A049820 -id:A049820 - cp's OEIS frontend

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

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.

A263091 Primes p for which A049820(x) = p has no solution.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 103, 109, 113, 131, 163, 167, 193, 229, 241, 251, 257, 271, 293, 307, 313, 353, 359, 379, 383, 397, 401, 439, 463, 479, 487, 491, 499, 503, 509, 563, 571, 647, 653, 661, 673, 701, 739, 743, 757, 761, 773, 823, 859, 863, 883, 887, 911, 937, 941, 953, 967, 971, 977, 983, 1009, 1093, 1103, 1109, 1171, 1181, 1193, 1217, 1279, 1283, 1291, 1297, 1307, 1321, 1361

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Primes p that there is no such k for which k - d(k) = p, where d(k) is the number of divisors of k (A000005).

Antti Karttunen, Table of n, a(n) for n = 1..15383

Complement among primes: A263090.
Intersection of A000040 and A045765.
Subsequence of A067774 (A049591).
Cf. A000005, A049820, A060990.

lim = 10000; s = Select[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], PrimeQ]; Take[s, 76] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem(123456789);
uplim1 = 2162160 + 320; \\ = A002182(41) + A002183(41).
v060990 = vector(uplim1);
for(n=3, uplim1, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
n=0; forprime(p=2, 524287, if((0 == A060990(p)), n++; write("b263091.txt", n, " ", p)));

;; With Antti Karttunen's IntSeq-library.
(define A263091 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 n)) (zero? (A060990 n))))))

A263257 a(n) = least number with distance n to the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Original entry on oeis.org

0, 1, 3, 5, 7, 19, 23, 27, 29, 31, 35, 37, 41, 43, 51, 53, 57, 59, 61, 65, 67, 71, 73, 77, 79, 223, 231, 233, 237, 239, 241, 249, 251, 263, 267, 269, 271, 277, 285, 291, 293, 299, 303, 315, 317, 321, 327, 331, 335, 337, 341, 347, 349, 357, 359, 369, 857, 859, 1077, 1081, 1087, 1095, 1097, 1101

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

a(n) = least number k for which A263254(k) = n.

Also positions of records in A263254, thus the sequence is strictly increasing.

Antti Karttunen, Table of n, a(n) for n = 0..1835

Cf. A049820, A259934, A263254, A263258.

First column of array A263255.

A325020 Numbers m 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, 2, 6, 22, 28, 76, 84, 90, 96, 170, 216, 248, 252, 496, 520, 532, 588, 672, 700, 852, 864, 1240, 2176, 2448, 2480, 2812, 3360, 6048, 7392, 7584, 8128, 9120, 11480, 12616, 12768, 13832, 14056, 14720, 15456, 19488, 20536, 21216, 27000, 30240, 31584, 31968

Offset: 1

Jaroslav Krizek, Mar 24 2019

nonn

Even perfect numbers from A000396 are terms.
Corresponding values of integers h: 0, 0, 1, 11, 11, 38, 27, 30, 32, 85, 72, 124, 81, ...
Supersequence of A325021 and A325023.

			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..1835

Cf. A000005, A000203, A000396, A001599, A049820, A325021, A325022, A325023, A325024.

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

Select[Range[10^5], IntegerQ[#1 (#1 - #2)/#3] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Michael De Vlieger, Mar 24 2019 *)

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

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325020_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-reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1))*n%s:
            yield n
A325020_list = list(islice(A325020_gen(),20)) # Chai Wah Wu, Feb 14 2023

A229210 Numbers k such that Sum_{i=1..k} (i-tau(i))^i == 0 (mod k), where tau(i) = A000005(i), the number of divisors of i, and i-tau(i) = A049820(i).

Original entry on oeis.org

1, 2, 5, 24, 36, 371, 445, 1578, 3616, 9292, 38123, 142815, 184097

Offset: 1

Paolo P. Lava, Sep 16 2013

a(12) > 50000.

a(14) > 200000. - Michel Marcus, Feb 25 2016

			(1 - tau(1))^1 + (2 - tau(2))^2 + ... + (5 - tau(5))^5 = 245 and 245 / 5 = 49.

Cf. A000010, A227427, A227429, A227502, A227848, A229095, A229207-A229209, A229211.

with(numtheory); P:=proc(q) local n, t; t:=0;
for n from 1 to q do t:=t+(n-tau(n))^n; if t mod n=0 then print(n);
fi; od; end: P(10^6);

isok(n) = sum(i=1, n, Mod(i-numdiv(i), n)^i) == 0; \\ Michel Marcus, Feb 25 2016

Name corrected by Michel Marcus, Feb 25 2016

a(12)-a(13) from Michel Marcus, Feb 25 2016

A262513 Numbers where A049820 takes a unique value; numbers n for which A060990(A049820(n)) = 1.

Original entry on oeis.org

5, 6, 7, 8, 11, 14, 17, 18, 20, 22, 23, 24, 27, 32, 34, 35, 40, 43, 46, 47, 50, 51, 57, 58, 61, 65, 72, 73, 77, 79, 81, 84, 86, 87, 88, 92, 93, 94, 96, 97, 98, 99, 101, 102, 103, 105, 107, 114, 116, 119, 120, 123, 125, 130, 135, 137, 143, 151, 154, 155, 158, 160, 163, 164, 173, 175, 177, 179, 184, 187, 191, 193, 194, 197, 198, 200, 203, 204, 206, 209, 210, 212

Offset: 1

Antti Karttunen, Sep 25 2015

nonn

Sequence A262512 sorted into ascending order.

Numbers n such that there is no other number k for which A049820(k) = A049820(n).

Antti Karttunen, Table of n, a(n) for n = 1..45700

Cf. A060990, A049820, A262511, A262512.

Cf. A262509 (a subsequence).

lim = 212; s = Table[n - DivisorSigma[0, n], {n, 2 lim + 3}]; t = Length@ Position[s, #] & /@ Range[0, lim]; Position[t[[# + 1]] & /@ Take[s, lim], 1] // Flatten (* Michael De Vlieger, Sep 29 2015, after Wesley Ivan Hurt at A049820 *)

A262676 Number of nonzero even numbers encountered when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = (1-A000035(n)) + a(A049820(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Number of even 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 even, but excludes the zero.

Antti Karttunen, Table of n, a(n) for n = 0..10080
A. Karttunen, Ratio a(n)/A262677(n) drawn with the help of OEIS Plot2-script

Cf. A000005, A000035, A049820, A155043, A262677, A262680.

a(0) = 0; for n >= 1, a(n) = (1-A000035(n)) + a(A049820(n)).
Other identities. For all n >= 0:
A155043(n) = a(n) + A262677(n).

A322974 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(1) = 0, f(2) = 1, f(n) = 3 if A009191(n) == 1 and f(n) = A049820(n) for all other numbers > 2.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 3, 5, 6, 6, 3, 6, 3, 7, 3, 3, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 12, 3, 12, 3, 13, 3, 14, 3, 15, 3, 16, 3, 17, 3, 16, 3, 18, 19, 20, 3, 18, 3, 21, 3, 22, 3, 22, 3, 23, 3, 24, 3, 23, 3, 25, 26, 3, 3, 25, 3, 27, 3, 27, 3, 28, 3, 29, 30, 29, 3, 29, 3, 29, 3, 31, 3, 32, 3, 33, 3, 34, 3, 31, 3, 35, 3, 36, 3, 37, 3, 38, 39, 3, 3, 40, 3, 41, 3

Offset: 1

Antti Karttunen, Jan 05 2019

nonn

For all i, j:
  A322810(i) = A322810(j) => a(i) = a(j),
  a(i) = a(j) => A323073(i) = A323073(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A009191, A049820, A323073.

Cf. also A319337, A322810, A322980.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A322974aux(n) = if(n<=2,n-1,my(u=(n-numdiv(n))); if(1==gcd(n,u),3,u));
v322974 = rgs_transform(vector(up_to,n,A322974aux(n)));
A322974(n) = v322974[n];

A262891 a(n) = A060990(A259934(n)); branching degree of node n in the infinite trunk of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

2, 1, 3, 1, 1, 4, 1, 2, 1, 3, 1, 2, 2, 4, 2, 1, 1, 3, 1, 2, 3, 1, 2, 3, 2, 2, 3, 4, 2, 2, 1, 1, 1, 2, 3, 2, 1, 1, 2, 1, 2, 2, 3, 3, 1, 1, 3, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 4, 2, 2, 2, 3, 2, 2, 3, 2, 1, 2, 2, 1, 3, 2, 2, 3, 1, 2, 3, 2, 2, 1

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10341

Cf. A049820, A060990, A259934.

Positions of ones: A262892.

nMax = 122; seq0 = {0}; seq = {1}; K = 1; While[seq != seq0, Print["K = ", K]; NN = K*nMax; Clear[A, B, S]; S[] = 0; For[n = NN + 1, n <= 2*NN, n++, k = n - DivisorSigma[0, n]; If[k <= NN, S[k] = S[k] + 1; B[k] = n]]; For[n = NN, n >= 3, n--, If[S[n] >= 1, k = n - DivisorSigma[0, n]; S[k] = S[k] + 1; B[k] = n]]; A[0] = 0; A[1] = 2; For[n = 2, True, n++, b = B[A[n - 1]]; If[b > NN || S[b] > 1, Break[]]; A[n] = b]; Clear[a0]; a0[] = 0; Do[n = x - DivisorSigma[0, x]; a0[n]++, {x, 1, NN}]; a[n_] := a0[A[n]]; seq0 = seq; seq = Table[a[n], {n, 0, nMax}]; K = 2K]; A262891 = seq (* Jean-François Alcover, Nov 16 2016, after Robert Israel for A259934 *)

(define (A262891 n) (A060990 (A259934 n)))

a(n) = A060990(A259934(n)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

A263091 Primes p for which A049820(x) = p has no solution.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A263257 a(n) = least number with distance n to the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Views

Author

Keywords

Comments

Links

Crossrefs

A325020 Numbers m 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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A229210 Numbers k such that Sum_{i=1..k} (i-tau(i))^i == 0 (mod k), where tau(i) = A000005(i), the number of divisors of i, and i-tau(i) = A049820(i).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

PARI

Extensions

A262513 Numbers where A049820 takes a unique value; numbers n for which A060990(A049820(n)) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A262676 Number of nonzero even numbers encountered when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = (1-A000035(n)) + a(A049820(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A322974 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(1) = 0, f(2) = 1, f(n) = 3 if A009191(n) == 1 and f(n) = A049820(n) for all other numbers > 2.

Views

Author