in context: A010396 A010431 A010438 * A005241 A199595 A028845

Adjacent sequences: A357471 A357472 A357473 * A357475 A357476 A357477

Keyword

nonn,easy,base

Author

Freddy Barrera, Sep 29 2022

Status

approved