A049820 -id:A049820 - cp's OEIS frontend

A262679 a(n) = largest k in A259934 which is an ancestor of n in a tree generated by edge-relation A049820(child) = parent. If n is itself in A259934, then a(n) = n.

0, 0, 2, 0, 0, 0, 6, 0, 0, 6, 6, 6, 12, 6, 6, 6, 6, 6, 18, 6, 6, 6, 22, 6, 6, 22, 22, 6, 22, 6, 30, 6, 22, 6, 34, 6, 6, 6, 34, 6, 22, 6, 42, 6, 34, 6, 46, 6, 34, 46, 34, 6, 46, 6, 54, 6, 34, 6, 58, 6, 34, 6, 62, 6, 6, 6, 58, 6, 62, 6, 70, 6, 34, 6, 70, 6, 70, 6, 78, 6, 70, 70, 78, 70, 34, 70, 78, 70, 70, 70, 90, 70, 78, 70, 94, 70, 34, 70, 78, 70, 70, 70, 102, 70, 34, 70, 106, 70, 34, 70, 102, 70, 102, 70, 114, 70, 102, 70, 118, 70, 34, 121, 118, 70

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

For the terms outside of A259934 the condition "largest k in A259934 which is an ancestor of n" is equivalent to the condition "nearest ancestor in A259934".

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

Cf. A000005, A049820, A259934, A262693.

Cf. A262522, A262695, A262696, A262697.

If A262693(n) = 1 [i.e., when n is in A259934], then a(n) = n, otherwise a(n) = a(A049820(n)).

A262892 Indices of nonbranching nodes in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 15, 16, 18, 21, 30, 31, 32, 36, 37, 39, 44, 45, 49, 50, 51, 53, 54, 55, 58, 60, 62, 65, 68, 71, 72, 73, 74, 75, 76, 80, 83, 84, 90, 91, 93, 96, 109, 112, 117, 122, 123, 124, 126, 127, 131, 134, 135, 137, 141, 142, 144, 145, 147, 149, 152, 154, 155, 161, 162, 165, 166, 170, 178, 187, 189, 190, 191, 193, 195, 199, 201, 203, 205, 211, 212, 213, 219, 223, 225

Offset: 1

Antti Karttunen, Oct 04 2015

nonn

Positions of zeros in A262890. Positions of ones in A262891.

Sequence b(n) = A262508(n)-1 = 9235, 9236, 9237, 9246, 9247, 9329, 9330, 9352, 9355, ..., is a subsequence.

Antti Karttunen, Table of n, a(n) for n = 1..4085

Cf. A049820, A259934, A262890, A262891, A262896.
Cf. A262897 (nonbranching nodes themselves).
Cf. A262508.

Other identities. For all n >= 1:

A262897(n) = A259934(a(n)) = A262896(a(n)).

A322996 Number of iterations of A049820(x) = x - A000005(x) needed to reach an odd number or zero, when starting from x = n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..110880

Cf. A000005, A049820, A155043, A259934, A322987, A322997 (bisection), A323073.

Cf. also A322983.

A322996[n_] := A322996[n] = If[n == 0 || OddQ[n], 0, 1 + A322996[n - DivisorSigma[0, n]]];
Array[A322996, 100, 0] (* Paolo Xausa, Jan 15 2025 *)

A322996(n) = if((!n)||(n%2),0,1+A322996(n-numdiv(n)));

A322996(n) = { for(j=0,oo,if((!n)||(n%2),return(j)); n -= numdiv(n)); };

a(0) = 0; for n >= 1, for n odd, a(n) = 0, and for n even, a(n) = 1 + a(n-A000005(n)).
a(n) <= A155043(n).
For n >= 83, a(2*n) = 1+A322987(2*n).

A323073 Number of iterations of A049820(x) = x - A000005(x) needed to reach either zero or such x that x and A049820(x) are coprime, when starting from x = n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..110880

Cf. A000005, A009191, A049820, A155043, A259934, A286540, A320009, A322974, A322987, A322996.

Cf. A046642 (positions of zeros after the initial a(0)=0).

A323073(n) = if(!n,0,my(nn=(n-numdiv(n))); if(1==gcd(n,nn),0,1+A323073(nn)));

A323073(n) = if(!n,0,for(j=0,oo,my(nn=(n-numdiv(n))); if((0==nn)||(1==gcd(n,nn)),return(j+(2==n)),n = nn)));

a(0) = 0; for n > 0, if A009191(n) == 1, a(n) = 0, otherwise a(n) = 1 + a(n-A000005(n)).

a(n) <= A155043(n).

A325022 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is not an integer, where k-tau(k) = the number of nondivisors of k (A049820), tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

140, 270, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 360360, 539400, 726180, 753480, 950976, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000, 2845800, 4358600, 4713984, 4754880, 5772200, 6051500

Offset: 1

Jaroslav Krizek, Mar 28 2019

nonn

Numbers m such that sigma(m) divides m*tau(m) but sigma(m) does not divide m*(m-tau(m)).

Complement of A325021 with respect to A001599.

			140 is a term because 140*(140-tau(140))/sigma(140) = 140*(140-12)/336 = 160/3.

Amiram Eldar, Table of n, a(n) for n = 1..682 (terms below 10^14)

Cf. A000005, A000203, A001599, A049820, A325020, A325021, A325023, A325024.

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and not IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]

Select[Range[10^5], And[IntegerQ@ HarmonicMean@ #4, ! IntegerQ[#1 (#1 - #2)/#3]] & @@ Append[{#}~Join~DivisorSigma[{0, 1}, #], Divisors@ #] &] (* Michael De Vlieger, Mar 30 2019 *)

isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && frac(m*(m-d)/s); \\ Michel Marcus, Mar 28 2019

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325022_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 n*n%s and not reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s:
            yield n
A325022_list = list(islice(A325022_gen(),10)) # Chai Wah Wu, Feb 14 2023

A325024 Multiply-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is not an integer where k-tau(k) is the number of the non-divisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

Original entry on oeis.org

120, 523776, 459818240, 1476304896, 31998395520, 51001180160, 518666803200, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, 87934476737668055040, 170206605192656148480, 1161492388333469337600, 1802582780370364661760

Offset: 1

Jaroslav Krizek, May 12 2019

nonn

Numbers m such that m divides sigma(m) but sigma(m) does not divide m*(m-tau(m)).

Complement of A325023 with respect to A007691.

			120 is a term because 120*(120-tau(120))/sigma(120) = 120*(120-16)/360 = 104/3.

Amiram Eldar, Table of n, a(n) for n = 1..218

Cf. A000005, A000203, A007691, A049820, A325020, A325021, A325022, A325023.

[n: n in [1..1000000] | not 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}, #], #] &] (* Amiram Eldar, Jul 10 2019 after Michael De Vlieger at A325023 *)

isA325024(m) = { my(s=sigma(m)); ((1==denominator(s/m)) && (1!=denominator(m*(m-numdiv(m))/s))); }; \\ Antti Karttunen, May 25 2019

A236561 Values taken by the A049820, sorted into ascending order.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 5, 6, 6, 6, 9, 10, 11, 11, 11, 12, 14, 15, 16, 17, 17, 18, 21, 22, 22, 22, 22, 23, 26, 27, 27, 29, 29, 30, 31, 32, 34, 34, 35, 35, 38, 38, 39, 39, 41, 42, 44, 45, 46, 46, 46, 47, 48, 48, 51, 51, 53, 54, 57, 57, 57, 58, 58, 59, 60, 61, 62

Offset: 1

Jaroslav Krizek, Feb 09 2014

nonn

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A049820, A236562, A236563, A236564, A236565.

A262888 a(n) = total number of nodes in the finite subtrees branching "left" (to the "smaller side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

6, 0, 41, 0, 0, 5, 0, 16, 0, 2, 0, 0, 1, 24, 4, 0, 0, 0, 0, 0, 0, 0, 105, 2, 0, 0, 0, 3, 18, 7, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 13, 1, 0, 0, 0, 0, 6, 1, 0, 0, 0, 47, 0, 0, 0, 90, 0, 0, 5, 0, 0, 0, 1, 0, 0, 12, 0, 0, 3, 61, 0, 0, 0, 0, 0, 0, 1, 117, 7, 0, 2, 10, 0, 0, 1, 23, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 568, 0, 1, 1, 4, 0, 5, 9, 0, 0, 0, 0, 0, 8, 0, 1, 1, 0, 2, 10, 1, 1, 0

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8107

Cf. A000005, A049820, A082284, A259934, A262686, A262697, A262889, A262890, A262894.

Cf. also A255328.

(define (A262888 n) (let ((t (A259934 n))) (let loop ((s 0) (k (A259934 (+ 1 n)))) (cond ((<= k t) s) ((= t (A049820 k)) (loop (+ s (A262697 k)) (- k 1))) (else (loop s (- k 1)))))))

a(n) = sum_{k = A082284(A259934(n)) .. A259934(n+1)} [A049820(k) = A259934(n)] * A262697(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:
A262890(n) = a(n) + A262889(n).

A262889 a(n) = total number of nodes in the finite subtrees branching "right" (to the "larger side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 13, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 5, 0, 4, 0, 1, 7, 0, 0, 7, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 5, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 3, 0, 22, 1, 0, 1, 2, 0, 6, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8107

Cf. A000005, A049820, A082284, A259934, A262686, A262697, A262888, A262890, A262894.

Cf. also A255329.

(define (A262889 n) (let ((t (A259934 n)) (u (A259934 (+ 1 n)))) (let loop ((s 0) (k (A262686 t))) (cond ((<= k u) s) ((= t (A049820 k)) (loop (+ s (A262697 k)) (- k 1))) (else (loop s (- k 1)))))))

a(n) = sum_{k = A259934(n+1) .. A262686(A259934(n))} [A049820(k) = A259934(n)] * A262697(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:
A262890(n) = A262888(n) + a(n).

A262897 Nonbranching nodes in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent: a(n) = A259934(A262892(n)).

Original entry on oeis.org

2, 12, 18, 30, 42, 54, 90, 94, 106, 121, 190, 194, 210, 236, 242, 254, 298, 302, 342, 346, 354, 366, 374, 390, 410, 426, 442, 466, 494, 530, 546, 558, 562, 566, 574, 606, 650, 658, 710, 716, 730, 746, 914, 942, 986, 1030, 1038, 1042, 1052, 1058, 1090, 1114, 1134, 1146, 1240, 1250, 1266, 1278, 1286, 1310, 1354, 1370, 1378, 1418, 1426, 1450, 1454, 1490, 1562, 1650, 1662, 1670, 1676, 1694, 1706

Offset: 1

Antti Karttunen, Oct 06 2015

nonn

Equally, numbers n for which A060990(n)*A262693(n) = 1, thus an intersection of A262511 and A259934.

The next odd term after a(10) = 121 occurs at a(3372) = 113569.

Antti Karttunen, Table of n, a(n) for n = 1..4085

Cf. A000005, A049820, A060990, A259934, A262511, A262693, A262892, A262896.

Cf. A262510 (a subsequence).

a(n) = A259934(A262892(n)).

cp's OEIS Frontend

A262679 a(n) = largest k in A259934 which is an ancestor of n in a tree generated by edge-relation A049820(child) = parent. If n is itself in A259934, then a(n) = n.

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A262892 Indices of nonbranching nodes in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

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A322996 Number of iterations of A049820(x) = x - A000005(x) needed to reach an odd number or zero, when starting from x = n.

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A323073 Number of iterations of A049820(x) = x - A000005(x) needed to reach either zero or such x that x and A049820(x) are coprime, when starting from x = n.

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A325022 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is not an integer, where k-tau(k) = the number of nondivisors of k (A049820), tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

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Magma

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A325024 Multiply-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is not an integer where k-tau(k) is the number of the non-divisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

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Magma

Mathematica

PARI

A236561 Values taken by the A049820, sorted into ascending order.

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A262888 a(n) = total number of nodes in the finite subtrees branching "left" (to the "smaller side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

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Formula

A262889 a(n) = total number of nodes in the finite subtrees branching "right" (to the "larger side") from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

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