A263087 -id:A263087 - cp's OEIS frontend

A263250 Even bisection of A263087; number of solutions to x - d(x) = 4(n^2), where d(x) is the number of divisors of x (A000005).

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

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A060990, A263087, A263251.

Cf. also A263252 (partial sums).

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);
A263250(n) = A263087(2*n);
p = 0; for(n=0, 10000, k = A263250(n); p += k; write("b263250.txt", n, " ", k); write("b263252.txt", n, " ", p)); \\ Compute A263250 and A263252 at the same time.

Scheme
```
(define (A263250 n) (A263087 (+ n n)))
```

a(n) = A263087(2*n).

A263251 Odd bisection of A263087; number of solutions to x - d(x) = (2n+1)^2, where d(x) is the number of divisors of x (A000005).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A060990, A263087, A263250.

Cf. also A263253 (partial sums).

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);
A263251(n) = A263087((2*n)+1);
p = 0; for(n=0, 10000, k = A263251(n); p += k; write("b263251.txt", n, " ", k); write("b263253.txt", n, " ", p)); \\ Compute A263251 and A263253 at the same time.

(define (A263251 n) (A263087 (+ n n 1)))

a(n) = A263087(2*n + 1).

A060990 Number of solutions to x - d(x) = n, where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

2, 2, 1, 1, 1, 1, 3, 0, 0, 1, 1, 3, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 4, 1, 0, 0, 1, 2, 0, 2, 1, 1, 1, 0, 2, 2, 0, 0, 2, 2, 0, 1, 1, 0, 1, 1, 3, 1, 2, 0, 0, 2, 0, 1, 1, 0, 0, 3, 2, 1, 1, 1, 2, 0, 0, 2, 0, 0, 0, 2, 4, 1, 1, 1, 0, 0, 1, 1, 2, 0, 1, 2, 1, 1, 1, 0, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 0, 1, 0, 1, 3, 0, 1, 1

Offset: 0

Labos Elemer, May 11 2001

nonn

If x-d(x) is never equal to n, then n is in A045765 and a(n) = 0.

Number of solutions to A049820(x) = n. - Jaroslav Krizek, Feb 09 2014

			a(11) = 3 because three numbers satisfy equation x-d(x)=11, namely {13,15,16} with {2,4,5} divisors respectively.

Antti Karttunen, Table of n, a(n) for n = 0..110880

Cf. A000005, A002183, A049820, A049816, A082284, A155043, A236561, A236565, A259934, A261100, A262507, A262513, A262686.
Cf. A045765 (positions of zeros), A236562 (positions of nonzeros), A262511 (positions of ones).
Cf. A263087 (computed for squares).

lim = 105; s = Table[n - DivisorSigma[0, n], {n, 2 lim + 3}]; Length@ Position[s, #] & /@ Range[0, lim] (* Michael De Vlieger, Sep 29 2015, after Wesley Ivan Hurt at A049820 *)

allocatemem(123456789);
uplim = 2162160; \\ = A002182(41).
v060990 = vector(uplim);
for(n=3, uplim, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
uplim2 = 110880; \\ = A002182(30).
for(n=0, uplim2, write("b060990.txt", n, " ", A060990(n)));
\\ Antti Karttunen, Sep 25 2015

(define (A060990 n) (if (zero? n) 2 (add (lambda (k) (if (= (A049820 k) n) 1 0)) n (+ n (A002183 (+ 2 (A261100 n)))))))
;; Auxiliary function add implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
;; Proof-of-concept code for the given formula, by Antti Karttunen, Sep 25 2015

a(0) = 2; for n >= 1, a(n) = Sum_{k = n .. n+A002183(2+A261100(n))} [A049820(k) = n]. (Here [...] denotes the Iverson bracket, resulting 1 when A049820(k) is n and 0 otherwise.) - Antti Karttunen, Sep 25 2015, corrected Oct 12 2015.
a(n) = Sum_{k = A082284(n) .. A262686(n)} [A049820(k) = n] (when tacitly assuming that A049820(0) = 0.) - Antti Karttunen, Oct 12 2015
Other identities and observations. For all n >= 0:
a(A045765(n)) = 0. a(A236562(n)) > 0. - Jaroslav Krizek, Feb 09 2014

Offset corrected by Jaroslav Krizek, Feb 09 2014

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)).

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))

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

A263250 Even bisection of A263087; number of solutions to x - d(x) = 4(n^2), where d(x) is the number of divisors of x (A000005).

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A263251 Odd bisection of A263087; number of solutions to x - d(x) = (2n+1)^2, where d(x) is the number of divisors of x (A000005).

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A060990 Number of solutions to x - d(x) = n, where d(n) is the number of divisors of n (A000005).

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A263093 Numbers whose squares are in A045765.

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

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