A141586 -id:A141586 - cp's OEIS frontend

A141900 a(n) = smallest term in A141586 that is divisible by 2^n but not by 2^(n+1).

1, 2, 12, 24, 240, 480, 6720, 13440, 241920, 483840, 10644480, 21288960, 553512960, 1107025920, 2214051840, 4428103680, 150555525120, 301111050240, 11442219909120, 22884439818240, 45768879636480

Offset: 0

David Applegate, Tim Kunisky (tkunisky(AT)gmail.com), Gerry Manoim (gerrymanoim(AT)gmail.com) and N. J. A. Sloane, Sep 18 2008

nonn

			a(5) = 480 = 2^5*3*5.

N. J. A. Sloane, Table of n, a(n) for n = 0..50 [Computed using David Applegate's program from A142593]

Cf. A141586, A142593 (factorizations).

A141756 Intersection of A141586 and A100933.

Original entry on oeis.org

115325637083043831900183479190311008528007516613207384396965600343647846400000, 3575094749574358788905687854899641264368233015009428916305933610653083238400000, 16347325453573190128511213579615837273221059920118876332835279076344751718400000

Offset: 1

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

For the prime factorizations of the first four terms (only three are shown above) see the Maple code.

Since all terms > 1 in A141586 are even, this is also the intersection of A141586 and A141757.

B0:=2^23*3^14*5^5*7^3*11^3*13^3*17^2*19^2*23^2*29;
a1:=B0*31^28; a2:=B0*31^29; a3:=B0*37^28; a4:=B0*350*31^28;
[seq(a1,a2,a3,a4)];

A141758 Elements n of A141586 with property that A100762(n) = n.

Original entry on oeis.org

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 4320, 6720, 13440, 20160, 21600, 40320, 60480, 64800, 120960, 194400, 241920, 302400, 423360, 483840, 604800, 846720, 907200, 1814400, 2721600, 3628800, 5443200, 6350400, 7257600, 10644480, 10886400, 12700800, 18144000

Offset: 1

David Applegate and N. J. A. Sloane, Sep 16 2008

nonn

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..66

A109802 Positive integers n such that 480*n is an element of A141586.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 14, 17, 19, 21, 23, 28, 29, 31, 33, 37, 39, 41, 42, 43, 45, 47, 51, 53, 57, 59, 61, 67, 69, 71, 73, 79, 83, 84, 87, 89, 93, 97, 101, 103, 107, 109, 111, 113, 123, 126, 127, 129, 131, 135, 137, 139, 141, 149, 151, 154, 157, 159, 163, 167, 173, 177, 179, 181

Offset: 1

N. J. A. Sloane and German Manoim (gerrymanoim(AT)gmail.com), Aug 27 2008, Sep 03 2008

nonn

Includes p and 3p, where p is any prime except 2 and 5.

Cf. A141586.

A062247 Let f(h) = A141900(h) = 2^h * 3^i * 5^j * ... be the smallest term in A141586 that is divisible by 2^h but not by 2^(h+1). Sequence gives values of h where i increases.

Original entry on oeis.org

0, 2, 8, 24, 26, 48, 80, 120, 168, 242, 288, 360, 528, 728, 840, 960, 1330, 1368, 1680, 1848, 2186, 2208, 2808, 3480, 3720, 4488, 5040, 5328, 6240, 6560, 6888, 7920, 9408

Offset: 1

N. J. A. Sloane, Nov 24 2008

nonn

			Pairs (h,i) where A141900 contains a term 2^h*3^i*... for the first time, with a dash if that value of i never occurs:
h i
0 0
2 1
- 2
8 3
24 4
26 5
48 6
80 7
- 8
120 9
168 10
242 11
288 12
360 13
528 14
728 15
840 16
960 17
1330 18
1368 19
1680 20
...
The missing values of i for h <= 10000 are 2, 8 and 26.

Cf. A141586, A141900.

A100786 Apply the function n -> A100762(n) to the terms of A141586.

Original entry on oeis.org

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 480, 4320, 480, 480, 6720, 480, 480, 1440, 480, 13440, 480, 480, 1440, 480, 1440, 480, 20160, 480, 21600, 480, 1440, 480, 1440, 480, 480, 480, 1440, 480, 480, 480, 480, 40320, 1440, 480, 1440, 480, 480, 480, 480, 480, 1440

Offset: 1

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

nonn

A253139 a(n) = lcm_{d|n} tau(d), where tau(d) represents the number of divisors of d (A000005(d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 12, 6, 4, 2, 12, 2, 4, 4, 60, 2, 12, 2, 12, 4, 4, 2, 24, 6, 4, 12, 12, 2, 8, 2, 60, 4, 4, 4, 36, 2, 4, 4, 24, 2, 8, 2, 12, 12, 4, 2, 120, 6, 12, 4, 12, 2, 24, 4, 24, 4, 4, 2, 24, 2, 4, 12, 420, 4, 8, 2, 12, 4, 8, 2, 72, 2, 4, 12, 12, 4, 8

Offset: 1

Matthew Vandermast, Dec 27 2014

A divisibility sequence (cf. Ward link and second formula).

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			The divisors of 20 are 1, 2, 4, 5, 10 and 20, which have 1, 2, 3, 2, 4 and 6 divisors respectively. The least common multiple of 1, 2, 3, 2, 4 and 6 is 12; therefore, a(20) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.

A250270 gives range of values. A141586 lists numbers n such that a(n) divides n.

Cf. A003418, A253141.

Table[LCM@@DivisorSigma[0,Divisors[n]],{n,100}] (* Harvey P. Dale, Sep 01 2017 *)
lcm[n_] := lcm[n] = LCM @@ Range[n]; a[1] = 1; a[n_] := Times @@ (lcm [Last[#] + 1] & /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

a(n) = my(d = divisors(n)); lcm(vector(#d, k, numdiv(d[k]))); \\ Michel Marcus, Jan 23 2015

If n = Product_ prime(i)^e(i), then a(n) = Product_ A003418(e(i)+1).

a(n) = Product_{d|n} A253141(d).

A212165 Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104

Offset: 1

Matthew Vandermast, May 22 2012

Union of A212164 and A212166.  Includes numerous subsequences that are subsequences of neither A212164 nor A212166.
Includes all factorials except A000142(3) = 6.
Observation: all terms in DATA section are also the first 65 numbers n whose difference between the arithmetic derivative of n and the sum of the divisors of n is nonnegative. - Omar E. Pol, Dec 19 2012

			10 = 2^1*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (although 1s are often left implicit).  2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 does not belong to the sequence. But 20 = 2^2*5^1 and 40 = 2^3*5^1 belong, since the largest exponents in their prime factorizations are 2 and 3 respectively.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Complement of A212168.
See also A212167.
Subsequences (none of which are subsequences of A212164 or A212166) include A000079, A001021, A066120, A087980, A130091, A141586, A166475, A181818, A181823, A181824, A182763, A212169. Also includes all terms in A181813 and A181814.
Cf. A001221, A051903, A188654, A225230.

import Data.List (findIndices)
a212165 n = a212165_list !! (n-1)
a212165_list = map (+ 1) $ findIndices (<= 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) >= #e;} \\ Amiram Eldar, Sep 08 2024

A225230(a(n)) <= 0. - Reinhard Zumkeller, May 03 2013

A100762 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n and let P(n) = A100549(n); then a(n) = Product_{ q <= P(n) } q^e_q; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 9, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 2, 1, 24, 1, 2, 27, 4, 1, 2, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 2, 1, 4, 9, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 8, 1, 2, 1, 12, 1, 2, 9, 64, 1, 2, 1, 4, 1, 2, 1, 72, 1, 2, 3, 4, 1, 2, 1, 80, 81, 2, 1, 12, 1, 2, 1, 8, 1, 18, 1, 4, 1, 2, 1, 96, 1

Offset: 1

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

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A100549, A100417, A141586, A082725.

# First load the procedure pp from A100549
# B = prod_{p <= pp(n)} p^e_p
B := proc(n) local v,f,pv; global pp; option remember;
pv := pp(n);
v := 1:
for f in op(2..-1,ifactors(n)) while f[1] <= pv do
v := v * f[1]^f[2];
end do;
return v;
end proc;

{1}~Join~Array[Function[{q, P}, Times @@ Power @@@ Select[q, First@# <= P &]] @@ {#, Prime@ PrimePi[1 + Max@ #[[All, -1]] ]} &@ FactorInteger[#] &, 96, 2] (* 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))); \\ Antti Karttunen, Nov 11 2018

A100549 Let n = 2^e_2 * 3^e_ * 5^e_ * ... be the prime factorization of n; then a(n) = largest prime <= 1 + max{e_2, e_3, e_5, ...}; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 5, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 5, 5, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 5, 2, 3, 3, 3, 2, 2, 2, 3, 2

Offset: 1

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

nonn

			If n = 8 = 2^3, a(n) = (largest prime <= 3+1) = 3.
If n = 480 = 2^5*3*5, a(n) = (largest prime <= 1 + max{5,1,1}) = 5.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A100762, A100417, A141586, A082725.

# if n = prod_p p^e_p, then
# pp = largest prime <= 1 + max e_p
with(numtheory):
pp := proc(n) local f,m; option remember;
if (n = 1) then
return 1;
end if;
m := 1:
for f in op(2..-1,ifactors(n)) do
if (f[2] > m) then
m := f[2]:
end if;
end do;
prevprime(m+2);
end proc;

{1}~Join~Array[Prime@PrimePi[1 + Max@FactorInteger[#][[All, -1]]] &, 105, 2] (* Michael De Vlieger, Nov 13 2018 *)

a(n) = if (n==1, 1, precprime(1 + vecmax(factor(n)[,2]~))); \\ Michel Marcus, Nov 14 2018

cp's OEIS Frontend

A141900 a(n) = smallest term in A141586 that is divisible by 2^n but not by 2^(n+1).

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A141756 Intersection of A141586 and A100933.

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Maple

A141758 Elements n of A141586 with property that A100762(n) = n.

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A109802 Positive integers n such that 480*n is an element of A141586.

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Comments

Crossrefs

A062247 Let f(h) = A141900(h) = 2^h * 3^i * 5^j * ... be the smallest term in A141586 that is divisible by 2^h but not by 2^(h+1). Sequence gives values of h where i increases.

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Author

Keywords

Examples

Crossrefs

A100786 Apply the function n -> A100762(n) to the terms of A141586.

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Author

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A253139 a(n) = lcm_{d|n} tau(d), where tau(d) represents the number of divisors of d (A000005(d)).

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Mathematica

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Formula

A212165 Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)).

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Haskell

Mathematica

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Formula

A100762 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n and let P(n) = A100549(n); then a(n) = Product_{ q <= P(n) } q^e_q; a(1) = 1 by convention.

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Programs

Maple

Mathematica

PARI

A100549 Let n = 2^e_2 * 3^e_ * 5^e_ * ... be the prime factorization of n; then a(n) = largest prime <= 1 + max{e_2, e_3, e_5, ...}; a(1) = 1 by convention.

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