A262902 -id:A262902 - cp's OEIS frontend

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

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

A262898 Square array A(row,col) read by antidiagonals: A(1,col) = A045765(col); for row > 1, if A(row-1,col) = 0 then A(row,col) = 0, otherwise A(row,col) = A049820(A(row-1,col)).

Original entry on oeis.org

7, 8, 5, 13, 4, 3, 19, 11, 1, 1, 20, 17, 9, 0, 0, 24, 14, 15, 6, 0, 0, 25, 16, 10, 11, 2, 0, 0, 28, 22, 11, 6, 9, 0, 0, 0, 33, 22, 18, 9, 2, 6, 0, 0, 0, 36, 29, 18, 12, 6, 0, 2, 0, 0, 0, 37, 27, 27, 12, 6, 2, 0, 0, 0, 0, 0, 40, 35, 23, 23, 6, 2, 0, 0, 0, 0, 0, 0, 43, 32, 31, 21, 21, 2, 0, 0, 0, 0, 0, 0, 0, 49, 41, 26, 29, 17, 17, 0, 0, 0, 0, 0, 0, 0, 0, 50, 46, 39, 22, 27, 15, 15, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 06 2015

The array is read by downwards antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
Column n gives the trajectory of iterates of A049820, when starting from A045765(n), thus stepping through successive parent-nodes when starting from the n-th leaf in the tree generated by edge-relation A049820(child) = parent, until finally reaching the fixed point 0, which is the root of the whole tree.
A portion of the hanging tail of each column (upward from the first encountered zero) converges towards A259934, although not in monotone fashion.

			The top left corner of the array:
7, 8, 13, 19, 20, 24, 25, 28, 33, 36, 37, 40, 43, 49, 50, 52, 55, 56
5, 4, 11, 17, 14, 16, 22, 22, 29, 27, 35, 32, 41, 46, 44, 46, 51, 48
3, 1,  9, 15, 10, 11, 18, 18, 27, 23, 31, 26, 39, 42, 38, 42, 47, 38
1, 0,  6, 11,  6,  9, 12, 12, 23, 21, 29, 22, 35, 34, 34, 34, 45, 34
0, 0,  2,  9,  2,  6,  6,  6, 21, 17, 27, 18, 31, 30, 30, 30, 39, 30
0, 0,  0,  6,  0,  2,  2,  2, 17, 15, 23, 12, 29, 22, 22, 22, 35, 22
0, 0,  0,  2,  0,  0,  0,  0, 15, 11, 21,  6, 27, 18, 18, 18, 31, 18
0, 0,  0,  0,  0,  0,  0,  0, 11,  9, 17,  2, 23, 12, 12, 12, 29, 12
0, 0,  0,  0,  0,  0,  0,  0,  9,  6, 15,  0, 21,  6,  6,  6, 27,  6
0, 0,  0,  0,  0,  0,  0,  0,  6,  2, 11,  0, 17,  2,  2,  2, 23,  2
0, 0,  0,  0,  0,  0,  0,  0,  2,  0,  9,  0, 15,  0,  0,  0, 21,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  6,  0, 11,  0,  0,  0, 17,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  2,  0,  9,  0,  0,  0, 15,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  6,  0,  0,  0, 11,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2,  0,  0,  0,  9,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  6,  0
...

Antti Karttunen, Table of n, a(n) for n = 1..5050; the first 100 antidiagonals of the array

Transpose: A262899.
Cf. A045765 (row 1), A262902 (row 2).
Cf. A049820, A259934.
Cf. also A257264.

(define (A262898 n) (A262898bi (A002260 n) (A004736 n)))
(define (A262898bi row col) (if (= 1 row) (A045765 col) (if (zero? (A262898bi (- row 1) col)) 0 (A049820 (A262898bi (- row 1) col)))))

A(1,col) = A045765(col), and for row > 1, if A(row-1,col) = 0 then A(row,col) = 0, otherwise A(row,col) = A049820(A(row-1,col)).

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.

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

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

A262898 Square array A(row,col) read by antidiagonals: A(1,col) = A045765(col); for row > 1, if A(row-1,col) = 0 then A(row,col) = 0, otherwise A(row,col) = A049820(A(row-1,col)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula