A316969 -id:A316969 - cp's OEIS frontend

A308917 Primes p such that the digits of p^2 are squares.

2, 3, 7, 97, 107, 701, 997, 1049, 7001, 10007, 10243, 20347, 70001, 97001, 100549, 202757, 306749, 379499, 700001, 997001, 1002247, 1070021, 3317257, 3346507, 9536249, 9970001, 10095247, 20470501, 21095021, 22144979, 94925771, 100000007, 100099501, 104933743, 202520347

Offset: 1

Bernard Schott, Jun 30 2019

The prime numbers of the form p = 7 * 10^k + 1 with k > = 2 are terms of the sequence. For example, for k = 2, 3, 4, 5, 8, 9, 45, 136, 142, 158, 243, 923, .... The squares have the form p^2 = 49 * 10^(2*k) + 14 * 10^k + 1 and the digits 0, 1, 4 and 9. - Marius A. Burtea, Jul 01 2019
Same remark with primes of the form p = 10^k + 7 and k > = 2 that are also terms of this sequence. For example, for k = 2, 4, 8, 9, ... The squares have the form p^2 = 100^k + 14 * 10^k + 49, so with only the digits 0, 1, 4 and 9. These primes are in A159031 \ {17}. - Bernard Schott, Jul 01 2019
From Chai Wah Wu, Jul 03 2019: (Start)
The prime numbers of the form p = (10^m-3)*10^k + 1 with k > m > 0 are terms of this sequence. Note that this includes primes of the form 7 * 10^k + 1 with k >=2 described in the first comment above. The squares are of the form p^2 = (10^m-3)^2*10^(2*k) + 2(10^m-3)*10^k + 1. Note that (10^m-3)^2 = 10^m(10^m-6)+9 which only contains the digits 0, 4 and 9. Similarly, 2*(10^m-3) = 2*10^m-6 which only contains the digits 1, 9 and 4 and has m+1 <= k decimal digits. Thus p^2 only contains the digits 0, 1, 4, 9. Some examples include (m,k) = (2,3), (2,8), (2,15), (3,4), (3,18), (4,71), (5,20), (6,7), ...
A similar argument shows that prime numbers of the form p = 10^k + (10^m-3) for k >= 2*m > 0 (which includes the primes of the form 10^k+7) are also terms of this sequence. In this case some examples include (m,k) = (2,9), (2,10), (3,12), (3,18), (4,10), (4,11), (5,14), ...
Some other sets of terms are:
1. primes of the form p = 20247*10^k+1 for k >= 5. Examples include k = 7, 25, 29, 31, ...
2. primes of the form p = 10^k + 20247 for k >= 9. Examples include k = 11, 15, 18, 19, 20, ...
3. primes of the form p = 2224745247*10^k+1 for k >= 10. Examples include k = 31, 57, 115, 163, ...
4. primes of the form p = 10^k + 2224745247 for k >= 19. Examples include k = 87, 257, 645, 819, ...
(End)

			997 is a term because 997 is prime and 997^2 = 994009 with 0, 4, 9 that are all squares.

Giovanni Resta, Table of n, a(n) for n = 1..512
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios!, 446653271.

Cf. A000290, A019544, A046030.
Cf. subsequences: A159031, A316969.
Intersection of A000040 and A285550.

p=primes(10000000); m=1;
for u=1:length(p) digit=dec2base(p(u).^2,10)-'0';
    if (mod(sqrt(digit), 1) == 0) sol(m)=p(u); m=m+1; end
end
sol % Marius A. Burtea, Jun 30 2019

ok[n_] := AllTrue[IntegerDigits[n], MemberQ[{0, 1, 4, 9}, #] &]; mo = Select[Range[1, 10^6, 2], ok@Mod[#^2, 10^6] &]; Reap[Sow@2; Do[x = 10^6 t + m; If[PrimeQ[x] && ok[x^2], Sow[x]], {t, 0, 202}, {m, mo}]][[2, 1]] (* Giovanni Resta, Jul 02 2019 *)
Select[Prime[Range[11211000]],AllTrue[Sqrt[IntegerDigits[#^2]],IntegerQ]&] (* Harvey P. Dale, Aug 17 2021 *)

isok(p) = isprime(p) && (d=digits(p^2)) && (#select(x->issquare(x), d) == #d); \\ Michel Marcus, Jun 30 2019

More terms from Michel Marcus, Jun 30 2019

A317579 Integers n such that the digit set of n^2 is {0,1,4,9}.

Original entry on oeis.org

138, 648, 701, 951, 1007, 1070, 1380, 1393, 3153, 3451, 3743, 3747, 4462, 6357, 6480, 7001, 7010, 7071, 9510, 9701, 10007, 10070, 10097, 10243, 10538, 10700, 13800, 13930, 20247, 20347, 22138, 31530, 34510, 37430, 37470, 37538, 38071, 38602, 44620, 63357, 63403, 63570, 64800

Offset: 1

Robert G. Wilson v, Jul 31 2018

n cannot end in the decimal digits 4, 5 or 6; but it most often ends in 0 since if n is present so is 10*n.
n cannot start with the decimal digits 5 or 8. It usually starts with either 3 or 1.
n must lie between 1*10^k & sqrt(2)*10^k, 2*10^k & sqrt(5)*10^k, 3 & sqrt(12)*10^k, sqrt(14)*10^k & sqrt(15)*10^k, sqrt(19)*10^k & sqrt(20)*10^k, sqrt(40)*10^k & sqrt(45)*10^k, sqrt(49)*10^k & sqrt(50)*10^k, sqrt(90)*10^k & sqrt(92)*10^k, sqrt(94)*10^k & sqrt(95)*10^k, sqrt(99)*10^k & sqrt(100)*10^k; for k>0.

			138 = 19044 which has only the decimal digits 0, 1, 4 & 9. Therefore it is in the sequence.

Cf. A285550, A316969.

fQ[n_] := Union[IntegerDigits[n^2]] == {0, 1, 4, 9}; Select[ Range@ 65000, fQ]

isok(n) = Set(digits(n^2)) == [0, 1, 4, 9]; \\ Michel Marcus, Aug 01 2018

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A308917 Primes p such that the digits of p^2 are squares.

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