A236562 -id:A236562 - 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))

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)).

A049820 a(n) = n - d(n), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 5, 4, 6, 6, 9, 6, 11, 10, 11, 11, 15, 12, 17, 14, 17, 18, 21, 16, 22, 22, 23, 22, 27, 22, 29, 26, 29, 30, 31, 27, 35, 34, 35, 32, 39, 34, 41, 38, 39, 42, 45, 38, 46, 44, 47, 46, 51, 46, 51, 48, 53, 54, 57, 48, 59, 58, 57, 57, 61, 58, 65

Offset: 1

Clark Kimberling

a(n) is the number of non-divisors of n in 1..n. - Jaroslav Krizek, Nov 14 2009
Also equal to the number of partitions p of n such that max(p)-min(p) = 1. The number of partitions of n with max(p)-min(p) <= 1 is n; there is one with k parts for each 1 <= k <= n. max(p)-min(p) = 0 iff k divides n, leaving n-d(n) with a difference of 1. It is easiest to see this by looking at fixed k with increasing n: for k=3, starting with n=3 the partitions are [1,1,1], [2,1,1], [2,2,1], [2,2,2], [3,2,2], etc. - Giovanni Resta, Feb 06 2006 and Franklin T. Adams-Watters, Jan 30 2011
Number of positive numbers in n-th row of array T given by A049816.
Number of proper non-divisors of n. - Omar E. Pol, May 25 2010
a(n+2) is the sum of the n-th antidiagonal of A225145. - Richard R. Forberg, May 02 2013
For n > 2, number of nonzero terms in n-th row of triangle A051778. - Reinhard Zumkeller, Dec 03 2014
Number of partitions of n of the form [j,j,...,j,i] (j > i). Example: a(7)=5 because we have [6,1], [5,2], [4,3], [3,3,1], and [2,2,2,1]. - Emeric Deutsch, Sep 22 2016

			a(7) = 5; the 5 non-divisors of 7 in 1..7 are 2, 3, 4, 5, and 6.
The 5 partitions of 7 with max(p) - min(p) = 1 are [4,3], [3,2,2], [2,2,2,1], [2,2,1,1,1] and [2,1,1,1,1,1]. - _Emeric Deutsch_, Mar 01 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. E. Andrews, M. Beck, N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv preprint arXiv:1406.3374 [math.NT], 2014-2015.

Cf. A000005.
One less than A062968, two less than A059292.
Cf. A161664 (partial sums).
Cf. A060990 (number of solutions to a(x) = n).
Cf. A045765 (numbers not occurring in this sequence).
Cf. A236561 (same sequence sorted into ascending order), A236562 (with also duplicates removed), A236565, A262901 and A262903.
Cf. A262511 (numbers that occur only once).
Cf. A055927 (positions of repeated terms).
Cf. A245388 (positions of squares).
Cf. A155043 (number of steps needed to reach zero when iterating a(n)), A262680 (number of nonzero squares encountered).
Cf. A259934 (an infinite trunk of the tree defined by edge-relation a(child) = parent, conjectured to be unique).
Cf. tables and arrays A047916, A051731, A051778, A173540, A173541.
Cf. also arrays A225145, A262898, A263255 and tables A263265, A263267.
Other related sequences: A006218, A006590, A051953, A070824, A094181, A062249, A067391, A076627, A128508, A131187, A134156, A140826, A161886, A177235, A177236, A227874, A228453, A230653, A230654, A231167, A245197, A253473.

List([1..80],n->n-Tau(n)); # Muniru A Asiru, Sep 28 2018

a049820 n = n - a000005 n  -- Reinhard Zumkeller, Feb 06 2012

A049820 := n->n-numtheory[tau](n):
seq(A049820(n), n=1..100);

Table[n - DivisorSigma[0, n], {n, 100}] (* Wesley Ivan Hurt, Nov 19 2014 *)
Array[(# - DivisorSigma[0, #])&, 70] (* Vincenzo Librandi, Dec 29 2015 *)

PARI
```
a(n)=n-numdiv(n)
```

(define (A049820 n) (- n (A000005 n))) ;; Antti Karttunen, Nov 27 2015

a(n) = Sum_{k=1..n} ceiling(n/k)-floor(n/k). - Benoit Cloitre, May 11 2003
G.f.: Sum_{k>0} x^(2*k+1)/(1-x^k)/(1-x^(k+1)). - Emeric Deutsch, Mar 01 2006
a(n) = A006590(n) - A006218(n) = A161886(n) - A000005(n) - A006218(n) + 1 for n >= 1. - Jaroslav Krizek, Nov 14 2009
a(n) = Sum_{k=1..n} A000007(A051731(n,k)). - Reinhard Zumkeller, Mar 09 2010
a(n) = A076627(n) / A000005(n). - Reinhard Zumkeller, Feb 06 2012
For n >= 2, a(n) = A094181(n) / A051953(n). - Antti Karttunen, Nov 27 2015
a(n) = Sum_{k=1..n} ((n mod k) + (-n mod k))/k. - Wesley Ivan Hurt, Dec 28 2015
G.f.: Sum_{j>=2} (x^(j+1)*(1-x^(j-1))/(1-x^j))/(1-x). - Emeric Deutsch, Sep 22 2016
Dirichlet g.f.: zeta(s-1)- zeta(s)^2. - Ilya Gutkovskiy, Apr 12 2017
a(n) = Sum_{i=1..n-1} sign(i mod n-i). - Wesley Ivan Hurt, Sep 27 2018

Edited by Franklin T. Adams-Watters, Jan 30 2012

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

A045765 k - d(k) never takes these values, where d(k) = A000005(k).

Original entry on oeis.org

7, 8, 13, 19, 20, 24, 25, 28, 33, 36, 37, 40, 43, 49, 50, 52, 55, 56, 63, 64, 66, 67, 68, 74, 75, 79, 85, 88, 98, 100, 103, 108, 109, 112, 113, 116, 117, 123, 124, 126, 131, 132, 133, 134, 136, 140, 143, 145, 150, 153, 156, 159, 160, 163, 164, 167, 168

Offset: 1

David W. Wilson

nonn

Complement of A236562. - Jaroslav Krizek, Feb 09 2014
Positions of zeros in A060990, leaf-nodes in the tree generated by edge-relation A049820(child) = parent. - Antti Karttunen, Oct 06 2015
Since A000005(x) <= 1 + x/2, k is in the sequence if there are no x <= 2*(k+1) with k = x - d(x). - Robert Israel, Oct 12 2015
This can be improved as: k is in the sequence if there are no x <= k + A002183(2+A261100(k)) with k = x - d(x). Cf. also A070319, A262686. - Antti Karttunen, Oct 12 2015
Luca (2005) proved that this seqeunce is infinite. - Amiram Eldar, Jul 26 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000
Florian Luca, On numbers not of the form n-omega(n), Acta Mathematica Hungarica, Vol. 106, No. 1 (2005), pp. 117-135.

Top row of A262898.
Cf. A000005, A002183, A049820, A060990, A070319, A236562, A236565, A261100, A262511, A262686, A262901, A262902, A262903, A262909, A263081.
Cf. A263091 (primes in this sequence), A263095 (squares).
Cf. A259934 (gives the infinite trunk of the same tree, conjectured to be unique).

N:= 1000: # to get all terms <= N
sort(convert({$1..N} minus {seq(x - numtheory:-tau(x), x=1..2*(1+N))},list)); # Robert Israel, Oct 12 2015

lim = 10000; Take[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], 57] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem((2^31)+(2^30));
uplim = 36756720 + 640; \\ = A002182(53) + A002183(53).
v060990 = vector(uplim);
for(n=3, uplim, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
uplim2 = 36756720;
n=0; k=1; while(n <= uplim2, if(0==A060990(n), write("b045765_big.txt", k, " ", n); k++); n++;);
\\ Antti Karttunen, Oct 09 2015

(define A045765 (ZERO-POS 1 1 A060990))
;; Using also IntSeq-library of Antti Karttunen, Oct 06 2015

A262511 Numbers k for which there is exactly one solution to x - d(x) = k, where d(k) is the number of divisors of k (A000005). Positions of ones in A060990.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 12, 14, 15, 16, 18, 21, 23, 26, 30, 31, 32, 41, 42, 44, 45, 47, 53, 54, 59, 60, 61, 71, 72, 73, 76, 77, 80, 82, 83, 84, 86, 89, 90, 92, 93, 94, 95, 97, 99, 101, 104, 105, 106, 110, 115, 119, 121, 122, 127, 135, 139, 146, 148, 149, 151, 154, 158, 161, 169, 171, 173, 176, 177, 183, 186, 188, 189, 190, 191, 192, 194, 195, 199, 200, 202

Offset: 1

Antti Karttunen, Sep 25 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000005, A049820, A060990.
Cf. A262512 (gives the corresponding x).
Cf. A262510 (a subsequence).
Subsequence of A236562.

allocatemem(123456789);
uplim = 14414400 + 504; \\ = A002182(49) + A002183(49).
v060990 = vector(uplim);
for(n=3, uplim, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
uplim2 = 14414400;
n=0; k=1; while(n <= uplim2, if(1==A060990(n), write("b262511_big.txt", k, " ", n); k++); n++;);

;; With Antti Karttunen's IntSeq-library.
(define A262511 (ZERO-POS 1 1 (COMPOSE -1+ A060990)))

Other identities. For all n >= 1:

a(n) = A049820(A262512(n)).

A262901 Numbers that have at least one leaf-child in the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

4, 5, 11, 14, 16, 17, 22, 27, 29, 32, 35, 41, 44, 46, 48, 51, 57, 58, 62, 65, 69, 70, 77, 80, 81, 91, 92, 96, 101, 102, 107, 110, 111, 114, 118, 119, 120, 128, 129, 130, 138, 139, 141, 144, 147, 148, 152, 155, 158, 161, 162, 165, 166, 169, 176, 181, 187, 191, 192, 199, 201, 214, 215, 216, 222, 224, 227, 231, 234, 238, 239, 247, 248, 249, 255, 258, 262, 264, 269, 277, 278, 282, 286, 291, 294, 296

Offset: 1

Antti Karttunen, Oct 06 2015

nonn

Positions of nonzeros in A262900.
Numbers n such that there is at least one k such that k - d(k) = n [where d(k) is the number of divisors of k, A000005(k)], but there is no such x that x - d(x) = k, in other words, k is one of the terms of A045765.
Sequence A262902 sorted into ascending order, with duplicates removed.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A045765, A049820, A262900, A262902.
Cf. A262903 (a subsequence).
Subsequence of A236562.
Cf. also A257508.

A236565 The smallest values m such that A049820(x) = m has exactly n solutions.

Original entry on oeis.org

7, 2, 0, 6, 22, 838, 17638, 192520, 3240114, 219476872, 2146772872, 24443168392, 1273061788552

Offset: 0

Jaroslav Krizek, Feb 09 2014

The 11 numbers x for which A049820(x) is equal to a(11) are a(11) + {12, 16, 24, 32, 36, 40, 56, 80, 96, 128, 512}. - Giovanni Resta, Feb 10 2014

			For n=4: 22 is the smallest identical value of A049820(x) for 4 distinct numbers x: 25, 26, 28, 30.

Cf. A049820, A236561, A236562, A236563, A060990.

a(7)-a(11) from Giovanni Resta, Feb 10 2014

a(12) from Ryan Tang, Jul 23 2025

A236561 Values taken by the A049820, sorted into ascending order.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 5, 6, 6, 6, 9, 10, 11, 11, 11, 12, 14, 15, 16, 17, 17, 18, 21, 22, 22, 22, 22, 23, 26, 27, 27, 29, 29, 30, 31, 32, 34, 34, 35, 35, 38, 38, 39, 39, 41, 42, 44, 45, 46, 46, 46, 47, 48, 48, 51, 51, 53, 54, 57, 57, 57, 58, 58, 59, 60, 61, 62

Offset: 1

Jaroslav Krizek, Feb 09 2014

nonn

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A049820, A236562, A236563, A236564, A236565.

A262903 Numbers that are not leaves but all of whose children are leaves in the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

4, 5, 14, 16, 32, 41, 44, 77, 80, 92, 101, 110, 119, 128, 139, 148, 158, 161, 169, 176, 191, 192, 199, 215, 224, 227, 234, 238, 249, 262, 264, 277, 296, 311, 317, 327, 350, 351, 352, 360, 363, 382, 385, 389, 392, 395, 396, 411, 427, 430, 437, 448, 449, 461, 464, 483, 488, 518, 523, 531, 532, 542, 552, 561, 568, 570, 577, 579, 600, 601, 613, 619, 632, 634, 636, 645, 648, 659, 665, 666, 671, 682, 683, 696, 705, 723

Offset: 1

Antti Karttunen, Oct 06 2015

nonn

Numbers n for which A060990(n) > 0 and A060990(n) = A262900(n).

Numbers n for which A262695(n) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A049820, A060990, A262695, A262900.
Subsequence of A262901 and A236562.
No common terms with A259934.
Cf. also A257512.

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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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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A049820 a(n) = n - d(n), where d(n) is the number of divisors of n (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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A045765 k - d(k) never takes these values, where d(k) = A000005(k).

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A262511 Numbers k for which there is exactly one solution to x - d(x) = k, where d(k) is the number of divisors of k (A000005). Positions of ones in A060990.

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A262901 Numbers that have at least one leaf-child in the tree generated by edge-relation A049820(child) = parent.

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