A049820 -id:A049820 - cp's OEIS frontend

A262904 If n = A259934(k) then a(n) = k, otherwise largest k such that A259934(k) is an ancestor of n in a tree generated by edge-relation A049820(child) = parent.

0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 5, 2, 2, 5, 5, 2, 5, 2, 6, 2, 5, 2, 7, 2, 2, 2, 7, 2, 5, 2, 8, 2, 7, 2, 9, 2, 7, 9, 7, 2, 9, 2, 10, 2, 7, 2, 11, 2, 7, 2, 12, 2, 2, 2, 11, 2, 12, 2, 13, 2, 7, 2, 13, 2, 13, 2, 14, 2, 13, 13, 14, 13, 7, 13, 14, 13, 13, 13, 15, 13, 14, 13, 16, 13, 7, 13, 14, 13, 13, 13, 17, 13, 7, 13, 18, 13, 7, 13, 17, 13, 17, 13, 19, 13, 17, 13, 20, 13, 7, 21

Offset: 0

Antti Karttunen, Oct 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32767

Cf. A049820, A262679, A262693, A262694, A262896, A262905, A262906, A262907.

If A262693(n) = 1 then a(n) = A262694(n) [i.e., when n = A259934(k), a(n) = k], otherwise a(n) = a(A049820(n)).
a(n) = A262694(A262679(n)).
Other identities. For all n >= 0:
a(A262896(n)) = n. [This sequence works as a left inverse for injection A262896.]

A262510 Parent nodes of nonzero terms of A262509: a(n) = A049820(A262509(n)).

Original entry on oeis.org

119139, 119143, 119147, 119213, 119225, 119919, 119921, 120073, 120091, 120095, 120097, 120277, 120291, 120347, 120391, 120703, 120739, 120883, 120891, 120895, 120915, 120917, 121435, 121543, 121819, 122075, 122257, 122261, 122271, 122273, 122809, 122953, 123197, 123205, 123219, 123231, 123251, 123749, 24660527, 24660543, 24662309, 24662321, 24663755, 24664989, 24665019, 24665347, 24665929, 24665977, 24669139, 24669833

Offset: 1

Antti Karttunen, Sep 25 2015

nonn

These numbers are one step nearer (than those of A262509) to the root (zero) of the tree where the parent-child relation is given by A049820(child) = parent. Like the terms of A262509, they are also vertices in the infinite trunk of that tree. Cf. A259934.

Antti Karttunen, Table of n, a(n) for n = 1..68

Cf. A049820, A262508, A262509.
Subsequence of A259934 and A262511.
Also a subsequence of A262517 (provided all terms are odd).

(define (A262510 n) (A049820 (A262509 n)))

a(n) = A049820(A262509(n)).

a(n) = A259934(A262508(n)-1).

A262696 a(n)=0 if n is in A259934, otherwise number of terminal nodes (including n itself if it is a leaf) in that finite subtree whose root is n and whose edge-relation is defined by A049820(child) = parent.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

			For n=1, its transitive closure (as defined by edge-relation A049820(child) = parent) is the union of {1} itself together with all its descendants: {1, 3, 4, 5, 7, 8}. We see that there are no other nodes in a subtree whose root is 1, because A049820(3) = 3 - d(3) = 1, A049820(4) = 1, A049820(5) = 3, A049820(7) = 5, A049820(8) = 4 and of these only 7 and 8 are terms of A045765. Thus a(1) = 2.
For n=9, its transitive closure is {9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79}, of which only thirteen members: {13, 19, 24, 33, 36, 37, 43, 55, 63, 64, 67, 75, 79} are leaves (in A045765), thus a(9) = 13.

Antti Karttunen, Table of n, a(n) for n = 0..17724

Cf. A000005, A045765, A049820, A060990, A082284, A259934, A262686, A262693.
Cf. A262522, A262695, A262697.
Cf. also A213726.

If A262693(n) = 1 [when n is in A259934],
  then a(n) = 0,
otherwise, if A060990(n) = 0 [when n is one of the leaves, A045765],
  then a(n) = 1,
otherwise:
  a(n) = Sum_{k = A082284(n) .. A262686(n)} [A049820(k) = n] * a(k).
(In the last clause [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = n, and 0 otherwise).
Other identities:
For any n in A262511 but not in A259934, a(n) = a(A082284(n)).

A262896 If n is in A262892, a(n) = A259934(n), otherwise the largest term in A045765 from which A259934(n) can be reached by iterating A049820, without visiting any other (larger) term of A259934.

Original entry on oeis.org

8, 2, 79, 12, 18, 40, 30, 140, 42, 52, 54, 66, 68, 123, 98, 90, 94, 116, 106, 126, 164, 121, 369, 133, 156, 168, 180, 184, 280, 229, 190, 194, 210, 218, 252, 246, 236, 242, 272, 254, 312, 324, 300, 364, 298, 302, 372, 356, 334, 342, 346, 354, 439, 366, 374, 390, 672, 414, 410, 438, 426, 460, 442, 452, 470, 466, 564, 496, 494, 524, 627, 530, 546, 558, 562, 566, 574, 592, 859, 660, 606, 642, 708, 650

Offset: 0

Antti Karttunen, Oct 06 2015

nonn

a(n) is the largest leaf-node among the finite subtrees branching from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent, and A259934(n) itself if it is one of the nonbranching nodes (A262897).

Note that without (so far undetected) regularity in A262509, there is no a priori upper bound for the value of a(n), and for some n this might not even be finite, if it happens that contrary to its conjectured nature, A259934 is not the unique infinite component, but just the lexicographically earliest instance of multiple infinite branches of the tree. In that case we might consider this sequence to be well-defined only up to the least such node branching to multiple infinite components, or alternatively, we might mark the nonfinite values at those points with -1.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Max Alekseyev & Antti Karttunen, Standalone C++-program for computing this sequence

Cf. A000005, A049820, A082284, A259934, A262509, A262522, A262679, A262686, A262890, A262892, A262897, A262904, A263081.

(define (A262896 n) (let ((t (A259934 n))) (let loop ((m t) (k (A262686 t))) (cond ((<= k t) m) ((= t (A049820 k)) (loop (max m (A262522 k)) (- k 1))) (else (loop m (- k 1)))))))

a(n) = max(A259934(n), Max_{k = A082284(A259934(n)) .. A262686(A259934(n))} [A049820(k) = A259934(n)] * A262522(k)).
(Here [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = A259934(n), and 0 otherwise).
Other identities. For all n >= 0:
A262904(a(n)) = n. [A262904 works as a left inverse for this sequence.]
A259934(n) = A262679(a(n)).
For all n >= 1:
a(A262892(n)) = A259934(A262892(n)) = A262897(n).

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.

A325021 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 6, 28, 496, 672, 8128, 30240, 32760, 332640, 695520, 2178540, 17428320, 23569920, 33550336, 45532800, 52141320, 142990848, 164989440, 318729600, 447828480, 481572000, 500860800, 540277920, 623397600, 644271264, 714954240, 995248800, 1047254400, 1307124000

Offset: 1

Jaroslav Krizek, Mar 27 2019

nonn

Numbers m such that m*tau(m)/sigma(m) is an integer g and simultaneously m*(m-tau(m))/sigma(m) is an integer h. Corresponding values of integers g: 1, 2, 3, 5, 8, 7, 24, 24, 44, 46, 54, 96, 80, 13, 96, ...
Corresponding values of integers h: 0, 1, 11, 243, 216, 4057, 7536, 8166, 76186, 166589, ...
Even perfect numbers from A000396 are terms.
Complement of A325022 with respect to A001599.
Intersection of A325020 and A001599.

			Harmonic number 28 is a term because 28*tau(28)/sigma(28) = 28*6/56 = 3 (integer) and simultaneously 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..255 (terms below 10^14)

Cf. A000005, A000203, A001599, A007691, A049820, A325020, A325022, A325023, A325024.

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]

Select[Range[10^6], And[IntegerQ@ HarmonicMean@ #2, IntegerQ[#1 (#1 - #3)/#4]] & @@ Join[{#}, {Divisors@ #}, DivisorSigma[{0, 1}, #]] &] (* Michael De Vlieger, Mar 27 2019 *)

isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && !frac(m*(m-d)/s); \\ Michel Marcus, Mar 27 2019

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325021_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
        if not (n*n%s or reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s):
            yield n
A325021_list = list(islice(A325021_gen(),10)) # Chai Wah Wu, Feb 14 2023

A325023 Multi-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 6, 28, 496, 672, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 14182439040, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 13661860101120, 181742883469056, 6088728021160320

Offset: 1

Jaroslav Krizek, Mar 24 2019

nonn

Numbers m such that sigma(m)/m is an integer f and simultaneously m*tau(m)/sigma(m) is an integer g. Corresponding values of integers f: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, ... Corresponding values of integers g: 0, 1, 11, 243, 216, 4057, 7536, 8166, ...
Complement of A325024 with respect to A007691.
Even perfect numbers from A000396 are terms.
Intersection of A325020 and A007691.
Conjecture: Numbers m such that all values of sigma(m)/m, m*tau(m)/sigma(m) and m*(m-tau(m))/sigma(m) are any integers (f, g, and h respectively). Corresponding values of integers f: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, ... Corresponding values of integers g: 0, 1, 11, 243, 216, 4057, 7536, 8166, ... Corresponding values of integers h: 1, 2, 3, 5, 8, 7, 24, 24, 54, 80, 13, 96, ...

			Multi-perfect number 28 is a term because 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..512

Cf. A000005, A000203, A000396, A007691, A049820, A325020, A325021, A325022, A325024.

[n: n in [1..1000000] | IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

Select[Range[10^6], And[Mod[#3, #1] == 0, IntegerQ[#1 (#1 - #2)/#3]] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Michael De Vlieger, Mar 24 2019 *)

isok(m) = my(s=sigma(m)); (frac(m*(m-numdiv(m))/s) == 0) && (frac(s/m) == 0); \\ Michel Marcus, Mar 25 2019

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

A262677 Number of odd numbers encountered when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = A000035(n) + a(A049820(n)).

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 4, 1, 1, 0, 2, 0, 3, 0, 3, 2, 4, 0, 5, 0, 5, 0, 6, 2, 1, 0, 7, 0, 8, 0, 9, 0, 9, 0, 10, 7, 11, 0, 11, 0, 12, 0, 13, 0, 12, 0, 13, 0, 1, 0, 14, 0, 15, 0, 15, 0, 16, 0, 17, 0, 18, 0, 17, 16, 19, 0, 20, 0, 20, 0, 21, 0, 22, 0, 21, 0, 23, 0, 24, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 0, 5, 0, 5, 0, 6, 0, 6, 4, 7, 0, 8, 0, 7, 0, 8

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Number of odd numbers encountered before zero is reached when starting from k = n and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005). This count includes n itself if it is odd.

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A000005, A000035, A049820, A155043, A262676, A262680.

a(0) = 0; for n >= 1, a(n) = A000035(n) + a(A049820(n)).
Other identities. For all n >= 0:
A155043(n) = A262676(n) + a(n).

cp's OEIS Frontend

A262904 If n = A259934(k) then a(n) = k, otherwise largest k such that A259934(k) is an ancestor of n in a tree generated by edge-relation A049820(child) = parent.

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A262510 Parent nodes of nonzero terms of A262509: a(n) = A049820(A262509(n)).

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A262696 a(n)=0 if n is in A259934, otherwise number of terminal nodes (including n itself if it is a leaf) in that finite subtree whose root is n and whose edge-relation is defined by A049820(child) = parent.

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A262896 If n is in A262892, a(n) = A259934(n), otherwise the largest term in A045765 from which A259934(n) can be reached by iterating A049820, without visiting any other (larger) term of A259934.

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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)).

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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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A325021 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of the divisors of k (A000203).

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A325023 Multi-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

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Magma

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