A344570 Number of pairs of n-digit squares such that the final (n-1) digits of the first square coincide with the initial (n-1) digits of the second.

0, 4, 6, 7, 10, 5, 10, 4, 15, 0, 13, 5, 16, 3, 57, 2, 8, 2, 5, 1, 119, 1, 13, 8, 39, 5, 55, 2, 53, 7, 12, 7, 76, 1, 193, 2, 21, 2, 59, 11, 35, 15, 42, 7, 541, 7, 17, 20, 37, 1, 233, 3, 32, 2, 373, 19, 65, 0, 15, 16, 181, 15, 8637, 5, 175, 15

Offset

1,2

Comments

For n=2, certain numbers (16 and 64) appear in more than one pair. No such numbers have been observed up to n=9, but so far there is no proof of this property.

a(10)=0 was found by Andrew Howroyd. Does this sequence contain infinitely many nonzero terms?

The sequence is inspired by a problem from the 2020 Polish Juniors' Mathematics Olympiad. It is problem 1: 'Is there a six-digit positive integer such that any two consecutive digits form a perfect square?'

Are there any other terms such that a(n) = 0 besides n=1 and n=10? - Chai Wah Wu, May 26 2021

Links

Table of n, a(n) for n=1..66.

Stowarzyszenie na rzecz Edukacji Matematycznej, Olimpiada Matematyczna Juniorów 2020/2021, etap 1 (in Polish).

Chai Wah Wu, pairs of squares for n = 1..46

Index to sequences related to Olympiads.

Example

For n=2: (81,16), (16,64), (36,64), (64,49).
For n=3: (144,441), (196,961), (225,256), (625,256), (484,841), (784,841).
For n=4: (3136,1369), (4624,6241), (5184,1849), (5476,4761), (7396,3969), (7921,9216), (9409,4096).
For n=20: (64764644930975528100, 47646449309755281001) is the only pair. - Andrew Howroyd, May 23 2021

Prog

(PARI) a(n)={sum(k=sqrtint(10^(n-1))+1, sqrtint(10^n-1), my(t=k^2*10%10^n); t>=10^(n-1) && sqrtint(t+9)^2\10==t\10)} \\ Andrew Howroyd, May 24 2021

Crossrefs

Cf. A343855.

Sequence in context: A243193 A087790 A278960 * A340833 A340847 A286494

Adjacent sequences: A344567 A344568 A344569 * A344571 A344572 A344573

Keyword

nonn,base,more

Author

Igor Trujnara, May 23 2021

Extensions

a(10)-a(20) from Andrew Howroyd, May 24 2021

a(21)-a(46) from Chai Wah Wu, May 26 2021

a(47)-a(66) from Chai Wah Wu, Jun 02 2021

Status

approved