A262901 -id:A262901 - cp's OEIS frontend

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

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

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

A262902 a(n) = A049820(A045765(n)); parent-nodes of the leaves of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 06 2015

nonn

The sequence is computed for each leaf of the tree (A045765), ordered by their magnitude, and it contains duplicates.

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

Row 2 of A262898.
Cf. A262901 (same sequence sorted into ascending order, with duplicates removed).
Cf. A049820, A045765.
Cf. also A257507.

(define (A262902 n) (A049820 (A045765 n)))

a(n) = A049820(A045765(n)).

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.

A262900 a(n) = number of leaf-children n has in the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 06 2015

nonn

a(n) = number of such terms k in A045765 for which k - d(k) = n [where d(k) is the number of divisors of k, A000005(k)].

			a(4) = 1, as there is only one such term k in A045765 which satisfies the condition A049820(k) = 4, namely 8 (8 - d(8) = 4).
a(5) = 1, as the only term in A045765 satisfying the condition is 7, as 7 - d(7) = 5.
a(22) = 2, as there are exactly two terms in A045765 satisfying the condition, namely 25 and 28, as 25 - d(25) = 28 - d(28) = 22.

Antti Karttunen, Table of n, a(n) for n = 0..65538

Cf. A000005, A045765, A049820, A060990, A082284, A262686.

Cf. A262901 (indices of nonzero terms), A262902.

(define (A262900 n) (let loop ((s 0) (k (A262686 n))) (cond ((<= k n) s) ((= n (A049820 k)) (loop (+ s (if (zero? (A060990 k)) 1 0)) (- k 1))) (else (loop s (- k 1))))))

a(n) = Sum_{k = A082284(n) .. A262686(n)} [A049820(k) = n] * [A060990(k) = 0].

In the above formula [ ] stands for Iverson bracket, giving in the first instance as its result 1 only when A049820(k) = n (that is, when k is really a child of n), and 0 otherwise, and in the second instance 1 only when A060990(k) = 0 (that is, when k itself has no children), and 0 otherwise. - Comment corrected by Antti Karttunen, Nov 27 2015

cp's OEIS Frontend

A049820 a(n) = n - d(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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A262902 a(n) = A049820(A045765(n)); parent-nodes of the leaves of the tree generated by edge-relation A049820(child) = parent.

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

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A262900 a(n) = number of leaf-children n has in the tree generated by edge-relation A049820(child) = parent.

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A263261 Row 1 of A263255.

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