cp's OEIS Frontend

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

%I A374024 #40 Jul 28 2024 00:31:02
%S A374024 199,289,379,388,496,559,568,595,739,775,838,955,1099,1189,1198,1468,
%T A374024 1495,1585,1738,1747,1765,1792,1855,1990,2098,2494,2665,2881,2890,
%U A374024 3169,3196,3259,3349,3466,3493,3745,3790,3880,4249,4519,4735,4951,4960,5149,5482
%N A374024 Integers k such that digsum(k) = digsum(k^2) = p, where p is prime and digsum(i) = A007953(i).
%C A374024 Subsequence of A058369.
%C A374024 If k is a term, then digsum(k) = 19, 37 or 73, for k < 10^9.
%C A374024 If k is an integer such that digsum(k) = digsum(k^2) = p, with p prime, then p == 1 (mod 9) (A061237).
%C A374024 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.
%C A374024 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
%e A374024 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.
%p A374024 ds:= n -> convert(convert(n,base,10),`+`):
%p A374024 filter:= proc(n) local p;
%p A374024   p:= ds(n);
%p A374024   isprime(p) and ds(n^2) = p
%p A374024 end proc:
%p A374024 select(filter, [seq(i,i=1..1000, 9)]); # _Robert Israel_, Jul 05 2024
%t A374024 Select[Range[5490],PrimeQ[dg=DigitSum[#]]&&(dg==DigitSum[#^2])&] (* _Stefano Spezia_, Jul 05 2024 *)
%o A374024 (PARI) isok(k) = my(s=sumdigits(k)); isprime(s) && (s==sumdigits(k^2)); \\ _Michel Marcus_, Jul 06 2024
%Y A374024 Cf. A007953, A058369, A061237, A067272, A254066.
%K A374024 nonn,base
%O A374024 1,1
%A A374024 _Gonzalo Martínez_, Jul 05 2024