A263094 -id:A263094 - cp's OEIS frontend

A263092 Numbers whose squares are in A236562; numbers n such that there is at least one such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

0, 1, 2, 3, 4, 9, 11, 12, 13, 15, 17, 19, 21, 23, 24, 25, 29, 30, 31, 32, 33, 36, 38, 39, 40, 41, 42, 43, 44, 45, 48, 49, 51, 52, 53, 55, 57, 61, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 91, 92, 93, 96, 97, 99, 101, 102, 105, 107, 108, 109

Offset: 0

Antti Karttunen, Oct 11 2015

nonn

Starting offset is zero, because a(0)=0 is a special case in this sequence.
Numbers n for which A060990(n^2) = A263087(n) > 0.
Numbers n for which A049820(x) = n^2 has a solution.

Antti Karttunen, Table of n, a(n) for n = 0..20763

Complement: A263093.
Cf. A000005, A000196, A049820, A060990, A236562, A263087, A263098.
Cf. A263094 (the squares of these numbers).
Cf. A262515 (a subsequence).

\\ Compute A263092 and A263094 at the same time:
A060990(n) = { my(k = n + 1440, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit 1440 good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
n = 0; k = 0; while((n^2)<1102701600, if((A060990(n*n) > 0), write("b263092.txt", k, " ", n); write("b263094.txt", k, " ", (n*n)); k++; ); n++; if(!(n%8192),print1(n,",k=", k, ", ")); );

;; With Antti Karttunen's IntSeq-library.
(define A263092 (MATCHING-POS 0 0 (lambda (n) (not (zero? (A060990 (* n n)))))))
(define A263092 (NONZERO-POS 0 0 A263087))

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

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 89, 97, 101, 107, 127, 137, 139, 149, 151, 157, 173, 179, 181, 191, 197, 199, 211, 223, 227, 233, 239, 263, 269, 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

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

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).

			2 is present, as we have 6 - d(6) = 6 - 4 = 2.
3 is present, as we have 5 - d(5) = 3. The same holds for all lesser twin primes (A001359).

Antti Karttunen, Table of n, a(n) for n = 1..7955

Complement among primes: A263091.
Intersection of A000040 and A236562.
Cf. A001359 (a subsequence).
Cf. A000005, A049820, A060990.
Cf. also A263094.

lim = 10000; s = Select[Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}], PrimeQ]; Take[s, 79] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem(123456789);
uplim1 = 2162160 + 320; \\ = A002182(41) + A002183(41).
v060990 = vector(uplim1);
for(n=3, uplim1, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
n=0; forprime(p=2, 131071, if((A060990(p) > 0), n++; write("b263090.txt", n, " ", p)));

;; With Antti Karttunen's IntSeq-library.
(define A263090 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 n)) (not (zero? (A060990 n)))))))

cp's OEIS Frontend

A263092 Numbers whose squares are in A236562; numbers n such that there is at least one such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

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