A049816 -id:A049816 - cp's OEIS frontend

A049826 a(n) = T(n,n) + T(n+1,n) + ... + T(2n-1,n) = sum over a period of n-th column of array T given by A049816.

0, 1, 3, 4, 8, 7, 13, 14, 16, 17, 27, 20, 32, 31, 31, 34, 46, 41, 55, 44, 50, 55, 73, 54, 68, 73, 75, 72, 96, 71, 101, 90, 96, 105, 101, 92, 124, 119, 123, 110, 146, 113, 155, 132, 132, 151, 177, 138, 164, 161, 169, 164, 204, 167, 183, 166, 192, 201, 231, 176

Offset: 1

Clark Kimberling

nonn

a(n) = sum of terms in row n of A051010.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A049816, A051010.

A049817 a(n) = Sum_{k=1..n} T(n,k), array T as in A049816.

Original entry on oeis.org

0, 0, 1, 1, 4, 2, 7, 7, 8, 8, 17, 9, 20, 18, 17, 19, 30, 24, 37, 25, 30, 34, 51, 31, 44, 48, 49, 45, 68, 42, 71, 59, 64, 72, 67, 57, 88, 82, 85, 71, 106, 72, 113, 89, 88, 106, 131, 91, 116, 112, 119, 113, 152, 114, 129, 111, 136, 144, 173

Offset: 1

Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A049816.

A049823 a(n)=number of 2's in n-th row of array T given by A049816.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 1, 4, 4, 4, 4, 7, 2, 6, 5, 6, 8, 8, 5, 7, 7, 8, 7, 10, 7, 10, 9, 9, 11, 12, 6, 11, 12, 10, 10, 12, 11, 10, 15, 12, 14, 14, 10, 14, 11, 16, 14, 12, 12, 17, 16, 16, 16, 14, 14, 10, 18, 15, 16, 20, 17, 16, 18, 19, 19

Offset: 1

Clark Kimberling

nonn

			There are 4 numbers k from 1 to 11 for which the Euclidean algorithm on (11,k) has exactly 2 nonzero remainders; hence a(11)=4.

A049818 a(n) = Sum_{k=1..n, m=1..n} T(m,k), array T as in A049816.

Original entry on oeis.org

0, 0, 1, 2, 6, 8, 15, 22, 30, 38, 55, 64, 84, 102, 119, 138, 168, 192, 229, 254, 284, 318, 369, 400, 444, 492, 541, 586, 654, 696, 767, 826, 890, 962, 1029, 1086, 1174, 1256, 1341, 1412, 1518, 1590, 1703, 1792, 1880, 1986, 2117

Offset: 1

Clark Kimberling

nonn

Partial sums of A049817.

Cf. A049816.

A049819 a(n) = Max_{k=1..n} T(n,k), array T as in A049816.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 3, 4, 3, 3, 3, 3, 4, 4, 3, 5, 3, 4, 3, 4, 4, 4, 4, 5, 5, 5, 4, 4, 6, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 6, 5, 5, 5, 4, 7, 5, 5, 5, 5, 6, 5, 5, 5, 6, 6, 6, 6, 6, 6, 5

Offset: 1

Clark Kimberling

nonn

A049824 a(n) = number of 3's in n-th row of array T given by A049816.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 1, 2, 1, 2, 4, 2, 4, 3, 0, 4, 8, 5, 3, 5, 6, 5, 7, 3, 6, 9, 7, 8, 9, 6, 8, 12, 9, 8, 9, 7, 12, 11, 5, 12, 18, 12, 10, 9, 11, 12, 17, 16, 12, 14, 10, 18, 22, 12, 11, 16, 11, 16

Offset: 1

Clark Kimberling

nonn

			There are 2 numbers k from 1 to 11 for which the Euclidean algorithm on (11,k) has exactly 3 nonzero remainders; hence a(11)=2.

Cf. A049816.

A049825 a(n)=number of 4's in n-th row of array T given by A049816.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 1, 0, 2, 0, 3, 3, 2, 2, 3, 1, 4, 2, 4, 2, 2, 4, 6, 4, 6, 3, 7, 3, 10, 2, 5, 6, 6, 5, 6, 7, 6, 7, 13, 8, 6, 5, 9, 10, 14, 3, 19, 10, 11, 8, 8, 7, 14, 8, 10, 9

Offset: 1

Clark Kimberling

nonn

A049827 a(n) = T(2n-1,n)+T(2n,n+1)+...+T(3n-3,2n-2) = sum over a period of n-th diagonal of array T given by A049816.

Original entry on oeis.org

0, 1, 3, 6, 8, 13, 13, 20, 22, 25, 27, 38, 32, 45, 45, 46, 50, 63, 59, 74, 64, 71, 77, 96, 78, 93, 99, 102, 100, 125, 101, 132, 122, 129, 139, 136, 128, 161, 157, 162, 150, 187, 155, 198, 176, 177, 197, 224, 186, 213, 211, 220, 216, 257, 221, 238, 222, 249

Offset: 1

Clark Kimberling

nonn

Cf. A049816.

More terms from Sean A. Irvine, Aug 07 2021

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

cp's OEIS Frontend

A049826 a(n) = T(n,n) + T(n+1,n) + ... + T(2n-1,n) = sum over a period of n-th column of array T given by A049816.

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A049817 a(n) = Sum_{k=1..n} T(n,k), array T as in A049816.

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A049823 a(n)=number of 2's in n-th row of array T given by A049816.

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A049818 a(n) = Sum_{k=1..n, m=1..n} T(m,k), array T as in A049816.

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A049819 a(n) = Max_{k=1..n} T(n,k), array T as in A049816.

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A049824 a(n) = number of 3's in n-th row of array T given by A049816.

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A049825 a(n)=number of 4's in n-th row of array T given by A049816.

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A049827 a(n) = T(2n-1,n)+T(2n,n+1)+...+T(3n-3,2n-2) = sum over a period of n-th diagonal of array T given by A049816.

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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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Mathematica

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