A263090 -id:A263090 - cp's OEIS frontend

A263091 Primes p for which A049820(x) = p has no solution.

7, 13, 19, 37, 43, 67, 79, 103, 109, 113, 131, 163, 167, 193, 229, 241, 251, 257, 271, 293, 307, 313, 353, 359, 379, 383, 397, 401, 439, 463, 479, 487, 491, 499, 503, 509, 563, 571, 647, 653, 661, 673, 701, 739, 743, 757, 761, 773, 823, 859, 863, 883, 887, 911, 937, 941, 953, 967, 971, 977, 983, 1009, 1093, 1103, 1109, 1171, 1181, 1193, 1217, 1279, 1283, 1291, 1297, 1307, 1321, 1361

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Primes p that there is no such k for which k - d(k) = p, where d(k) is the number of divisors of k (A000005).

Antti Karttunen, Table of n, a(n) for n = 1..15383

Complement among primes: A263090.
Intersection of A000040 and A045765.
Subsequence of A067774 (A049591).
Cf. A000005, A049820, A060990.

lim = 10000; s = Select[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], PrimeQ]; Take[s, 76] (* 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, 524287, if((0 == A060990(p)), n++; write("b263091.txt", n, " ", p)));

;; With Antti Karttunen's IntSeq-library.
(define A263091 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 n)) (zero? (A060990 n))))))

A263094 Squares in A236562; numbers n^2 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).

Original entry on oeis.org

0, 1, 4, 9, 16, 81, 121, 144, 169, 225, 289, 361, 441, 529, 576, 625, 841, 900, 961, 1024, 1089, 1296, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2304, 2401, 2601, 2704, 2809, 3025, 3249, 3721, 3969, 4096, 4225, 4356, 4624, 4761, 4900, 5041, 5184, 5476, 5625, 5776, 5929, 6241, 6400, 6561, 6889, 7056, 7396, 7569, 7744, 8281, 8464, 8649, 9216, 9409, 9801, 10201, 10404, 11025

Offset: 0

Antti Karttunen, Oct 11 2015

nonn

Starting offset is zero, because a(0)=0 is a special case in this sequence.

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

Cf. A000005, A049820, A060990.
Intersection of A000290 and A236562.
Cf. A263092 (gives the square roots of these terms).
Cf. A263095 (complement among squares).
Cf. A262514 (a subsequence).
Cf. also A263090, A263098.

Take[Select[Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, 20000}], IntegerQ@ Sqrt@ # &], 68] (* Michael De Vlieger, Oct 13 2015 *)

PARI
```
\\ See code in A263092.
```

(define (A263094 n) (A000290 (A263092 n)))

a(n) = A000290(A263092(n)).

cp's OEIS Frontend

A263091 Primes p for which A049820(x) = p has no solution.

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