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A263090 Primes p for which A049820(x) = p has a solution.

%I A263090 #17 Mar 16 2025 02:38:24
%S A263090 2,3,5,11,17,23,29,31,41,47,53,59,61,71,73,83,89,97,101,107,127,137,
%T A263090 139,149,151,157,173,179,181,191,197,199,211,223,227,233,239,263,269,
%U A263090 277,281,283,311,317,331,337,347,349,367,373,389,409,419,421,431,433,443,449,457,461,467,521,523,541,547,557,569,577,587,593,599,601,607,613,617,619,631,641,643
%N A263090 Primes p for which A049820(x) = p has a solution.
%C A263090 Primes p that there is at least one such k for which k - d(k) = p, where d(k) is the number of divisors of k (A000005).
%H A263090 Antti Karttunen, <a href="/A263090/b263090.txt">Table of n, a(n) for n = 1..7955</a>
%e A263090 2 is present, as we have 6 - d(6) = 6 - 4 = 2.
%e A263090 3 is present, as we have 5 - d(5) = 3. The same holds for all lesser twin primes (A001359).
%t A263090 lim = 10000; s = Select[Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}], PrimeQ]; Take[s, 79] (* _Michael De Vlieger_, Oct 13 2015 *)
%o A263090 (PARI)
%o A263090 allocatemem(123456789);
%o A263090 uplim1 = 2162160 + 320; \\ = A002182(41) + A002183(41).
%o A263090 v060990 = vector(uplim1);
%o A263090 for(n=3, uplim1, v060990[n-numdiv(n)]++);
%o A263090 A060990 = n -> if(!n,2,v060990[n]);
%o A263090 n=0; forprime(p=2, 131071, if((A060990(p) > 0), n++; write("b263090.txt", n, " ", p)));
%o A263090 (Scheme)
%o A263090 ;; With _Antti Karttunen_'s IntSeq-library.
%o A263090 (define A263090 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 n)) (not (zero? (A060990 n)))))))
%Y A263090 Complement among primes: A263091.
%Y A263090 Intersection of A000040 and A236562.
%Y A263090 Cf. A001359 (a subsequence).
%Y A263090 Cf. A000005, A049820, A060990.
%Y A263090 Cf. also A263094.
%K A263090 nonn
%O A263090 1,1
%A A263090 _Antti Karttunen_, Oct 11 2015