A374024 Integers k such that digsum(k) = digsum(k^2) = p, where p is prime and digsum(i) = A007953(i).

199, 289, 379, 388, 496, 559, 568, 595, 739, 775, 838, 955, 1099, 1189, 1198, 1468, 1495, 1585, 1738, 1747, 1765, 1792, 1855, 1990, 2098, 2494, 2665, 2881, 2890, 3169, 3196, 3259, 3349, 3466, 3493, 3745, 3790, 3880, 4249, 4519, 4735, 4951, 4960, 5149, 5482

Offset: 1

Author

Gonzalo Martínez, Jul 05 2024

Keywords

• nonn
• base

Comments

Subsequence of A058369.

If k is a term, then digsum(k) = 19, 37 or 73, for k < 10^9.

If k is an integer such that digsum(k) = digsum(k^2) = p, with p prime, then p == 1 (mod 9) (A061237).

This sequence has infinitely many terms of the form 1999...9 (A067272). If p is a prime with p == 1 (mod 9), i.e., p = 9m + 1 for some m, then t = 2*10^m - 1 = 1999...9, i.e., 1 followed by m 9's, is in this sequence since digsum(t) = 9m + 1 = p and t^2 = 39...960...01, where there are (m - 1) 9's and (m - 1) 0's, so digsum(t^2) = 3 + 9*(m - 1) + 6 + 1 = 9m + 1 = p. Dirichlet's theorem guarantees the existence of infinitely many primes of the form 9w + 1 and hence infinitely many terms of this sequence.

2*10^m - 1 is the least number with digit sum 9*m + 1. Since the next prime congruent to 1 (mod 9) after 73 is 109 = 9*12 + 1, the first term with digit sum other than 19, 37 or 73 is 2*10^12 - 1. - Robert Israel, Jul 07 2024

Examples

			199 is a term, because its digital sum is 1 + 9 + 9 = 19 and 199^2 = 39601, whose digital sum is 3 + 9 + 6 + 0 + 1 = 19, which is prime.
		

Crossrefs

Cf. A007953, A058369, A061237, A067272, A254066.

Programs

Maple

ds:= n -> convert(convert(n,base,10),`+`):
filter:= proc(n) local p;
  p:= ds(n);
  isprime(p) and ds(n^2) = p
end proc:
select(filter, [seq(i,i=1..1000, 9)]); # Robert Israel, Jul 05 2024

Mathematica

Select[Range[5490],PrimeQ[dg=DigitSum[#]]&&(dg==DigitSum[#^2])&] (* Stefano Spezia, Jul 05 2024 *)

PARI

isok(k) = my(s=sumdigits(k)); isprime(s) && (s==sumdigits(k^2)); \\ Michel Marcus, Jul 06 2024