A084114 -id:A084114 - cp's OEIS frontend

A084115 A084113(n) minus A084114(n).

0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 2, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 2, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 3, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 3, 4, 1, 3, 1, 3, 3

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = A084113(n) - A084114(n) = 2*A084113(n) - A032741(n) = A032741(n) - 2*A084114(n);

a(A084116(n)) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A066423, A084116.

Cf. A084110, A084113, A084114.

a084115 n = a084113 n - a084114 n -- Reinhard Zumkeller, Jul 31 2014

Definition fixed by Reinhard Zumkeller, Jul 31 2014

A000430 Primes and squares of primes.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223

Offset: 1

R. Muller

Also numbers n such that the product of proper divisors is < n.
See A050216 for lengths of blocks of consecutive primes. - Reinhard Zumkeller, Sep 23 2011
Numbers q > 1 such that d(q) < 4. Numbers k such that the number of ways of writing k = m + t is equal to the number of ways of writing k = r*s, where m|t and r|s. - Juri-Stepan Gerasimov, Oct 14 2017
Called multiplicatively deficient numbers by Chau (2004). - Amiram Eldar, Jun 29 2022

F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House 2000
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

T. D. Noe, Table of n, a(n) for n = 1..1000
William Chau, The tau, sigma, rho functions, and some related numbers, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; entire issue.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Union of A000040 and A001248.

Cf. A007422, A010051, A010055, A032741, A046951, A050216, A056595, A058080, A064911, A084110, A084114, A293575.

a000430 n = a000430_list !! (n-1)
a000430_list = m a000040_list a001248_list where
   m (x:xs) (y:ys) | x < y = x : m xs (y:ys)
                   | x > y = y : m (x:xs) ys
-- Reinhard Zumkeller, Sep 23 2011

nn = 223; t = Union[Prime[Range[PrimePi[nn]]], Prime[Range[PrimePi[Sqrt[nn]]]]^2] (* T. D. Noe, Apr 11 2011 *)
Module[{upto=250,prs},prs=Prime[Range[PrimePi[upto]]];Select[Join[ prs,prs^2], #<=upto&]]//Sort (* Harvey P. Dale, Oct 08 2016 *)

is(n)=isprime(n) || (issquare(n,&n) && isprime(n)) \\ Charles R Greathouse IV, Sep 04 2013

from math import isqrt
from sympy import primepi
def A000430(n):
    def f(x): return n+x-primepi(x)-primepi(isqrt(x))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

A084114(a(n)) = 0, see also A084110. - Reinhard Zumkeller, May 12 2003
A109810(a(n)) = 2. - Reinhard Zumkeller, May 24 2010
A010051(a(n)) + A010055(a(n))*A064911(a(n)) = 1;
A056595(a(n)) = 1. - Reinhard Zumkeller, Aug 15 2011
A032741(a(n)) = A046951(a(n)); A293575(a(n)) = 0. - Juri-Stepan Gerasimov, Oct 14 2017
The number of terms not exceeding x is N(x) ~ (x + 2*sqrt(x))/log(x) (Chau, 2004). - Amiram Eldar, Jun 29 2022

A080257 Numbers having at least two distinct or a total of at least three prime factors.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Feb 10 2003

Complement of A000430; A080256(a(n)) > 3.
A084114(a(n)) > 0, see also A084110.
Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));
a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.
Numbers with at least 4 divisors. - Franklin T. Adams-Watters, Jul 28 2006
Union of A024619 and A033942; A211110(a(n)) > 2. - Reinhard Zumkeller, Apr 02 2012
Also numbers > 1 that are neither prime nor a square of a prime. Also numbers whose omega-sequence (A323023) has sum > 3. Numbers with omega-sequence summing to m are: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7). - Gus Wiseman, Jul 03 2019
Numbers n such that sigma_2(n)*tau(n) = A001157(n)*A000005(n) >= 4*n^2. Note that sigma_2(n)*tau(n) >= sigma(n)^2 = A072861 for all n. - Joshua Zelinsky, Jan 23 2025

			8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
From _Gus Wiseman_, Jul 03 2019: (Start)
The sequence of terms together with their prime indices begins:
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.

Cf. A060687, A118914, A323014, A323023, A325249.

a080257 n = a080257_list !! (n-1)
a080257_list = m a024619_list a033942_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y  = x : m xs ys'
                           | x == y = x : m xs ys
                           | x > y  = y : m xs' ys
-- Reinhard Zumkeller, Apr 02 2012

Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* Harvey P. Dale, Jul 23 2013 *)

PARI
```
is(n)=omega(n)>1 || isprimepower(n)>2
```

is(n)=my(k=isprimepower(n)); if(k, k>2, !isprime(n)) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = n + O(n/log n). - Charles R Greathouse IV, Sep 14 2015

Definition clarified by Harvey P. Dale, Jul 23 2013

A084110 Let L(n) = ordered list of divisors of n = {d_1=1, d_2, ..., d_k=n}; set e_1=1, e_i = e_{i-1}/d_i if that is an integer otherwise e_i = e_{i-1}*d_i; then a(n) = e_k.

Original entry on oeis.org

1, 2, 3, 8, 5, 1, 7, 1, 27, 1, 11, 48, 13, 1, 1, 16, 17, 162, 19, 80, 1, 1, 23, 16, 125, 1, 1, 112, 29, 25, 31, 512, 1, 1, 1, 1944, 37, 1, 1, 25, 41, 49, 43, 176, 405, 1, 47, 48, 343, 1250, 1, 208, 53, 324, 1, 49, 1, 1, 59, 9, 61, 1, 567, 8, 1, 121, 67, 272, 1, 49, 71, 9, 73, 1

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = r(n,tau(n)), where r is defined as follows:
let d(n,j) = j-th divisor of n, 1 <= j <= tau(n) = A000005(n), r(n,1)=d(n,1), r(n,j) = if d(n,j) divides r(n,j-1) then r(n,j-1)/d(n,j) else r(n,j-1)*d(n,j), 1 < j <= tau(n);
p prime: a(p)=p, a(p^2)=p^3, a(p^3)=1, a(p^k)=p^A008344(k+1);
a(m)=1 iff m multiplicatively perfect: a(A007422(k))=1.
a(A084111(n)) = A084111(n). - Reinhard Zumkeller, Jul 31 2014

			Divisors of 48 = {1,2,3,4,6,8,12,16,24,48}: 1*2*3 = 6 -> 6*4 = 24 -> 24/6 = 4 -> 4*8 = 32 -> 32*12 = 384 -> 384/16 = 24 -> 24/24 = 1 -> 1*48 = a(48);
divisors of 49 = {1,7,49}: 1*7 = 7 -> 7*49 = 343 = a(49);
divisors of 50 = {1,2,5,10,25,50}: 1*2*5 = 10 -> 10/10 = 1 -> 1*25 = 25 -> 25*50 = 1250 = a(50).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product.
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A027750, A084111 (fixed points), A084113, A084114.

a084110 = foldl (/*) 1 . a027750_row where
   x /* y = if m == 0 then x' else x*y where (x',m) = divMod x y
-- Reinhard Zumkeller, Feb 21 2012, Oct 25 2010

a[n_] := Module[{d = Divisors[n], e}, e[i_] := e[i] = If[i == 1, 1, If[Divisible[e[i-1], d[[i]]], e[i-1]/d[[i]], e[i-1] d[[i]]]]; e[Length[d]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 10 2021 *)

Corrected and extended by David Wasserman, Dec 14 2004

A084116 Numbers m such that A084115(m) = 1.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

A084113(a(n)) = A084114(a(n)) + 1.
Union of primes and multiplicatively perfect numbers (A000040, A007422).
A084115(a(n)) = 1; A066729(a(n)) = a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A084110, A066729, A084113, A084114, A084115, A066423 (complement).

a084116 n = a084116_list !! (n-1)
a084116_list = filter ((== 1) . a084115) [1..]
-- Reinhard Zumkeller, Jul 31 2014

Select[Range[2, 200], PrimeQ[DivisorSigma[0, #]^DivisorSigma[0, #] + 1] &] (* Carl Najafi, Oct 19 2011 *)

is(n)=isprime(n) || numdiv(n) == 4 \\ Charles R Greathouse IV, Oct 19 2015

It appears that a(n) = n such that A000005(n)^A000005(n)+1 is prime. - Carl Najafi, Oct 19 2011

Corrected and edited by Carl Najafi, Oct 19 2011

Revised by Reinhard Zumkeller, Jul 31 2014

A084113 Number of multiplications when calculating A084110(n).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = A000005(n)-1-A084114(n) = A032741(n)-A084114(n) = (A032741(n)+A084115(n))/2;

a(n) = 1 iff n is prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product.

Cf. A027750, A084110, A084114.

a084113 = f 0 1 . a027750_row where
   f c _ []     = c
   f c x (d:ds) = if r == 0 then f c x' ds else f (c + 1) (x * d) ds
                  where (x', r) = divMod x d
-- Reinhard Zumkeller, Jul 31 2014

cp's OEIS Frontend

A084115 A084113(n) minus A084114(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Extensions

A000430 Primes and squares of primes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A080257 Numbers having at least two distinct or a total of at least three prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

A084110 Let L(n) = ordered list of divisors of n = {d_1=1, d_2, ..., d_k=n}; set e_1=1, e_i = e_{i-1}/d_i if that is an integer otherwise e_i = e_{i-1}*d_i; then a(n) = e_k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A084116 Numbers m such that A084115(m) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A084113 Number of multiplications when calculating A084110(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell