Matthew Campbell - cp's OEIS frontend

A316346 a(n) = A316297(n+1)/A316297(n).

4, 9, 8, 25, 2, 49, 16, 81, 10, 121, 12, 169, 14, 15, 32, 289, 6, 361, 4, 7, 22, 529, 72, 625, 26, 243, 28, 841, 30, 961, 64, 3, 34, 35, 36, 1369, 38, 39, 40, 1681, 6, 1849, 484, 45, 46, 2209, 48, 2401, 50, 51, 52, 2809, 18, 55, 56, 57, 58, 3481, 60, 3721, 62

Offset: 1

Matthew Campbell, Jun 29 2018

nonn

			a(3) = A316297(4)/A316297(3) = 288/36 = 8.

Cf. A316297.

#[[2]]/#[[1]]&/@Partition[Array[#! Denominator[HarmonicNumber[#]]&,70], 2,1] (* Harvey P. Dale, Jun 12 2019 *)

A316297 a(n) = n! times the denominator of the n-th harmonic number H(n).

Original entry on oeis.org

1, 4, 36, 288, 7200, 14400, 705600, 11289600, 914457600, 9144576000, 1106493696000, 13277924352000, 2243969215488000, 31415569016832000, 471233535252480000, 15079473128079360000, 4357967734014935040000, 26147806404089610240000, 9439358111876349296640000

Offset: 1

Matthew Campbell, Jun 29 2018

nonn

			a(4) = 4! * A002805(4) = 24 * 12 = 288.

Cf. A000142, A001008, A002805, A027611, A027612, A124837, A124838.

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> denom(H(n))*n!:
seq(a(n), n=1..20);  # Alois P. Heinz, Jul 21 2018

a[n_] := n! Denominator@HarmonicNumber@n; Array[a, 18] (* Robert G. Wilson v, Jun 30 2018 *)

a(n) = n! * denominator(sum(k=1, n, 1/k)); \\ Michel Marcus, Aug 12 2018

a(n) = A000142(n) * A002805(n).

A280420 Product of divisors of n!.

Original entry on oeis.org

1, 1, 2, 36, 331776, 42998169600000000, 7244150201408990671659859968000000000000000, 1182813011613388022005884215741990164001544397058025540221953280041975323323006976000000000000000000000000000000

Offset: 0

Matthew Campbell, Jan 02 2017

nonn

Matthew Campbell, Table of n, a(n) for n = 0..10

Cf. A000005, A000142, A007955, A027423.

A280420 := proc(n)
    mul(d,d=numtheory[divisors](n!)) ;
end proc: # R. J. Mathar, Jan 04 2017

Table[(n!)^(DivisorSigma[0, n!]/2), {n, 0, 10}]

from math import isqrt, factorial
from sympy import divisor_count
def A280420(n): return (lambda m:isqrt(m)**c if (c:=divisor_count(m)) & 1 else m**(c//2))(factorial(n)) # Chai Wah Wu, Jun 25 2022

a(n) = A007955(A000142(n)).

a(n) = (n!)^(d(n!)/2) = (A000142(n))^(A000005(A000142(n))/2).

A280409 Primes in the order that they appear in A280408, without repetitions.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 13, 23, 53, 19, 29, 41, 31, 47, 71, 107, 137, 103, 233, 263, 593, 167, 251, 283, 479, 719, 1619, 911, 1367, 577, 433, 61, 37, 59, 89, 67, 101, 43, 83, 73, 113, 79, 179, 269, 131, 197, 97, 149, 109, 173, 139, 157, 127, 191, 431, 647, 971

Offset: 1

Matthew Campbell, Jan 02 2017

nonn

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A280408, A000040, A070165.

DeleteDuplicates@ Flatten@ Table[Select[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, n, # > 1 &], PrimeQ], {n, 200}] (* Michael De Vlieger, Jan 02 2017 *)

A280408 Irregular triangle read by rows listing the prime numbers that appear from the trajectory of n in Collatz Problem.

Original entry on oeis.org

2, 2, 3, 5, 2, 2, 5, 2, 3, 5, 2, 7, 11, 17, 13, 5, 2, 2, 7, 11, 17, 13, 5, 2, 5, 2, 11, 17, 13, 5, 2, 3, 5, 2, 13, 5, 2, 7, 11, 17, 13, 5, 2, 23, 53, 5, 2, 2, 17, 13, 5, 2, 7, 11, 17, 13, 5, 2, 19, 29, 11, 17, 13, 5, 2, 5, 2, 2, 11, 17, 13, 5, 2, 23, 53, 5, 2, 3, 5, 2, 19, 29, 11, 17, 13, 5, 2

Offset: 1

Matthew Campbell, Jan 02 2017

			The irregular array a(n,k) starts:
n\k   1   2   3   4   5   6
...
1:    2
2:    2
3:    3   5   2
4:    2
5:    5   2
6:    3   5   2
7:    7  11  17  13   5   2
8:    2
9:    7  11  17  13   5   2
10:   5  2
11:  11  17  13   5   2
12:   3   5   2
13:  13   5   2
14:   7  11  17  13   5   2
15:  23  53   5   2

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A070165, A280409.

Table[Select[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, n, # > 1 &], PrimeQ], {n, 2, 30}] // Flatten (* Michael De Vlieger, Jan 02 2017 *)

from sympy import isprime
def a(n):
    if n==1: return [2]
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        l+=[n, ]
        if n<2: break
    return list(filter(lambda i: isprime(i), l))
for n in range(1, 21): print(a(n)) # Indranil Ghosh, Apr 14 2017

A280327 a(n) is obtained by applying the map k -> composite(k) n times, starting at n.

Original entry on oeis.org

4, 12, 25, 39, 60, 94, 133, 183, 236, 320, 415, 520, 640, 805, 1007, 1212, 1463, 1800, 2144, 2562, 3021, 3523, 4135, 4840, 5747, 6630, 7701, 9057, 10392, 11812, 13519, 15400, 17534, 19827, 22564, 25624, 29206, 32998, 37041, 41819, 46659, 53223, 59345, 66104, 73368, 81897, 91157, 100827, 112045

Offset: 1

Matthew Campbell, Dec 31 2016

nonn

			a(3) is 25 because the third composite is 8, the eighth composite is 15, and for the 3rd iteration, the fifteenth composite is 25.
To get a(4): 4 -> 9 -> 16 -> 26 -> 39.

Cf. A002808.

For primes, see A058009.

c = Select[Range[10^6], CompositeQ]; Table[Nest[c[[#]] &, n, n], {n, 50}] (* Michael De Vlieger, Dec 31 2016 *)

More terms from Michael De Vlieger, Dec 31 2016

A274472 Number of iterations of the Collatz recursion required to reach a prime number.

Original entry on oeis.org

2, 0, 0, 1, 0, 1, 0, 2, 3, 1, 0, 2, 0, 1, 2, 3, 0, 4, 0, 2, 6, 1, 0, 3, 3, 1, 2, 2, 0, 3, 0, 4, 6, 1, 2, 5, 0, 1, 2, 3, 0, 7, 0, 2, 4, 1, 0, 4, 3, 4, 6, 2, 0, 3, 2, 3, 3, 1, 0, 4, 0, 1, 17, 5, 6, 7, 0, 2, 5, 3, 0, 6, 0, 1, 2, 2, 4, 3, 0, 4, 3, 1, 0, 8, 8, 1, 2

Offset: 1

Matthew Campbell, Jun 24 2016

nonn

If n is prime then a(n)=0. If n is composite then a(n)=A280929(n). - Dmitry Kamenetsky, Jan 11 2017

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000040, A006877, A280929.

Table[Length@ NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, n, ! PrimeQ@ # &] - 1, {n, 120}] (* Michael De Vlieger, Jun 26 2016 *)

a(n) = my(i=0, k=n); while(!ispseudoprime(k), if(k%2==0, k=k/2, k=3*k+1); i++); i
for(n=1, 87, print1(a(n), ", ")) \\ Felix Fröhlich, Jun 24 2016

More terms from Felix Fröhlich, Jun 24 2016

A260624 a(n) = arithmetic derivative of the n-th composite number.

Original entry on oeis.org

4, 5, 12, 6, 7, 16, 9, 8, 32, 21, 24, 10, 13, 44, 10, 15, 27, 32, 31, 80, 14, 19, 12, 60, 21, 16, 68, 41, 48, 39, 25, 112, 14, 45, 20, 56, 81, 16, 92, 22, 31, 92, 33, 51, 192, 18, 61, 72, 26, 59, 156, 39, 55, 80, 18, 71, 176, 108, 43, 124, 22, 45, 32, 140, 123

Offset: 1

Matthew Campbell, Oct 06 2015

			The second composite number is 6. 6 = 2 * 3. 6' = 2*3' + 3*2' = 3 * 1 + 2 * 1 = 3 + 2 = 5, so a(2) = 5.

Matthew Campbell, Table of n, a(n) for n = 1..10000; used code by _Altug Alkan_.

Cf. A002808 (composite), A003415 (n').

Cf. A001787 ((2^n)'), A068719 ((2*n)'), A068720 ((n^2)'), A068721 ((n^3)').

lim = 120; f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; f /@ Rest@ Complement[Range@ lim, Prime@ Range@ PrimePi@ lim] (* Michael De Vlieger, Oct 07 2015, after Michael Somos at A003415 *)

forcomposite(n=1, 100, if((a(n) = local(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))), print1(a(n)", "))); \\ Altug Alkan, Oct 06 2015

a(n) = A003415(A002808(n)).

More terms from Altug Alkan, Oct 06 2015

A260623 Decimal expansion of the real solution x to zeta(x) - primezeta(x) = 2.

Original entry on oeis.org

1, 4, 2, 5, 7, 1, 0, 4, 1, 1, 6, 1, 3, 1, 8, 1, 6, 5, 1, 7, 8, 2, 3, 6, 8, 3, 6, 7, 5, 4, 8, 5, 5, 0, 5, 6, 9, 3, 3, 9, 1, 8, 6, 2, 0, 5, 3, 4, 6, 2, 4, 7, 3, 5, 9, 4, 9, 4, 9, 4, 7, 6, 7, 4, 3, 6, 6, 8, 7, 3, 0, 4, 5, 6, 7, 5, 6, 1, 7, 5, 0, 1, 6, 7, 7, 8, 6

Offset: 1

Matthew Campbell, Oct 06 2015

This is also the solution x to Sum_{c composite}(1/c^x) = 1.

			1.4257...

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100

For the prime analog, see A243350.

x /. FindRoot[Zeta[x] - PrimeZetaP[x] == 2, {x, 3/2}, WorkingPrecision -> 100] // RealDigits // First (* Jean-François Alcover, May 07 2021 *)

solve(x=1.1, 2, zeta(x) - sumeulerrat(1/p, x) - 2) \\ Michel Marcus, May 07 2021

a(6) corrected and a(7)-a(87) added by Hiroaki Yamanouchi, Nov 12 2015

A260621 Let b(k, n) = number obtained when the map x->A002808(x) is applied k times to n; a(n) is the smallest k such that b(k, n) + 1 is prime.

Original entry on oeis.org

1, 1, 12, 2, 1, 1, 3, 11, 1, 1, 7, 9, 1, 2, 10, 4, 2, 1, 1, 6, 8, 3, 3, 1, 9, 3, 1, 1, 18, 3, 1, 5, 7, 2, 2, 1, 4, 8, 2, 14, 1, 1, 6, 17, 2, 6, 1, 4, 6, 1, 1, 2, 2, 3, 7, 1, 13, 6, 1, 4, 16, 5, 16, 1, 5, 31, 35, 3, 5, 2, 1, 2, 3, 1, 1, 2, 6, 1, 1, 12, 5, 1, 2

Offset: 1

Matthew Campbell, Sep 25 2015

nonn

a(n) is also the smallest value of k at which b(k, n+1) - b(k, n) > 1.

			When n = 3, writing Composite(x) for A002808(x):
1. Composite(3) = 8. 8 + 1 = 9 = 3^2. 9 is not prime.
2. Composite(8) = 15. 15 + 1 = 16 = 2^4. 16 is not prime.
3. Composite(15) = 25. 25 + 1 = 26 = 2*13. 26 is not prime.
4. Composite(25) = 38. 38 + 1 = 39 = 3*13. 39 is not prime.
5. Composite(38) = 55. 55 + 1 = 56 = 2^3*7. 56 is not prime.
6. Composite(55) = 77. 77 + 1 = 78 = 2*3*13. 78 is not prime.
7. Composite(77) = 105. 105 + 1 = 106 = 2*53. 106 is not prime.
8. Composite(105) = 140. 140 + 1 = 141 = 3*47. 141 is not prime.
9. Composite(140) = 183. 183 + 1 = 184 = 2^3*23. 184 is not prime.
10. Composite(183) = 235. 235 + 1 = 236 = 2^2*59. 236 is not prime.
11. Composite(235) = 298. 298 + 1 = 299 = 13*23. 299 is not prime.
12. Composite(298) = 372. 372 + 1 = 373. 373 is prime.
--------------------------------------------------------------
Since the composite function was applied 12 times, a(3)=12.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Primes and nonprimes: A000040, A002808, A008578, A018252.
a(1) = p, a(n+1) = a(n)-th composite number: A006508, A022450, A022451, A025010, A025011, A059407, A059408.
Composites with order n > 1: A050435, A050436, A050438, A050439, A050440.
Composites with order n = b, n >= 1: A022449.
Composites with prime subscripts: A065858.
Composites without prime subscripts: A175251.
Order of compositeness: A059981, A236536.
Prime(n)-1: A006093.

c = Select[Range[10^5], CompositeQ]; Table[k = 1; While[! PrimeQ[Nest[c[[#]] &, n, k] + 1], k++]; k, {n, 120}] (* Michael De Vlieger, Jul 15 2016 *)

Terms from a(12) onward from Jon E. Schoenfield, Sep 27 2015

cp's OEIS Frontend

User: Matthew Campbell

A316346 a(n) = A316297(n+1)/A316297(n).

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A316297 a(n) = n! times the denominator of the n-th harmonic number H(n).

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A280420 Product of divisors of n!.

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A280409 Primes in the order that they appear in A280408, without repetitions.

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A280408 Irregular triangle read by rows listing the prime numbers that appear from the trajectory of n in Collatz Problem.

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Python

A280327 a(n) is obtained by applying the map k -> composite(k) n times, starting at n.

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Mathematica

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A274472 Number of iterations of the Collatz recursion required to reach a prime number.

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PARI

Extensions

A260624 a(n) = arithmetic derivative of the n-th composite number.

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PARI