A263092 -id:A263092 - cp's OEIS frontend

A263087 a(n) = A060990(n^2); number of solutions to x - d(x) = n^2, where d(x) is the number of divisors of x (A000005).

2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 1, 0, 0, 0, 1, 1, 2, 1, 1, 0, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 3, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0

Offset: 0

Antti Karttunen, Oct 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10082

Cf. A000005, A000290, A002182, A002183, A060990.
Cf. A263093 (positions of zeros), A263092 (nonzeros).
Cf. A263250, A263251 (bisections) and A263252, A263253 (their partial sums).
Cf. also A261088, A263088.

A060990(n) = { my(k = n + 2400, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit A002183(77)=2400 good for at least up to A002182(77) = 10475665200.
A263087(n) = A060990(n^2);
for(n=0, 10082, write("b263087.txt", n, " ", A263087(n)));

(define (A263087 n) (A060990 (A000290 n)))

a(n) = A060990(n^2) = A060990(A000290(n)).

A263093 Numbers whose squares are in A045765.

Original entry on oeis.org

5, 6, 7, 8, 10, 14, 16, 18, 20, 22, 26, 27, 28, 34, 35, 37, 46, 47, 50, 54, 56, 58, 59, 60, 62, 67, 73, 78, 82, 85, 89, 90, 94, 95, 98, 100, 103, 104, 106, 110, 114, 116, 118, 122, 124, 125, 126, 127, 128, 130, 135, 140, 141, 142, 148, 150, 155, 158, 161, 164, 170, 172, 174, 177, 178, 182, 184, 188, 190, 199, 202, 205, 207

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Numbers n such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Numbers n for which A060990(n^2) = A263087(n) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..12443
A. Karttunen, Ratio a(n)/A263092(n) drawn with the help of OEIS Plot2-utility

Cf. A000005, A000196, A045765, A049820, A060990, A263098.
Complement: A263092.
Positions of zeros in A263087 and positions of ones in A263088.
Cf. A263095 (the squares of these numbers).

\\ Compute A263093 and A263095 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 is good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
n = 1; k = 0; while((n^2)<1102701600, if((0 == A060990(n*n)), k++; write("b263093.txt", k, " ", n); write("b263095.txt", k, " ", (n*n)); ); n++; if(!(n%8192),print1(n,",k=", k, ", ")); );

;; With Antti Karttunen's IntSeq-library.
(define A263093 (MATCHING-POS 1 1 (lambda (n) (zero? (A060990 (* n n))))))
(define A263093 (ZERO-POS 1 0 A263087))

a(n) = A000196(A263095(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

A263087 a(n) = A060990(n^2); number of solutions to x - d(x) = n^2, where d(x) is the number of divisors of x (A000005).

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A263093 Numbers whose squares are in A045765.

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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).

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