A141586 -id:A141586 - cp's OEIS frontend

A082725 a(n) = n/A100762(n).

1, 1, 3, 1, 5, 3, 7, 1, 1, 5, 11, 1, 13, 7, 15, 1, 17, 1, 19, 5, 21, 11, 23, 1, 25, 13, 1, 7, 29, 15, 31, 1, 33, 17, 35, 1, 37, 19, 39, 5, 41, 21, 43, 11, 5, 23, 47, 1, 49, 25, 51, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 7, 1, 65, 33, 67, 17, 69, 35, 71, 1, 73, 37, 25, 19, 77, 39, 79, 1, 1, 41

Offset: 1

N. J. A. Sloane, Nov 17 2008

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A100762, A100549, A141586.

{1}~Join~Table[Function[{q, P}, n/Times @@ Power @@@ Select[q, First@ # <= P &]] @@ {#, Prime@ PrimePi[1 + Max@ #[[All, -1]] ]} &@ FactorInteger[n], {n, 2, 82}] (* Michael De Vlieger, Nov 13 2018 *)

A100549(n) = if(1==n,1,prime(primepi(1+vecmax(factor(n)[,2]))));
A100762(n) = if(1==n,1,my(u = A100549(n), f=factor(n)); prod(i=1, #f~, if(f[i, 1]<=u, f[i, 1]^f[i, 2], 1)));
A082725(n) = (n/A100762(n)); \\ Antti Karttunen, Nov 11 2018

A135130 Let n = Product_{p} p ^ e_p be the prime factorization of n and let M = max{e_p + 1 }. Then n is in the sequence iff for all primes q in the range 3 <= q <= M we have e_q >= Sum_{r} floor( log_q (e_r + 1) ).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97

Offset: 1

N. J. A. Sloane, Nov 29 2008

nonn

A variant of A141586, which is a subsequence.

N. J. A. Sloane, Maple program

Cf. A141586, A143718.

A142593 Irregular triangle read by rows: row n gives successive exponents in prime factorization of A141900(n).

Original entry on oeis.org

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

Offset: 1

David Applegate, Tim Kunisky (tkunisky(AT)gmail.com), Gerry Manoim (gerrymanoim(AT)gmail.com) and N. J. A. Sloane, Sep 25 2008

			Triangle begins:
1
2 1
3 1
4 1 1
5 1 1
6 1 1 1
7 1 1 1
8 3 1 1
9 3 1 1
10 3 1 1 1
11 3 1 1 1
12 3 1 1 1 1
Row 8 = [8,3,1,1] because A141900(8) = 2^8*3^3*5*7 = 241920.

David Applegate, Maple program which generates this triangle [The command hpat(n); returns the n-th row in an encoded form.]
N. J. A. Sloane, Rows 1 through 50 of the triangle

Cf. A141586, A141900, A142594.

A100933 Complement of A100417.

Original entry on oeis.org

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 50, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 98, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 143, 145

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2008

nonn

Cf. A100417, A141757, A141586, A141756, A141757.

A141551 Numbers k with property that if d divides k, then tau(tau(d)) also divides k.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 180, 192, 204, 228, 240, 252, 276, 288, 300, 324, 348, 360, 372, 396, 444, 468, 480, 492, 516, 564, 576, 588, 600, 612, 636, 684, 708, 720, 732, 804, 828, 840, 852, 876, 900, 948, 960, 972, 996, 1044, 1068

Offset: 1

N. J. A. Sloane, Sep 12 2008

nonn

tau() = A000005().

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

Cf. A000005, A010553, A141586.

with(numtheory); isA141551 := proc(n) local dvs,d; dvs := divisors(n) ; for d in dvs do if not tau(tau(d)) in dvs then RETURN(false): fi; od: RETURN(true); end:
t1:=[]; for n from 1 to 60000 do if isA141551(n) then t1:=[op(t1),n]; fi; od:

aQ[n_] := AllTrue[Divisors[n], Divisible[n, DivisorSigma[0, DivisorSigma[0, #]]] &]; Select[Range[1000], aQ] (* Amiram Eldar, Jul 08 2019 *)

A143719 Let n = Product_{p} p ^ e_p be the prime factorization of n and let M = max{e_p}. Then n is in the sequence iff for all primes q in the range 2 <= q <= M we have e_q >= Sum_{r} floor( log_q (e_r + 1) ).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 100

Offset: 1

N. J. A. Sloane, Nov 29 2008

nonn

A variant of A141586, which is a subsequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A141586, A135130, A143720.

filter:= proc(n) local F,M;
  F:= ifactors(n)[2];
  M:= max(F[..,2]);
  andmap(proc(q) local e; padic:-ordp(n,q) >= add(floor(log[q](e+1)),e=F[..,2]) end proc, select(isprime, [$2..M]))
end proc:
filter(1):= true:
select(filter, [$1..200]); # Robert Israel, Jan 29 2025

A143720 Complement of A143719.

Original entry on oeis.org

8, 9, 16, 18, 25, 27, 32, 40, 45, 49, 50, 54, 56, 60, 63, 64, 75, 80, 81, 84, 88, 90, 96, 98, 99, 104, 108, 112, 117, 120, 121, 125, 126, 128, 132, 135, 136, 140, 147, 150, 152, 153, 156, 160, 162, 168, 169, 171, 175, 176, 180, 184, 189, 192, 198, 200, 204, 207, 208, 216

Offset: 1

N. J. A. Sloane, Nov 29 2008

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A143719, A141586.

filter:= proc(n) local F,M;
  F:= ifactors(n)[2];
  M:= max(F[..,2]);
  ormap(proc(q) local e; padic:-ordp(n,q) < add(floor(log[q](e+1)),e=F[..,2]) end proc, select(isprime, [$2..M]))
end proc:
select(filter, [$2..300]); # Robert Israel, Jan 29 2025

A208251 Number of refactorable numbers less than or equal to n.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Jan 12 2013

A number is refactorable if it is divisible by the number of its divisors.

			a(1) = 1 since 1 is the first refactorable number, a(2) = 2 since there are two refactorable numbers less than or equal to 2, a(3) through a(7) = 2 since the next refactorable number is 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Refactorable Number.

Cf. A033950, A000005, A141586, A057265, A036896, A036898, A114617.

Partial sums of A336040.

with(numtheory) a:=n->sum((1 + floor(i/tau(i)) - ceil(i/tau(i))), i=1..n);

Accumulate[Table[If[Divisible[n, DivisorSigma[0, n]], 1, 0], {n, 1,100}]] (* Amiram Eldar, Oct 11 2023 *)

a(n) = sum(i=1, n, q = i/numdiv(i); 1+ floor(q) - ceil(q)); \\ Michel Marcus, Sep 10 2018

a(n) = Sum_{i=1..n} 1 + floor(i/d(i)) - ceiling(i/d(i)), where d(n) is the number of divisors of n.

A306263 Numbers k such that, for any divisor d of k, the Hamming weight of d divides k.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 32, 34, 36, 40, 42, 48, 60, 64, 66, 68, 72, 80, 84, 92, 96, 108, 116, 120, 126, 128, 132, 136, 144, 156, 160, 168, 172, 180, 184, 192, 204, 212, 216, 222, 228, 232, 240, 246, 252, 256, 264, 272, 276, 284, 288, 300, 310

Offset: 1

Rémy Sigrist, Mar 02 2019

The Hamming weight of a number is given by A000120.
This sequence is a binary variant of A285815.
This sequence is infinite as it contains all powers of 2 (A000079).
All terms belong to A049445.
If k belongs to the sequence, then 2*k belongs to the sequence.
All terms except 1 are even. - Robert Israel, Mar 05 2019

			For n = 108:
- the divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108,
- the corresponding Hamming weights are 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 4,
- they all divide 108,
- hence 108 belongs to the sequence.
For n = 98:
- the divisors of 98 are 1, 2, 7, 14, 49, 98,
- the correspond Hamming weights are 1, 1, 3, 3, 3, 3,
- 3 does not divide 98,
- hence 98 does not belong to the sequence.

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

Cf. A000079, A000120, A049445, A141586, A285815.

Positions of zeros in A324393.

[k:k in [1..310]| forall{d:d in Divisors(k)| k mod &+Intseq(d,2) eq 0}]; // Marius A. Burtea, Dec 30 2019

filter:= proc(n) local F;
  F:= map(convert,map(convert,numtheory:-divisors(n),base,2),`+`);
  andmap(t -> n mod t = 0, F)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 05 2019

Select[Range@ 310, With[{k = #}, AllTrue[Divisors@ k, Mod[k, DigitCount[#, 2, 1]] == 0 &]] &] (* Michael De Vlieger, Mar 05 2019 *)

is(n) = fordiv(n,d,if (n%hammingweight(d), return (0))); return ( )

A174457 Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 72, 84, 96, 108, 132, 156, 180, 204, 228, 240, 252, 276, 288, 348, 360, 372, 396, 444, 468, 480, 492, 504, 516, 564, 600, 612, 636, 640, 672, 684, 708, 720, 732, 792, 804, 828, 852, 864, 876, 936, 948, 972, 996, 1044, 1056, 1068, 1116, 1152

Offset: 1

Matthew Vandermast, Dec 04 2010

nonn

In other words, let d^1(n) = A000005(n) and, for all positive integers k, let d^(k+1)(n) = A000005(d^k(n)). Sequence lists numbers n with the property that every such value of d^k(n) divides n.
A141586 is a subsequence. Is A110821 a subsequence?
Not a subsequence of A141551: 504 is the smallest term in this sequence not member of A141551.
a(n) is even for all n, since for any n >= 2, d^k(n) = 2 for some k. Proof: {d^k(n)} is a nonincreasing sequence of k, so it must stablize at a fixed point of the map x -> A000005(x), namely x = 1 or 2. But d^k(n) = 1 for some k implies that n = 1. - Jianing Song, Apr 20 2022

			9 has 3 divisors, and 9 is a multiple of 3. But 3 has 2 divisors, and 9 is not a multiple of 2. Hence, 9 does not belong to this sequence.
36 has 9 divisors, 9 has 3 divisors, 3 has 2 divisors, and 9, 3, and 2 are all divisors of 36. (Since 2 has 2 divisors, all further steps produce a value of 2.) Hence, 36 belongs to this sequence.

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

Cf. A036459 (number of steps of the map), A000005 (d(n): number of divisors).
Cf. A010553 (d(d(n))), A036450 (d^3(n)), A036452 (d^4(n)), A036453 (d^5(n)).
Subsequence of A033950 (refactorable numbers: d(n) | n) and A141113 (d(d(n))| n).

is_A174457(n, d=n)=!until(d<3, n%(d=numdiv(d)) && return) \\ M. F. Hasler, Dec 05 2010, updated PARI syntax Apr 16 2022

Edited by M. F. Hasler, Apr 16 2022

cp's OEIS Frontend

A082725 a(n) = n/A100762(n).

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A135130 Let n = Product_{p} p ^ e_p be the prime factorization of n and let M = max{e_p + 1 }. Then n is in the sequence iff for all primes q in the range 3 <= q <= M we have e_q >= Sum_{r} floor( log_q (e_r + 1) ).

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A142593 Irregular triangle read by rows: row n gives successive exponents in prime factorization of A141900(n).

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A100933 Complement of A100417.

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A141551 Numbers k with property that if d divides k, then tau(tau(d)) also divides k.

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Maple

Mathematica

A143719 Let n = Product_{p} p ^ e_p be the prime factorization of n and let M = max{e_p}. Then n is in the sequence iff for all primes q in the range 2 <= q <= M we have e_q >= Sum_{r} floor( log_q (e_r + 1) ).

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Maple

A143720 Complement of A143719.

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Maple

A208251 Number of refactorable numbers less than or equal to n.

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Maple

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Formula

A306263 Numbers k such that, for any divisor d of k, the Hamming weight of d divides k.

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Magma

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