A263093 -id:A263093 - 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)).

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).

Original entry on oeis.org

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))

A263095 Squares in A045765; numbers n^2 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).

Original entry on oeis.org

25, 36, 49, 64, 100, 196, 256, 324, 400, 484, 676, 729, 784, 1156, 1225, 1369, 2116, 2209, 2500, 2916, 3136, 3364, 3481, 3600, 3844, 4489, 5329, 6084, 6724, 7225, 7921, 8100, 8836, 9025, 9604, 10000, 10609, 10816, 11236, 12100, 12996, 13456, 13924, 14884, 15376, 15625, 15876, 16129, 16384, 16900, 18225, 19600, 19881, 20164, 21904, 22500, 24025, 24964, 25921, 26896

Offset: 1

Antti Karttunen, Oct 10 2015

nonn

Some of the terms are shared with A262687, but none with A262514.

Antti Karttunen, Table of n, a(n) for n = 1..12443

Cf. A000005, A049820, A010052, A060990, A262514, A262687, A263098.
Cf. A263093 (gives the square roots).
Intersection of A000290 and A045765.
Cf. also A263091.

lim = 40000; Take[Select[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], IntegerQ@ Sqrt@ # &], 60] (* Michael De Vlieger, Oct 13 2015 *)

PARI
```
\\ See code in A263093.
```

;; With Antti Karttunen's IntSeq-library.
(define A263095 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010052 n)) (zero? (A060990 n))))))

A263088 a(n) = A262697(n^2).

Original entry on oeis.org

0, 6, 2, 38, 2, 1, 1, 1, 1, 22, 1, 0, 0, 2, 1, 3, 1, 9, 1, 39, 1, 47, 1, 51, 4, 114, 1, 1, 1, 529, 2, 6, 2, 3, 1, 1, 22, 1, 11, 3, 2, 4, 7, 93, 7, 967, 1, 1, 3, 4, 1, 3, 2, 4, 1, 3, 1, 3, 1, 1, 1, 2, 1, 139, 2, 265, 2, 1, 6, 464, 12, 4, 22, 1, 2, 1503, 2, 6, 1, 5, 2, 2, 1, 2, 5, 1, 2, 4, 2, 1, 1, 6, 3, 386, 1, 1, 3, 800, 1, 2, 1, 7, 5, 1, 1, 3353, 1, 2, 21, 3, 1, 17, 3, 3, 1, 4, 1, 5, 1, 3, 9, 2

Offset: 0

Antti Karttunen, Oct 12 2015

nonn

a(n)=0 if n^2 is in A259934, otherwise number of nodes in that finite subtree whose root is n^2 and edge-relation is defined by A049820(child) = parent. This count includes also leaves and n^2 itself.

Antti Karttunen, Table of n, a(n) for n = 0..222

Cf. A000290, A049820, A259934, A262697.
Cf. also A261088, A263087.
Cf. A262515 (positions of zeros), A263093 (positions of ones).

(define (A263088 n) (A262697 (A000290 n)))

a(n) = A262697(A000290(n)) = A262697(n^2).
Other identities. For all n >= 0:
If A263087(n) = 0, a(n) = 1.

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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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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A263095 Squares in A045765; numbers n^2 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).

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A263088 a(n) = A262697(n^2).

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