A085546 Starting term of the smallest n-chain of numbers whose squares are permutations of the same digits.

12, 13, 102, 128, 103, 1023, 1003, 3206, 3514, 3774, 10013, 10012, 10172, 10126, 10209, 10117, 10215, 11093, 10023, 31968, 10128, 31822, 31647, 32157, 31723, 32946, 32238, 32625, 11826, 32013, 33567, 32673, 100762, 101213, 35574, 100215, 100374, 31708, 100674, 101177, 100394

Offset

2,1

Comments

Chains must have exactly n elements (else terms would be nondecreasing). - Michael S. Branicky, Oct 07 2025

Links

Michael S. Branicky, Table of n, a(n) for n = 2..11055 (first 58 terms from Michael S. Branicky, then next 1680 terms from David A. Corneth)

Michael S. Branicky, Table of n, a(n) for n = 2..18981 and all other terms < 10^10

Michael S. Branicky, Python program for producing b-file

David A. Corneth, PARI program

S. S. Gupta, Fascinating Squares

Example

a(5) = 128 is the starting term of the 5-tuple(128,178,191,196,209) having squares (16384,31684,36481,38416,43681).
From Jeffrey Shallit, Aug 21 2020, corrections by Michael S. Branicky, Oct 07 2025: (Start)
Here are the n-chains corresponding to the first few terms:
2: 12, 21;
3: 13, 14, 31;
4: 102, 120, 201, 210;
5: 128, 178, 191, 196, 209;
6: 103, 130, 140, 247, 301, 310;
7: 1023, 1302, 1395, 2013, 2301, 2493, 3105;
8: 1003, 1030, 1300, 1400, 2470, 3001, 3010, 3100;
9: 3206, 3292, 4804, 5699, 6181, 6199, 6509, 6901, 8494;
10: 3514, 4396, 4406, 5593, 5909, 6556, 6944, 9013, 9121, 9572;
11: 3774, 5592, 6132, 6432, 6561, 6639, 6858, 6879, 7812, 8568, 8571;
12: 10013, 10031, 10130, 10310, 11003, 11030, 13001, 13010, 30011, 30110, 31001, 31010;
13: 10012, 10021, 10120, 10210, 11002, 11020, 12001, 12010, 14179, 20011, 20110, 21001, 21010;
14: 10172, 12688, 13952, 17521, 20837, 21553, 22259, 22457, 22592, 23321, 24707, 26138, 28312, 30571;
... (End)

Prog

(Python) # see links for an alternate program suitable for generating b-file
from math import isqrt
from itertools import count, islice, permutations
def agen(verbose=False): # generator of terms
    adict, n, seen = dict(), 2, set()
    for m in count(10):
        k = "".join(sorted(str(m**2)))
        if k in seen: continue
        seen.add(k)
        ub = isqrt(int(k[::-1]))
        v = 1 + sum(1 for i in range(m+1, ub+1) if "".join(sorted(str(i**2))) == k)
        if v > 0 and v not in adict:
            if verbose: print("FOUND", v, m)
            adict[v] = m
            while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 10))) # Michael S. Branicky, Oct 07 2025, edited Oct 09 2025

Crossrefs

Cf. A045879.

Sequence in context: A041302 A041304 A396061 * A041306 A058952 A058950

Adjacent sequences: A085543 A085544 A085545 * A085547 A085548 A085549

Keyword

nonn,base

Author

Lekraj Beedassy, Jul 03 2003

Extensions

a(7)-a(31) from Jeffrey Shallit, Aug 21 2020

Corrected terms and a(32) and beyond from Michael S. Branicky, Oct 07 2025

Status

approved