A051451 -id:A051451 - cp's OEIS frontend

A057825 Values of k for which A003418(k) - 1 is prime.

3, 4, 5, 6, 7, 8, 19, 20, 21, 22, 23, 24, 29, 30, 32, 33, 34, 35, 36, 47, 48, 61, 62, 63, 97, 98, 99, 100, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 233, 234, 235, 236, 237, 238, 307, 308, 309, 310, 401, 402, 403, 404, 405, 406, 407, 408, 887, 888, 889, 890

Offset: 1

Labos Elemer, Nov 08 2000

nonn

Fewer distinct primes than distinct values of a(n) are generated. So, e.g., k = 97, 98, 99, 100 all correspond to lcm([1..97]) - 1 = 69720375229712477164533808935312303556799, a prime.

Jinyuan Wang, Table of n, a(n) for n = 1..125

Cf. A003418, A049536, A049537, A051451.

isok(k) = ispseudoprime(lcm(vector(k, i, i))-1); \\ Jinyuan Wang, May 02 2020

A076244 Distinct values arising in A051547, sequence of a(n) = lcm(phi(1), ..., phi(n)).

Original entry on oeis.org

1, 2, 4, 12, 60, 120, 240, 720, 7920, 55440, 1275120, 2550240, 33153120, 961440480, 2884321440, 118257179040, 236514358080, 1182571790400, 20103720436800, 1065497183150400, 39423395776564800, 118270187329694400

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..365

Cf. A003418, A051451, A051547.

Union@ FoldList[LCM @@ {#1, EulerPhi@ #2} &, Range@ 165] (* Michael De Vlieger, Dec 09 2018 *)

lista(nn) = {last = -1; for (n=1, nn, new = lcm(vector(n, k, eulerphi(k))); if (new != last, print1(new, ", "); last = new););} \\ Michel Marcus, Mar 18 2018

A076245 Positions of records in A051547.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 23, 29, 47, 51, 53, 59, 81, 83, 85, 101, 103, 107, 149, 163, 167, 173, 179, 191, 197, 227, 251, 255, 257, 263, 269, 283, 293, 311, 317, 347, 359, 367, 383, 389, 467, 479, 487, 503, 509, 557, 563, 569, 587, 607, 619, 643, 653, 677

Offset: 1

Labos Elemer, Oct 08 2002

nonn

Observe that both primes and composites (including 15, 51, 81, 85, and 255) occur.

In totients of consecutive terms some prime-factor appears at higher power than in preceding ones: see A076246 and A051451.

Michael De Vlieger, Table of n, a(n) for n = 1..13256 (First 1924 terms from _Vincenzo Librandi_; a(n) <= 10^6.)

Cf. A003418, A051451, A051547, A076244.

With[{s = FoldList[LCM @@ {#1, EulerPhi@ #2} &, Range[700]]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Dec 09 2018 *)

lista(nn) = {least = 1; print1(1, ", "); for (n=2, nn, nleast = lcm(least, eulerphi(n)); if (nleast > least, print1(n, ", ")); least = nleast;);} \\ Michel Marcus, Jul 30 2017

A208768 The distinct values of A070198.

Original entry on oeis.org

0, 1, 5, 11, 59, 419, 839, 2519, 27719, 360359, 720719, 12252239, 232792559, 5354228879, 26771144399, 80313433199, 2329089562799, 72201776446799, 144403552893599, 5342931457063199, 219060189739591199, 9419588158802421599, 442720643463713815199

Offset: 1

Reinhard Zumkeller, Mar 01 2012

nonn

The terms of A070198, and duplicates removed.
a(n) = A051451(n) - 1 = A051452(n) - 2.
From Daniel Forgues, Apr 27 2014: (Start)
Factorizations:
  5, 11, 59, 419, 839 are primes;
  2519 = 11*229, 27719 = 53*523, 360359 = 173*2083,
  720719 = 31*67*347, 12252239 = 29*647*653;
  232792559, 5354228879 are primes;
  26771144399 = 47*12907*44131, 80313433199 = 29*61*45400471;
  2329089562799 is prime;
  72201776446799 = 37*149*239*1091*50227;
  144403552893599 is prime;
Very likely contains an infinite number of primes (see A057824). (End)
A more natural (compare with A051452) name for the sequence: lcm(1, ..., k) - 1, where k is the n-th prime power A000961(n). - Daniel Forgues, May 09 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

import Data.List (nub)
a208768 n = a208768_list !! (n-1)
a208768_list = nub a070198_list

from math import prod
from sympy import primepi, integer_nthroot, integer_log, primerange
def A208768(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod(p**integer_log(m, p)[0] for p in primerange(m+1))-1 # Chai Wah Wu, Aug 15 2024

A305325 Irregular triangle read by rows T(n, k), n >= 1 and 1 <= k <= A305215(n): T(n, k) is the k-th positive number with largest prime power factor equal to A000961(n).

Original entry on oeis.org

1, 2, 3, 6, 4, 12, 5, 10, 15, 20, 30, 60, 7, 14, 21, 28, 35, 42, 70, 84, 105, 140, 210, 420, 8, 24, 40, 56, 120, 168, 280, 840, 9, 18, 36, 45, 63, 72, 90, 126, 180, 252, 315, 360, 504, 630, 1260, 2520, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 132, 154, 165

Offset: 1

Rémy Sigrist, May 30 2018

The largest prime power factor of a number n is given by A034699(n).

When interpreted as a flat sequence we obtain a permutation of the natural numbers.

			Triangle begins:
  1: [1]
  2: [2]
  3: [3, 6]
  4: [4, 12]
  5: [5, 10, 15, 20, 30, 60]
  6: [7, 14, 21, 28, 35, 42, 70, 84, 105, 140, 210, 420]
  7: [8, 24, 40, 56, 120, 168, 280, 840]
  8: [9, 18, 36, 45, 63, 72, 90, 126, 180, 252, 315, 360, 504, 630, 1260, 2520]

Rémy Sigrist, Table of n, a(n) for n = 1..15360
Rémy Sigrist, PARI program for A305325
Index entries for sequences that are permutations of the natural numbers

Cf. A000961, A034699, A051451, A305215, A336816 (inverse).

PARI
```
See Links section.
```

T(n, 1) = A000961(n).

T(n, A305215(n)) = A051451(n).

A025543 Least common multiple of the first n composite numbers.

Original entry on oeis.org

1, 4, 12, 24, 72, 360, 360, 2520, 2520, 5040, 5040, 5040, 5040, 55440, 55440, 277200, 3603600, 10810800, 10810800, 10810800, 21621600, 21621600, 367567200, 367567200, 367567200, 6983776800, 6983776800, 6983776800, 6983776800, 6983776800

Offset: 0

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index to divisibility sequences

Differs from A003418 and A051451
Cf. A002808, A003418, A051451, A064354.
Cf. A036691.

a025543 n = a025543_list !! n
a025543_list = scanl lcm 1 a002808_list
-- Reinhard Zumkeller, Nov 10 2013

Table[Apply[LCM, Take[Select[Range[2, 300], !PrimeQ[#] &], n]], {n, 1, 100}]  (* Clark Kimberling, Nov 12 2016 *)

A305215 a(n) is the number of numbers whose largest prime power factor equals A000961(n).

Original entry on oeis.org

1, 1, 2, 2, 6, 12, 8, 16, 48, 96, 48, 240, 480, 960, 960, 960, 3840, 7680, 3072, 18432, 36864, 73728, 147456, 147456, 442368, 884736, 1769472, 589824, 4128768, 8257536, 16515072, 33030144, 16515072, 82575360, 165150720, 330301440, 660602880, 1321205760

Offset: 1

Rémy Sigrist, May 27 2018

nonn

The largest prime power factor of a number n is given by A034699(n).

			The first terms, alongside A000961(n) and the set of numbers k such that A034699(k) = A000961(n), are:
  n   a(n)  A000961(n)    S(n)
  --  ----  ----------    ----
   1     1           1    { 1 }
   2     1           2    { 2 }
   3     2           3    { 3, 6 }
   4     2           4    { 4, 12 }
   5     6           5    { 5, 10, 15, 20, 30, 60 }
   6    12           7    { 7, 14, 21, 28, 35, 42, 70, 84, 105, 140, 210, 420 }
   7     8           8    { 8, 24, 40, 56, 120, 168, 280, 840 }

Cf. A000005, A000961, A034699, A051451.
First differences of A056795.
Row lengths of A305325.

my(l=1); for (k=1, 103, if (omega(k) <= 1, l = lcm(l, k); print1 (numdiv(l/k) ", ")))

a(n) = A000005(A051451(n) / A000961(n)).

A308471 Lowest outliers for A057660.

Original entry on oeis.org

1, 4, 6, 12, 24, 30, 60, 120, 180, 210, 360, 420, 840, 1260, 2520, 4620, 9240, 13860, 27720, 55440, 60060, 120120, 180180, 360360, 720720, 1441440, 1801800, 2042040, 3063060, 6126120, 12252240, 24504480, 30630600, 36756720, 38798760

Offset: 1

Charlie Neder, May 29 2019

nonn

A057660(n) is a multiplicative function bounded above by n*(n-1)+1, which is reached whenever n is 1 or prime. These numbers are the n such that the ratio between A057660(n) and the upper bound reaches a record low.
Motivated by Daniel Forgues's conjecture that this sequence consists of 4 and A051451.
A subsequence of A025487.

			A057660(60060)/(60060*60059+1) = 1211716737/3607143541 ~ 0.3359214, and every number less than 60060 has a ratio > 0.34, so 60060 is in this sequence.

Charlie Neder, Table of n, a(n) for n = 1..215

Cf. A057660, A025487, A051451.

A354418 a(n) is the denominator of the sum of the reciprocals of the first n squarefree numbers.

Original entry on oeis.org

1, 2, 6, 30, 5, 35, 70, 770, 10010, 5005, 15015, 255255, 4849845, 1616615, 3233230, 74364290, 37182145, 1078282205, 6469693230, 200560490130, 6077590610, 3038795305, 607759061, 22487085257, 44974170514, 134922511542, 5531822973222, 921970495537, 39644731308091

Offset: 1

Ilya Gutkovskiy, May 26 2022

			1, 3/2, 11/6, 61/30, 11/5, 82/35, 171/70, 1951/770, 26133/10010, 13424/5005, 41273/15015, ...

Cf. A002110, A002805, A005117, A034386, A051451, A072983, A354417 (numerators).

Accumulate[1/Select[Range[43], SquareFreeQ]] // Denominator

a(n) = my(i=0, s=0); for(x=1, oo, if(core(x)==x, s+=1/x; i++; if(i==n, return(denominator(s))))) \\ Felix Fröhlich, May 26 2022

A056795 Number of divisors of k as k runs through sequence of distinct values of LCM(1,..,n).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 32, 48, 96, 192, 240, 480, 960, 1920, 2880, 3840, 7680, 15360, 18432, 36864, 73728, 147456, 294912, 442368, 884736, 1769472, 3538944, 4128768, 8257536, 16515072, 33030144, 66060288, 82575360, 165150720, 330301440

Offset: 1

Labos Elemer, Aug 28 2000

nonn

Values of LCM's in A003418 and accordingly their number of divisors jump at powers of primes (A000961). Here divisor-numbers of LCM's are displayed without repetition.

			For x = 19,20,21,22 the value of A003418(x) = A051451(13) = LCM(1,..,x) = 232792560, of which the total number of divisors is 960, so a(13) = 960.

Amiram Eldar, Table of n, a(n) for n = 1..3365

Cf. A000005, A003418, A051451, A000961.

Partial sums of A305215.

f(n) = lcm(vector(n, i, i)); \\ A003418
lista(nn) = {my(last = 0); for (n=1, nn, my(new = f(n)); if (new != last, print1(numdiv(new), ", "); last = new););} \\ Michel Marcus, Oct 08 2020

a(n) = A000005(A051451(n)).

cp's OEIS Frontend

A057825 Values of k for which A003418(k) - 1 is prime.

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A076244 Distinct values arising in A051547, sequence of a(n) = lcm(phi(1), ..., phi(n)).

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Mathematica

PARI

A076245 Positions of records in A051547.

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Mathematica

PARI

A208768 The distinct values of A070198.

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A305325 Irregular triangle read by rows T(n, k), n >= 1 and 1 <= k <= A305215(n): T(n, k) is the k-th positive number with largest prime power factor equal to A000961(n).

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A025543 Least common multiple of the first n composite numbers.

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Mathematica

A305215 a(n) is the number of numbers whose largest prime power factor equals A000961(n).

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PARI

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A308471 Lowest outliers for A057660.

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A354418 a(n) is the denominator of the sum of the reciprocals of the first n squarefree numbers.

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