A045765 -id:A045765 - cp's OEIS frontend

A264970 If A262686(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A262686(n)), where A262686(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

12, 2, 11, 2, 1, 1, 10, 0, 0, 3, 2, 2, 9, 0, 1, 5, 1, 4, 8, 0, 0, 3, 7, 2, 0, 0, 2, 1, 0, 1, 6, 6, 1, 0, 5, 5, 0, 0, 6, 4, 0, 1, 4, 0, 1, 3, 3, 2, 5, 0, 0, 1, 0, 2, 2, 0, 0, 1, 1, 4, 4, 3, 3, 0, 0, 2, 0, 0, 0, 1, 2, 3, 3, 2, 0, 0, 2, 1, 4, 0, 1, 1, 3, 3, 2, 0, 2, 2, 0, 4, 3, 1, 1, 3, 2, 5, 1, 4, 0, 2, 0

Offset: 0

Antti Karttunen, Nov 29 2015

nonn

a(n) = number of iterations of A262686 needed before zero is reached. In the context of tree (A263267) defined by edge-relation A049820(child) = parent, this is the number of hops we make before reaching one of the leaves (A045765), when we start from n and always select the largest child at each iteration.

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

Cf. A000005, A049820, A060990, A262686, A263267.
Cf. A045765 (positions of zeros).
One less than A264971.

If A060990(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A262686(n)).
Other identities. For all n >= 0:
a(n) = A264971(n) - 1.

A266110 If A082284(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A082284(n)), where A082284(n) = smallest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

Original entry on oeis.org

4, 3, 4, 2, 1, 1, 3, 0, 0, 2, 2, 1, 3, 0, 1, 2, 1, 1, 2, 0, 0, 6, 1, 5, 0, 0, 2, 4, 0, 3, 4, 2, 1, 0, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 1, 8, 1, 7, 1, 0, 0, 6, 0, 5, 3, 0, 0, 4, 2, 3, 7, 2, 1, 0, 0, 1, 0, 0, 0, 4, 1, 3, 6, 2, 0, 0, 8, 1, 4, 0, 1, 7, 3, 6, 5, 0, 2, 5, 0, 4, 12, 5, 1, 3, 11, 4, 4, 3, 0, 2, 0, 1, 10, 0, 3, 2, 9, 1, 0, 0, 1, 1, 0, 0, 8, 2, 0, 0, 7, 1, 2, 6

Offset: 0

Antti Karttunen, Dec 21 2015

nonn

Starting from n, search for a smallest number k such that k - d(k) = n, and if found such a number, replace n with k and repeat the procedure. When eventually such k is no longer found, then (new) n must be one of the terms of A045765. The number of times the procedure can be repeated before that happens is the value of a(n). Sequence A266116 gives the stopping value of n.

			Starting from n = 21, we get the following chain: 21 -> 23 -> 27 -> 29 -> 31 -> 35 -> 37, with A082284 iterated 6 times before the final term 37 (for which A060990(37) = A082284(37) = 0) is encountered. Thus a(21) = 6.

Antti Karttunen, Table of n, a(n) for n = 0..124340

One less than A266111.
Cf. A000005, A060990, A082284, A266116.
Cf. A045765 (positions of zeros).
Cf. tree A263267 (and its illustration).
Cf. also A264970.

A266116 The last nonzero term on each row of A265751.

Original entry on oeis.org

7, 7, 13, 7, 8, 7, 13, 7, 8, 13, 20, 13, 25, 13, 20, 19, 24, 19, 25, 19, 20, 37, 25, 37, 24, 25, 40, 37, 28, 37, 50, 37, 40, 33, 50, 37, 36, 37, 50, 43, 40, 43, 49, 43, 50, 67, 49, 67, 56, 49, 50, 67, 52, 67, 68, 55, 56, 67, 68, 67, 136, 67, 68, 63, 64, 67, 66, 67, 68, 79, 74, 79, 136, 79, 74, 75, 103, 79, 98, 79, 88, 103, 98, 103, 136, 85

Offset: 0

Antti Karttunen, Dec 21 2015

nonn

Starting from j = n, search for a smallest number k such that k - d(k) = j, and if found such a number, replace j with k and repeat the procedure. When eventually such k is no longer found, then the (last such) j must be one of the terms of A045765, and it is set as the value of a(n).

			Starting from n = 21, we get the following chain: 21 -> 23 -> 27 -> 29 -> 31 -> 35 -> 37, with A082284 iterated 6 times before the final term 37 (for which A060990(37) = A082284(37) = 0) is encountered. Thus a(21) = 37.

Antti Karttunen, Table of n, a(n) for n = 0..124340

Cf. A000005, A045765, A060990, A082284, A265751.
Cf. A266110 (gives the number of iterations of A082284 needed before a(n) is found).
Cf. also tree A263267 (and its illustration).

(definec (A266116 n) (cond ((A082284 n) => (lambda (lad) (if (zero? lad) n (A266116 lad))))))
;; Alternatively:
(define (A266116 n) (A265751bi n (A266110 n))) ;; Code for A265751bi given in A265751.

a(n) = A265751(n, A266110(n)).
If A060990(n) = 0, a(n) = n, otherwise a(n) = a(A082284(n)), where A082284(n) = smallest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).
Other identities and observations. For all n >= 0:
a(n) >= n.
A060990(a(n)) = 0. [All terms are in A045765.]

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

A386571 Numbers that are not of the form k - omega(k), where omega(k) is the number of distinct primes dividing k.

Original entry on oeis.org

5, 9, 11, 14, 17, 21, 23, 25, 29, 35, 41, 45, 47, 51, 59, 62, 64, 65, 68, 69, 71, 76, 77, 79, 95, 101, 103, 105, 118, 119, 125, 128, 149, 152, 154, 155, 161, 163, 184, 188, 189, 191, 193, 197, 202, 208, 209, 218, 220, 221, 229, 236, 239, 241, 244, 253, 254, 258

Offset: 1

Amiram Eldar, Jul 26 2025

nonn

Luca (2005) proved that this sequence is infinite.

Amiram Eldar, 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.

Cf. A001221, A378993.

Numbers not of the form k-f(k): A005278 (phi), A045765 (d), this sequence (omega), A386572 (Omega).

seq[lim_] := Complement[Range[lim], Table[k - PrimeNu[k], {k, 1, lim + Log2[lim]}]]; seq[300]

list(lim) = setminus(vector(lim, i, i), Set(vector(lim + logint(lim, 2), i, i - omega(i))));

A386572 Numbers that are not of the form k - Omega(k), where Omega(k) is the number of prime factors of k counted with multiplicity.

Original entry on oeis.org

3, 11, 14, 21, 26, 29, 34, 35, 38, 45, 48, 51, 54, 57, 59, 61, 62, 64, 68, 69, 71, 74, 76, 79, 81, 87, 94, 97, 98, 101, 105, 110, 118, 123, 124, 125, 129, 133, 134, 137, 142, 147, 149, 155, 158, 160, 165, 170, 173, 174, 177, 182, 184, 186, 188, 189, 191, 193, 197

Offset: 1

Amiram Eldar, Jul 26 2025

nonn

Luca (2005) proved that this sequence is infinite, and Kátai (2006) proved that it has a positive lower density.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Imre Kátai, A remark on a paper of Luca, Acta Mathematica Hungarica, Vol. 113, No. 4 (2006), pp. 313-318.
Florian Luca, On numbers not of the form n-omega(n), Acta Mathematica Hungarica, Vol. 106, No. 1 (2005), pp. 117-135.

Cf. A001222, A069345.

Numbers not of the form k-f(k): A005278 (phi), A045765 (d), A386571 (omega), this sequence (Omega).

seq[lim_] := Complement[Range[lim], Table[k - PrimeOmega[k], {k, 1, lim + Log2[lim]}]]; seq[200]

list(lim) = setminus(vector(lim, i, i), Set(vector(lim + logint(lim, 2), i, i - bigomega(i))));

A348093 Numbers k >= 1 such that there is no pair (x,y) such that x - d(x) = k or y + d(y) = k, where d = A000005 = number of divisors.

Original entry on oeis.org

8, 20, 36, 40, 67, 68, 79, 88, 100, 116, 117, 131, 132, 134, 140, 156, 164, 167, 180, 185, 196, 204, 228, 244, 252, 268, 276, 284, 300, 308, 312, 321, 324, 341, 348, 370, 372, 379, 388, 401, 405, 408, 420, 425, 436, 439, 453, 460, 476, 479

Offset: 1

Ctibor O. Zizka, Sep 29 2021

nonn

Numbers k >= 1 such that A060990(k) + A036431(k) = 0.

			k = 8 is a term: there are no x,y such that x - d(x) = 8, y + d(y) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A036431, A049820, A060990, A062249.

Intersection of A036434 and A045765.

With[{max = 480}, Complement[Range[max], Select[Union[Flatten[Table[n + DivisorSigma[0, n]*{-1, 1}, {n, 1, max + 2 + 2*Ceiling[Sqrt[2*max+4]]}]]], # <= max &]]] (* Amiram Eldar, Mar 04 2023 *)

okp(k) = sum(i=1, k, i+numdiv(i) == k) == 0;
okm(k) = sum(i=1, 2*k+2, i-numdiv(i) == k) == 0;
isok(k) = okp(k) && okm(k); \\ Michel Marcus, Oct 01 2021

A060991 a(n) is the smallest positive integer c such that the equation A049820(x) = c has exactly n solutions.

Original entry on oeis.org

7, 2, 1, 6, 22, 838, 17638, 192520, 3240114, 219476872, 2146772872, 24443168392

Offset: 0

Labos Elemer, May 11 2001

Essentially same as A236565, except here for n=2 we have a(2) = 1 instead of A236565(2) = 0, because this sequence requires its terms to be strictly positive. - Antti Karttunen, Oct 09 2015

			The solution sets of smallest values of x-d(x) deviations with 1, 2, 3, 4, 5, 6 terms are as follows: {6}, {3, 4}, {9, 10, 12}, {25, 26, 28, 30}, {841, 842, 844, 848, 850}, {17642, 17648, 17650, 17654, 17658, 17670}. Thus difference x-d(x) for x={25, 26, 28, 30} with d(x)={3, 4, 6, 8} divisors is equally 22, so a(4)=22.

Cf. A000005, A049820, A045765, A049816, A236565.

s = Array[# - DivisorSigma[0, #] &, {20000}]; t = Length@ Position[s, #] & /@ Range@ Max@ s; Table[FirstPosition[t, n], {n, 0, 6}] // Flatten (* Michael De Vlieger, Oct 09 2015 *)

a(9)-a(11) from Donovan Johnson, Jan 08 2009

cp's OEIS Frontend

A264970 If A262686(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A262686(n)), where A262686(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

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A266110 If A082284(n) = 0, a(n) = 0, otherwise a(n) = 1 + a(A082284(n)), where A082284(n) = smallest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

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A266116 The last nonzero term on each row of A265751.

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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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A386571 Numbers that are not of the form k - omega(k), where omega(k) is the number of distinct primes dividing k.

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A386572 Numbers that are not of the form k - Omega(k), where Omega(k) is the number of prime factors of k counted with multiplicity.

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A348093 Numbers k >= 1 such that there is no pair (x,y) such that x - d(x) = k or y + d(y) = k, where d = A000005 = number of divisors.

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A060991 a(n) is the smallest positive integer c such that the equation A049820(x) = c has exactly n solutions.

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