A072873 -id:A072873 - cp's OEIS frontend

A048102 Numbers k such that if k = Product p_i^e_i then p_i = e_i for all i.

1, 4, 27, 108, 3125, 12500, 84375, 337500, 823543, 3294172, 22235661, 88942644, 2573571875, 10294287500, 69486440625, 277945762500, 285311670611, 1141246682444, 7703415106497, 30813660425988, 302875106592253, 891598970659375, 1211500426369012, 3566395882637500

Offset: 1

N. J. A. Sloane

			3^3*5^5 = 84375.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1038 terms from T. D. Noe)
Daniel Mondot, Table of n, a(n) for n = 1..10000 with factorizations

Cf. A027748, A048103, A048104, A051674, A072873, A124010.

import Data.Set (empty, fromList, deleteFindMin, union)
import qualified Data.Set as Set (null, map)
a048102 n = a048102_list !! (n-1)
a048102_list = 1 : f empty [1] a051674_list where
  f s ys pps'@(pp:pps)
    | Set.null s = f (fromList (map (* pp) ys)) (pp:ys) pps
    | pp < m     = f (s `union` Set.map (* pp) s `union`
                      fromList (map (* pp) ys)) ys pps
    | otherwise  = m : f s' (m:ys) pps'
    where (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Jan 21 2012

isok(n) = my(f = factor(n)); for (k=1, #f~, if (f[k,1] != f[k,2], return(0))); 1; \\ Michel Marcus, Apr 29 2016

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

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^p) = 1.2967126856... - Amiram Eldar, Oct 13 2020

More terms from Naohiro Nomoto, Jun 28 2001

A083346 Denominator of r(n) = Sum(e/p: n=Product(p^e)).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 2, 3, 10, 11, 3, 13, 14, 15, 1, 17, 6, 19, 5, 21, 22, 23, 6, 5, 26, 1, 7, 29, 30, 31, 2, 33, 34, 35, 3, 37, 38, 39, 10, 41, 42, 43, 11, 15, 46, 47, 3, 7, 10, 51, 13, 53, 2, 55, 14, 57, 58, 59, 15, 61, 62, 21, 1, 65, 66, 67, 17, 69, 70, 71, 6, 73, 74, 15, 19, 77, 78

Offset: 1

Reinhard Zumkeller, Apr 25 2003

Multiplicative with a(p^e) = 1 iff p|e, p otherwise. For f(n) = A083345(n)/A083346(n), f(p^i*q^j*...) = f(p^i)+f(q^j)+ ... The denominator of each term is 1 or the prime, thus the denominator of the sum is the product of the denominators of the components. - Christian G. Bower, May 16 2005

n divided by the greatest common divisor of n and its arithmetic derivative, i.e., a(n) = n/gcd(n,n') = A000027(n)/A085731(n). - Giorgio Balzarotti, Apr 14 2011

			n=12 = 2*2*3 = 2^2 * 3^1 -> r(12) = 2/2 + 1/3 = (6+2)/6, therefore a(12)=3, A083345(12)=4;
n=18 = 2*3*3 = 2^1 * 3^2 -> r(18) = 1/2 + 2/3 = (3+4)/6, therefore a(18)=6, A083345(18)=7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A083345 (numerator).
Cf. A035263 (parity of terms), A003159 (positions of odd terms), A036554 (of even terms).
Cf. A065463, A072873, A083347, A083348, A359588 (Dirichlet inverse).

a[n_] := Product[Module[{p, e}, {p, e} = pe; If[Divisible[e, p], 1, p]], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Oct 06 2021 *)

A083346(n) = { my(f=factor(n)); denominator(vecsum(vector(#f~,i,f[i,2]/f[i,1]))); }; \\ Antti Karttunen, Mar 01 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463 * Product_{p prime} (p^(2*p)*(p^2+p-1)-p^3)/((p^2+p-1)*(p^(2*p)-1)) = 0.3374565531... . - Amiram Eldar, Nov 18 2022

Incorrect formula removed by Antti Karttunen, Jan 09 2023

A327939 Multiplicative with a(p^e) = p^(e-(e mod p)).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 27, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 27, 1, 4, 1, 1, 1, 4, 1, 1, 1, 64, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 27, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 108, 1, 1, 1, 16

Offset: 1

Antti Karttunen, Oct 01 2019

Fixed points of the map x -> gcd(x, A003415(x)), i.e., if we start iterating with A085731 from any x = n (>= 1), we will eventually reach a(n), after which the result does not change anymore. This was found by LODA miner (see C. Krause link), and is easily seen to be true by Eric M. Schmidt's multiplicative formula for A085731. Note also that this sequence is idempotent, meaning a(a(n)) = a(n) for all n. - Antti Karttunen, Apr 05 2021

The largest divisor of n that is a term of A072873. - Amiram Eldar, Sep 14 2023

Antti Karttunen, Table of n, a(n) for n = 1..65537
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences.

Cf. A003415, A072873, A085731, A327938.

Differs from A234957 for the first time at n=27.

f[p_, e_] := p^(e - Mod[e, p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2023 *)

A327939(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]-(f[k,2]%f[k,1]))); factorback(f); };

Multiplicative with a(p^e) = p^(e-(e mod p)).

a(n) = n / A327938(n).

A083347 Numbers k such that Sum(e/p: k=Product(p^e)) < 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 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, 91, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 1

Reinhard Zumkeller, Apr 25 2003

nonn

Numbers k whose arithmetic derivative (A003415) k' < k. - T. D. Noe, Apr 24 2011

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A003415, A072873, A051674, A083348.

Cf. A168036.

a083347 n = a083347_list !! (n-1)
a083347_list = filter ((< 0) . a168036) [1..]
-- Reinhard Zumkeller, May 22 2015, May 10 2011

Select[Range@ 102, If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger@ Abs@ #]] < # &] (* Michael De Vlieger, Feb 02 2019 *)

A083345(a(n)) < A083346(a(n));
A083345(A051674(n)) = A083346(A051674(n));
A083345(A083348(n)) > A083346(A083348(n)).
A168036(a(n)) < 0. - Reinhard Zumkeller, May 22 2015

A083348 Numbers k such that r(k) = Sum(e/p: k = Product(p^e)) > 1.

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 198

Offset: 1

Reinhard Zumkeller, Apr 25 2003

nonn

The number of terms not exceeding 10^m, for m = 1, 2, ..., are 1, 29, 318, 3174, 31763, 317813, 3177179, 31774009, 317745099, 3177373819, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3177... . - Amiram Eldar, Jun 24 2022

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A003415, A072873, A051674, A083345, A083346, A083347, A168036, A369048 (characteristic function), A369049.

Subsequence of A100717.

a083348 n = a083348_list !! (n-1)
a083348_list = filter ((> 0) . a168036) [1..]
-- Reinhard Zumkeller, May 22 2015, May 10 2011

ad[n_] := Switch[n, 0 | 1, 0, _, If[PrimeQ[n], 1, Sum[Module[ {p, e}, {p, e} = pe; n e/p], {pe, FactorInteger[n]}]]];
Select[Range[1000], ad[#] > # &] (* Jean-François Alcover, May 04 2023 *)

A083345(a(n)) > A083346(a(n)).
A083345(A051674(n)) = A083346(A051674(n)).
A083345(A083347(n)) < A083346(A083347(n)).
A168036(a(n)) > 0. - Reinhard Zumkeller, May 22 2015

A342014 Arithmetic derivative of n, taken modulo n: a(n) = A003415(n) mod n.

Original entry on oeis.org

0, 1, 1, 0, 1, 5, 1, 4, 6, 7, 1, 4, 1, 9, 8, 0, 1, 3, 1, 4, 10, 13, 1, 20, 10, 15, 0, 4, 1, 1, 1, 16, 14, 19, 12, 24, 1, 21, 16, 28, 1, 41, 1, 4, 39, 25, 1, 16, 14, 45, 20, 4, 1, 27, 16, 36, 22, 31, 1, 32, 1, 33, 51, 0, 18, 61, 1, 4, 26, 59, 1, 12, 1, 39, 55, 4, 18, 71, 1, 16, 27, 43, 1, 40, 22, 45, 32, 52, 1, 33

Offset: 1

Antti Karttunen, Mar 04 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A072873 (positions of zeros), A085731 [= gcd(n, a(n))], A342015 [= A342014(A276086(n))].

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342014(n) = (A003415(n)%n);

a(n) = A003415(n) mod n.

A342090 Numbers with at least one prime power p^e in their prime factorization such that p|e.

Original entry on oeis.org

4, 12, 16, 20, 27, 28, 36, 44, 48, 52, 54, 60, 64, 68, 76, 80, 84, 92, 100, 108, 112, 116, 124, 132, 135, 140, 144, 148, 156, 164, 172, 176, 180, 188, 189, 192, 196, 204, 208, 212, 216, 220, 228, 236, 240, 244, 252, 256, 260, 268, 270, 272, 276, 284, 292, 297, 300

Offset: 1

Amiram Eldar, Feb 27 2021

nonn

Numbers with a unitary divisor of the form p^(m*p) where p is a prime and m > 0.
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 19, 188, 1883, 18825, 188244, 1882429, 18824297, 188242957, 1882429628, ...
The asymptotic density of this sequence is 1 - Product_{p prime} 1 - (p - 1)/(p*(p^p - 1)) = 0.18824296270011399086...

			4 = 2^2 is a term since 2 divides 2.
8 = 2^3 is not a term since 2 does not divide 3.

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

Subsequence of A013929.
Subsequences: A000302, A000312, A009971, A017113, A051674.
Cf. A072873, A369070 (characteristic function).

q[n_] := AnyTrue[FactorInteger[n], Divisible[Last[#], First[#]] &]; Select[Range[2, 300], q]

Wrong term 1 removed by Amiram Eldar, Jan 16 2024

A368334 The number of terms of A054744 that are unitary divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 21 2023

First differs from A081117 at n = 28.

Also, the number of terms of A072873 that are unitary divisors of n.

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

Cf. A034444, A048103, A054744, A072873, A327939, A368330, A368332, A368333, A368335.

Cf. A081117.

f[p_, e_] := If[e < p, 1, 2]; 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, 2));}

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = 2 if e >= p.
a(n) = A034444(A368333(n)).
a(n) = A034444(A327939(n)).
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A034444(n), with equality if and only if n is in A054744.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(p*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^p) = 1.29671268566745796443... .

A165558 Integers that are half of their arithmetic derivatives.

Original entry on oeis.org

0, 16, 108, 729, 12500, 84375, 3294172, 9765625, 22235661, 2573571875, 678223072849, 1141246682444, 7703415106497, 891598970659375, 1211500426369012, 8177627877990831, 234966429149994773, 946484708100790625

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Sep 22 2009

nonn

All integers of the form p^p*q^q, with q and p two distinct primes, are in the sequence. [R. J. Mathar, Sep 26 2009]
6*10^8 < a(10) <= 2573571875. a(11) <= 678223072849. [Donovan Johnson, Nov 03 2010]
By a result of Ufnarovski and Ahlander, an integer is in this sequence if and only if it has the form p^(2p) or p^p*q^q, with p and q distinct primes. See comments from A072873. [Nathaniel Johnston, Nov 22 2010]

			For k =84375 = 3^3*5^5, so A003415(k)/2 = 84375*(3/3+5/5)/2 = 84375 = k, which adds k=84375 to the sequence.

Cf. A003415, A072873.

with(numtheory);
P:=proc(n)
local a,i,p,pfs;
for i from 1 to n do
  pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs);  if a=2*i then print(i); fi; od;
end:
P(100000000);

d[0] = d[1] = 0; d[n_] := n*Total[f = FactorInteger[n]; f[[All, 2]]/f[[All, 1]] ]; Join[{0}, Reap[Do[p = Prime[n]; ip = p^(2*p); If[ip == d[ip]/2, Sow[ip]]; Do[q = Prime[k]; iq = p^p*q^q; If[iq == d[iq]/2, Sow[iq]], {k, n+1, 6}], {n, 1, 5}]][[2, 1]] // Union][[1 ;; 18]] (* Jean-François Alcover, Apr 22 2015, after Nathaniel Johnston *)

{n: A003415(n) = 2*n}.

Entries checked by R. J. Mathar, Sep 26 2009
a(7)-a(9) from Donovan Johnson, Nov 03 2010
a(10)-a(18) and general form from Nathaniel Johnston, Nov 22 2010

A348284 Numbers k such that k | k" where k" is the 2nd arithmetic derivative of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 31, 37, 41, 43, 47, 48, 53, 54, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 108, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 162, 163, 167, 168, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Wesley Ivan Hurt, Oct 09 2021

nonn

As 1" = 0 and p" = 0 when p is prime, 1 and every prime are terms, hence A008578 is a subsequence. - Bernard Schott, Oct 12 2021

			8 is in the sequence since 8" = 16 and 8 | 16.

Cf. A003415 (1st derivative), A068346 (2nd derivative).

Cf. A008578, A072873.

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
q:= n-> is(irem(d(d(n)), n)=0):
select(q, [$1..250])[];  # Alois P. Heinz, Oct 15 2021

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = !(ad(ad(k)) % k); \\ Michel Marcus, Oct 10 2021

cp's OEIS Frontend

A048102 Numbers k such that if k = Product p_i^e_i then p_i = e_i for all i.

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A083346 Denominator of r(n) = Sum(e/p: n=Product(p^e)).

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A327939 Multiplicative with a(p^e) = p^(e-(e mod p)).

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A083347 Numbers k such that Sum(e/p: k=Product(p^e)) < 1.

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A083348 Numbers k such that r(k) = Sum(e/p: k = Product(p^e)) > 1.

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A342014 Arithmetic derivative of n, taken modulo n: a(n) = A003415(n) mod n.

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A342090 Numbers with at least one prime power p^e in their prime factorization such that p|e.

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A368334 The number of terms of A054744 that are unitary divisors of n.