A370362 Numbers k such that any two consecutive decimal digits of k^2 differ by 1 after arranging the digits in decreasing order.

0, 1, 2, 3, 18, 24, 66, 74, 152, 179, 3678, 3698, 4175, 4616, 5904, 5968, 6596, 7532, 8082, 8559, 9024, 10128, 10278, 11826, 12363, 12543, 12582, 13278, 13434, 13545, 13698, 14442, 14676, 14766, 15681, 15963, 16854, 17529, 17778, 18072, 19023, 19377, 19569, 19629

Offset: 1

Author

Jianing Song, Feb 16 2024

Keywords

• nonn
• base
• fini
• full

Comments

Numbers k such that k^2 is in A215014. There are 160 terms in this sequence.

Examples

			18^2 = 324 consists of the consecutive digits 2, 3 and 4;
24^2 = 576 consists of the consecutive digits 5, 6 and 7;
66^2 = 4356 consists of the consecutive digits 3, 4, 5 and 6;
74^2 = 5476 consists of the consecutive digits 4, 5, 6 and 7.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..160

Crossrefs

Cf. A215014, A370370. Supersequence of A156977.

The actual squares are given by A370610.

Programs

PARI

isconsecutive(m, {b=10})=my(v=vecsort(digits(m, b))); for(i=2, #v, if(v[i]!=1+v[i-1], return(0))); 1 \\ isconsecutive(k, b) == 1 if and only if any two consecutive digits of the base-n expansion of m differ by 1 after arranging the digits in decreasing order
a(n) = isconsecutive(n^2)

Python

from math import isqrt
from sympy.ntheory import digits
def afull(): return([i for i in range(isqrt(10**10)+1) if len(d:=sorted(str(i*i))) == ord(d[-1])-ord(d[0])+1 == len(set(d))])
print(afull()) # Michael S. Branicky, Feb 23 2024