A048102 -id:A048102 - cp's OEIS frontend

A365634 The number of divisors of n that are terms of A048102.

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 14 2023

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

Cf. A048102, A365632, A365635.

f[p_, e_] := If[e == p, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == f[i,1], 2, 1));}

Multiplicative with a(p^e) = 1 + [e = p], where [] is the Iverson bracket.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (p-1)/p^(p+1)) = 1.153074089009... .

A365635 The largest divisor of n that is a term of A048102.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 27, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 27, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 27, 1, 1, 4, 1, 1

Offset: 1

Amiram Eldar, Sep 14 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graphs - the asymptotic ratio (10^7, 10^8 and 10^9 terms).

Cf. A048102, A048103, A327939, A365634.

f[p_, e_] := p^If[e < p, 0, p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i, 1]^f[i,1]));}

Multiplicative with a(p^e) = 1 if e < p and p^p otherwise.
a(n) <= n with equality if and only if n is in A048102.
a(n) >= 1 with equality if and only if n is in A048103.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + (p^p-1)/p^(p*s)).

A051674 a(n) = prime(n)^prime(n).

Original entry on oeis.org

4, 27, 3125, 823543, 285311670611, 302875106592253, 827240261886336764177, 1978419655660313589123979, 20880467999847912034355032910567, 2567686153161211134561828214731016126483469, 17069174130723235958610643029059314756044734431

Offset: 1

Asher Auel

Numbers k such that bigomega(k)^(bigomega(k)) = k, where bigomega = A001222. - Lekraj Beedassy, Aug 21 2004
Positive k such that k' = k, where k' is the arithmetic derivative of k. - T. D. Noe, Oct 12 2004
David Beckwith proposes (in the AMM reference): "Let n be a positive integer and let p be a prime number. Prove that (p^p) | n! implies that (p^(p + 1)) | n!". - Jonathan Vos Post, Feb 20 2006
Subsequence of A100716; A003415(m*a(n)) = A129283(m)*a(n), especially A003415(a(n)) = a(n). - Reinhard Zumkeller, Apr 07 2007
A168036(a(n)) = 0. - Reinhard Zumkeller, May 22 2015

			a(1) = 2^2 = 4.
a(2) = 3^3 = 27.
a(3) = 5^5 = 3125.

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 740 pp. 95; 312, Ellipses Paris 2004.

T. D. Noe, Table of n, a(n) for n = 1..40
David Beckwith, Problem 11158, American Mathematical Monthly, Vol. 112, No. 5 (May 2005), p. 468.
Jurij Kovic, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences, Vol. 15 (2012), #12.3.8.

Cf. A000040, A000312, A003415 (arithmetic derivative of n), A129150, A129151, A129152, A048102, A072873 (multiplicative closure), A104126.
Subsequence of A100717; A203908(a(n)) = 0.
Subsequence of A097764.
Cf. A168036, A094289 (decimal expansion of Sum(1/p^p)).

a051674_list = map (\p -> p ^ p) a000040_list
-- Reinhard Zumkeller, Jan 21 2012

[p^p: p in PrimesUpTo(30)]; // Vincenzo Librandi, Mar 27 2014

A051674:=n->ithprime(n)^ithprime(n): seq(A051674(n), n=1..10); # Wesley Ivan Hurt, Jun 25 2016

Array[Prime[ # ]^Prime[ # ] &, 12] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
#^#&/@Prime[Range[10]] (* Harvey P. Dale, May 17 2024 *)

a(n)=n=prime(n);n^n \\ Charles R Greathouse IV, Mar 20 2013

from gmpy2 import mpz
[mpz(prime(n))**mpz(prime(n)) for n in range(1,100)] # Chai Wah Wu, Jul 28 2014

a(n) = A000312(A000040(n)). - Altug Alkan, Sep 01 2016

Sum_{n>=1} 1/a(n) = A094289. - Amiram Eldar, Oct 13 2020

A048103 Numbers not divisible by p^p for any prime p.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98

Offset: 1

N. J. A. Sloane

If a(n) = Product p_i^e_i then p_i > e_i for all i.
Complement of A100716; A129251(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2007
Density is 0.72199023441955... = Product_{p>=2} (1 - p^-p) where p runs over the primes. - Charles R Greathouse IV, Jan 25 2012
A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012
Range of A276086. Also numbers not divisible by m^m for any natural number m > 1. - Antti Karttunen, Nov 18 2024

			6 = 2^1 * 3^1 is OK but 12 = 2^2 * 3^1 is not.
625 = 5^4 is present because it is not divisible by 5^5.

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

Complement: A100716.
Positions of 0's in A129251, A342023, A376418, positions of 1's in A327936, A342007, A359550 (characteristic function).
Cf. A048102, A048104, A051674 (p^p), A054743, A054744, A377982 (a left inverse, partial sums of char. fun, see also A328402).
Cf. A276086 (permutation of this sequence, see also A376411, A376413).
Subsequences: A002110, A005117, A006862, A024451 (after its initial 0), A057588, A099308 (after its initial 0), A276092, A328387, A328832, A359547, A370114, A371083, A373848, A377871, A377992.
Disjoint union of {1}, A327934 and A358215.
Also A276078 is a subsequence, from which this differs for the first time at n=451 where a(451)=625, while that value is missing from A276078.

a048103 n = a048103_list !! (n-1)
a048103_list = filter (\x -> and $
   zipWith (>) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012

{1}~Join~Select[Range@ 120, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] > 0 &] (* Michael De Vlieger, Aug 19 2016 *)

isok(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,1] <= f[i,2], return(0))); return(1); \\ Michel Marcus, Nov 13 2020

A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); }; \\ (A359550 is the characteristic function for A048103) - Antti Karttunen, Nov 18 2024

from itertools import count, islice
from sympy import factorint
def A048103_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda d:d[1]A048103_list = list(islice(A048103_gen(),30)) # Chai Wah Wu, Jan 05 2023

;; With Antti Karttunen's IntSeq-library.
(define A048103 (ZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016

a(n) ~ kn with k = 1/Product_{p>=2}(1 - p^-p) = Product_{p>=2}(1 + 1/(p^p - 1)) = 1.3850602852..., where the product is over all primes p. - Charles R Greathouse IV, Jan 25 2012

For n >= 1, A377982(a(n)) = n. - Antti Karttunen, Nov 18 2024

More terms from James Sellers, Apr 22 2000

A072873 Numbers k such that Sum_i ( e(i)/p(i) ) is an integer, where the prime factorization of k is Product_i ( p(i)^e(i) ).

Original entry on oeis.org

1, 4, 16, 27, 64, 108, 256, 432, 729, 1024, 1728, 2916, 3125, 4096, 6912, 11664, 12500, 16384, 19683, 27648, 46656, 50000, 65536, 78732, 84375, 110592, 186624, 200000, 262144, 314928, 337500, 442368, 531441, 746496, 800000, 823543

Offset: 1

Benoit Cloitre, Jul 28 2002

nonn

Also, numbers k such that k divides k', the arithmetic derivative of k. As shown by Ufnarovski and Ahlander, all terms in this sequence have the form Product_{j=1..r} (pj^pj)^ej, where the pj are primes. The quotient k'/k equals Sum_{j=1..r} ej. - T. D. Noe, Jan 04 2006
Multiplicative closure of A051674. - Reinhard Zumkeller, Jan 21 2012
The number of terms < 10^k: 2, 5, 9, 15, 25, 36, 52, 73, 98, 128, 167, 213, 270, 338, 421, 517, 632, 768, 920, 1101, ..., . - Robert G. Wilson v, Jan 19 2016

			108 is in the sequence because 108 = 2^2*3^3 and 2/2 + 3/3 = 2 is an integer.

See A003415.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Nathaniel Johnston)

Cf. A003415, A051674, A027748, A085731, A048102, A124010.

import Data.Set (empty, fromList, deleteFindMin, union)
import qualified Data.Set as Set (null)
a072873 n = a072873_list !! (n-1)
a072873_list = 1 : h empty [1] a051674_list where
   h s mcs xs'@(x:xs)
    | Set.null s || x < m = h (s `union` fromList (map (* x) mcs)) mcs xs
    | otherwise = m : h (s' `union` fromList (map (* m) $ init (m:mcs)))
                        (m:mcs) xs'
    where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jan 21 2012

Select[Range[1000000],IntegerQ[Total[#[[2]]/#[[1]]&/@FactorInteger[#]]]&] (* Harvey P. Dale, Jul 04 2014 *)
lst = {}; Do[n = 2^e2*3^e3*5^e5*7^e7; If[n < 10^11, AppendTo[lst, n]], {e2, 0, 36, 2}, {e3, 0, 23, 3}, {e5, 0, 15, 5}, {e7, 0, 13, 7}]; Take[ Sort@ lst, 40] (* Robert G. Wilson v, Jan 19 2016 *)

is(n)=my(f=factor(n)); for(i=1,#f~,if(f[i,2]%f[i,1], return(0))); 1 \\ Charles R Greathouse IV, Oct 28 2014

from itertools import count, islice
from sympy import factorint
def A072873_gen(startvalue=1): # generator of terms >= startvalue
    return (k for k in count(max(startvalue,1)) if not any(e%p for p, e in factorint(k).items()))
A072873_list = list(islice(A072873_gen(),20)) # Chai Wah Wu, Sep 15 2023

A124010(a(n),k) mod A027748(a(n),k) = 0 for k = 1 .. A001221(a(n)). - Reinhard Zumkeller, Jan 21 2012

Sum_{n>=1} 1/a(n) = Product_{p prime} p^p/(p^p-1) = 1.38506028520448917638... - Amiram Eldar, Sep 27 2020

More terms from T. D. Noe, Jan 04 2006

A008478 Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.

Original entry on oeis.org

1, 4, 16, 27, 72, 108, 432, 800, 3125, 6272, 12500, 21600, 30375, 50000, 84375, 121500, 169344, 225000, 247808, 337500, 486000, 750141, 823543, 1350000, 1384448, 3000564, 3294172, 6690816, 12002256, 13176688, 19600000, 22235661, 37380096, 37879808, 59295096, 88942644

Offset: 1

Olivier Gérard

nonn

Fixed points of A008477.

a(3) = 16 is the only term of the form p^q with p <> q. - Bernard Schott, Mar 28 2021

			16 = 2^4 = 4^2.
27 = 3^3.
108 = 2^2*3^3.
6272 = 2^7*7^2.
121500 = 2^2 * 3^5*5^3.

Cf. A000040, A008477.

Some subsequences: p_i^p_i (A051674), Product_i {p_i^p_i} (A048102), Product_(j,k)(p_j^p_k * p_k^p_j) with p_j < p_k (A082949) (see examples).

f[n_] := Product[{p, e} = pe; e^p, {pe, FactorInteger[n]}];
Reap[For[n = 1, n <= 10^8, n++, If[f[n] == n, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 29 2021 *)

for(n=2,10^8,if(n==prod(i=1,omega(n), component(component(factor(n),2),i)^component(component(factor(n),1),i)),print1(n,",")))

More terms from David W. Wilson

a(34)-a(36) from Jean-François Alcover, Mar 29 2021

A100717 Numbers k having a prime divisor p such that p^p is the highest power of p that divides k.

Original entry on oeis.org

4, 12, 20, 27, 28, 36, 44, 52, 54, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 135, 140, 148, 156, 164, 172, 180, 188, 189, 196, 204, 212, 216, 220, 228, 236, 244, 252, 260, 268, 270, 276, 284, 292, 297, 300, 308, 316, 324, 332, 340, 348, 351, 356, 364, 372

Offset: 1

Leroy Quet, Dec 10 2004

nonn

For each prime p, the sequence includes all k*p^p for k such that gcd(k,p)=1. - T. D. Noe

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/p^p + 1/p^(p+1)) = 0.14682429539560371215... . - Amiram Eldar, Jun 25 2022

			54 is included because 3^3, but not 3^4, divides 54.

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

Cf. A100716, A203908.

Subsequences: A051674, A048102 \ {1}.

a100717 n = a100717_list !! (n-1)
a100717_list = filter ((== 0) . a203908) [1..]
-- Reinhard Zumkeller, Dec 24 2013

fQ[n_] := Union[ Table[ #[[1]] == #[[2]]] & /@ FactorInteger[n]][[ -1]] == True; Select[ Range[2, 375], fQ[ # ] &] (* Robert G. Wilson v, Dec 14 2004 *)

A203908(a(n)) = 0. - Reinhard Zumkeller, Dec 24 2013

More terms from T. D. Noe and Robert G. Wilson v, Dec 14 2004

A048104 If n = Product p_i^e_i (e_i >= 1) then for some i, p_i > e_i and for some j, p_j < e_j.

Original entry on oeis.org

24, 40, 48, 56, 72, 80, 88, 96, 104, 112, 120, 136, 144, 152, 160, 162, 168, 176, 184, 192, 200, 208, 224, 232, 240, 248, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 405, 408, 416, 424, 440, 448, 456, 464, 472

Offset: 1

N. J. A. Sloane

The asymptotic density of this sequence is 1 - Product_{p prime} (1-1/p^(p+1)) = 0.13585792767780221591... . - Amiram Eldar, Feb 14 2023

Verified up to a(120) = 1000, except for a(16) = 162 and a(55) = 486, every a(n) is also the order of an isomorphism class for which there exists at least one nonabelian nilpotent group G such that |Aut(G)|/a(n) is nonintegral. Within the same range there are 26 group orders not in a(n), which, except for 3^4*2^3 = 648, all have the form 3^3*m or 5^3*k, with m and k being prime, squarefree, or nonsquarefree. - Miles Englezou, Jul 16 2024

			48 = 2^4*3^1 is a term but 12 = 2^2*3^1 is not.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A048102, A048103.

Select[Range[500], AnyTrue[(f = FactorInteger[#]), First[#1] > Last[#1] &] && AnyTrue[f, First[#1] < Last[#1] &] &] (* Amiram Eldar, Nov 13 2020 *)

isok(n) = my(f=factor(n), b1=0, b2=0); for (i=1, #f~, if (f[i,1] < f[i,2], b1=1, if (f[i,1] > f[i,2], b2=1))); return(b1 && b2); \\ Michel Marcus, Nov 13 2020

More terms from Reiner Martin, Jul 07 2001

A272818 Numbers such that (sum + product) of all their prime factors equals (sum + product) of all exponents in their prime factorization.

Original entry on oeis.org

1, 4, 27, 72, 96, 108, 486, 800, 1280, 3125, 6272, 10976, 12500, 14336, 21600, 30375, 36000, 48600, 51840, 54675, 69120, 84375, 121500, 134456, 169344, 174960, 192000, 225000, 240000, 247808, 337500, 340736, 395136, 435456, 451584, 703125, 750141, 781250, 787320, 823543, 857304, 885735

Offset: 1

Giuseppe Coppoletta, May 08 2016

nonn

For p prime, p^p satisfy the condition, hence A051674 (and also A048102) is a subsequence. Moreover, if p and q are primes and i and j are positive integers, if p^i * q^j verify the condition, then the same is true for p^j * q^i. So A122406 is also a subsequence. More generally, if a number is a term, then any permutation of the exponents in its prime factorization (i.e. any permutation of its prime signature) gives also a term. In addition, any number having no more than two distinct prime factors (apart their multiplicity) is a term iff it belongs also to A272858.

			885735 = 3^11 * 5 is included because (3+5) + 3*5 = (11+1) + 11*1.
2^10 * 3^6 * 19^2 is included because (2+3+19)+ 2*3*19 = (10+6+2)+ 10*6*2.

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..1413 (terms < 10^19, first 100 terms from G. Coppoletta)

Cf. A048102, A054411, A054412, A071174, A122406, A272858, A272859.

Select[Range[10^6], Total@ First@ # + Times @@ First@ # == Total@ Last@ # + Times @@ Last@ # &@ Transpose@ FactorInteger@ # &] (* Michael De Vlieger, May 08 2016 *)

spp(v) = vecsum(v) + prod(k=1, #v, v[k]);
isok(n) = my(f = factor(n)); spp(f[,1]) == spp(f[,2]); \\ Michel Marcus, May 08 2016

def d(n):
    v = factor(n)
    d1 = sum(w[0] for w in v) + prod(w[0] for w in v)
    d2 = sum(w[1] for w in v) + prod(w[1] for w in v)
    return d1 == d2
[k for k in (1..10000) if d(k)]

A272858 Numbers m such that Product(1 + p_i) = Product(1 + e_i), where m = Product((p_i)^e_i).

Original entry on oeis.org

1, 4, 27, 72, 96, 108, 486, 800, 1280, 3125, 6272, 10976, 12500, 14336, 21600, 28800, 30375, 34560, 36000, 38880, 48600, 54675, 84375, 92160, 96000, 121500, 134456, 153600, 169344, 217728, 218700, 225000, 247808, 262440, 296352, 300000, 337500, 340736, 387072, 395136, 489888, 666792, 703125, 750141, 781250, 823543, 857304, 885735

Offset: 1

Giuseppe Coppoletta, May 08 2016

nonn

A048102 is clearly a subsequence, as for any prime p, p^p satisfy the herein condition. Similarly, A122406 is also a subsequence. More generally, if a number is a term, then any permutation of the exponents in its prime factorization (i.e., any permutation of its prime signature) gives also a term.

The condition defining this sequence coincides with the condition in A272859 at least for the terms of A114129.

			92160 is included because 92160 = 2^11 * 3^2 * 5 and (2+1)*(3+1)*(5+1) = (11+1)*(2+1)*(1+1).

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..6058 (terms < 10^18, first 100 terms from G. Coppoletta)

Cf. A048102, A054411, A054412, A071174, A114129, A122406, A272818, A272859.

ok[n_] := Block[{p,e}, {p,e} = Transpose@ FactorInteger@ n; Times @@ (1+p) == Times @@ (1+e)]; Select[Range[10^6], ok] (* Giovanni Resta, May 08 2016 *)

is(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]+1)==prod(i=1,#f~,f[i,2]) \\ Charles R Greathouse IV, Sep 08 2016

def d(n):
    v = factor(n)
    d1 = prod(1 + w[0] for w in v)
    d2 = prod(1 + w[1] for w in v)
    return d1 == d2
[k for k in (1..10000) if d(k)]

If N is a positive integer and N = Product_{i=1..k} (p_i)^e_i is its prime factorization, then N is in A272858 iff Product_{i=1..k} (1 + p_i) = Product_{i=1..k} (1 + e_i).

For a number with three different prime factors N = p1^e1 * p2^e2 * p3^e3, the defining condition can be expressed as: p1 + p2 + p3 + p1*p2 + p1*p3 + p2*p3 + p1*p2*p3 = e1 + e2 + e3 + e1*e2 + e1*e3 + e2*e3 + e1*e2*e3.

cp's OEIS Frontend

A365634 The number of divisors of n that are terms of A048102.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A365635 The largest divisor of n that is a term of A048102.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A051674 a(n) = prime(n)^prime(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

A048103 Numbers not divisible by p^p for any prime p.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Python

Scheme

Formula

Extensions

A072873 Numbers k such that Sum_i ( e(i)/p(i) ) is an integer, where the prime factorization of k is Product_i ( p(i)^e(i) ).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

Extensions

A008478 Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica