A322791 -id:A322791 - cp's OEIS frontend

A335218 Exponential Zumkeller numbers: numbers whose exponential divisors can be partitioned into two disjoint subsets of equal sum.

36, 180, 252, 396, 468, 612, 684, 828, 900, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1764, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3600, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4356, 4500, 4572, 4716, 4788, 4900

Offset: 1

Amiram Eldar, May 27 2020

nonn

First differs from A318100 at n = 49: 4900 is a term that is not an exponential pseudoperfect number.

			36 is a term since its exponential divisors, {6, 12, 18, 36}, can be partitioned into 2 disjoint sets whose sum is equal: 6 + 12 + 18 = 36.

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

The exponential version of A083207.
Subsequence of A129575.
A054979 is a subsequence.
Cf. A318100, A322791, A323343.

dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; eDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expDivQ[n, #] &]]; ezQ[n_] := Module[{d = eDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[10^4], ezQ]

A340233 a(n) is the least number with exactly n exponential divisors.

Original entry on oeis.org

1, 4, 16, 36, 65536, 144, 18446744073709551616, 576, 1296, 589824

Offset: 1

Amiram Eldar, Jan 01 2021

nonn

a(11) = 2^(2^10) has 309 digits and is too large to be included in the data section.

See the link for more values of this sequence.

			a(2) = 4 since 4 is the least number with 2 exponential divisors, 2 and 4.

Amiram Eldar, Table of n, a(n) for n = 1..12
Amiram Eldar, Table of n, a(n) for n = 1..100 (given by prime factorizations)

Subsequence of A025487.
Cf. A049419, A318278, A322791.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340232 (bi-unitary).

f[p_, e_] := DivisorSigma[0, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 6; s = Table[0, {max}]; c = 0; n = 1;  While[c < max, i = d[n]; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s (* ineffective for n > 6 *)

A049419(a(n)) = n and A049419(k) != n for all k < a(n).

A379027 Irregular table read by rows in which the n-th row lists the modified exponential divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 6, 8, 24, 1, 25, 1, 2, 13, 26, 1, 3, 27, 1, 4, 7, 28

Offset: 1

Amiram Eldar, Dec 14 2024

If the prime factorization of n is Product_{i} p_i^e_i, then the modified exponential divisors of n are all the divisors of n that are of the form Product_{i} p_i^b_i such that 1 + b_i | 1 + e_i for all i.

			The table starts:
  1;
  1, 2;
  1, 3;
  1, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 3, 4, 12;

Amiram Eldar, Table of n, a(n) for n = 1..11627 (first 2000 rows)
Andrew V. Lelechenko, The Quest for the Generalized Perfect Numbers, Theoretical and Applied Aspects of Cybernetics, Proceedings, The 4th International Scientific Conference of Students and Young Scientists, Kyiv, 2014.
David Moews, Perfect, amicable and sociable numbers.

Cf. A379028 (row lengths), A241405 (row sums).

Similar tables: A027750 (all divisors), A077609 (infinitary), A077610 (unitary), A222266 (bi-unitary), A322791 (exponential), A361255 (exponential unitary).

modexpDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[Divisible, {f[[;; , 2]] + 1, IntegerExponent[d, f[[;; , 1]]] + 1}]]; row[1] = {1}; row[n_] := Select[Divisors[n], modexpDivQ[n, #] &]; Table[row[n], {n, 1, 28}] // Flatten

ismodexpdiv(f, d) = {my(e); for(i=1, #f~, e = valuation(d, f[i, 1]); if((f[i, 2]+1) % (e+1), return(0))); 1; }
row(n) = {my(f = factor(n), d = divisors(f), mediv = [1]); if(n == 1, return(mediv)); for(i=2, #d, if(ismodexpdiv(f, d[i]), mediv = concat(mediv, d[i]))); mediv; }

A383760 Irregular triangle read by rows in which the n-th row lists the exponential infinitary divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 16, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 24, 5, 25, 26, 3, 27, 14, 28, 29, 30, 31, 2, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 48, 7, 49, 10, 50

Offset: 1

Amiram Eldar, May 09 2025

First differs from A322791 and A383761 at rows 16, 48, 80, 81, 112, 144, 162, ... and from A361255 at rows 256, 768, 1280, 1792, ... .

An exponential infinitary divisor d of a number n is a divisor d of n such that for every prime divisor p of n, the p-adic valuation of d is an infinitary divisor of the p-adic valuation of n.

			The first 10 rows are:
  1
  2
  3
  2, 4
  5
  6
  7
  2, 8
  3, 9
  10

Amiram Eldar, Table of n, a(n) for n = 1..15379 (first 10000 rows, flattened)
Andrew V. Lelechenko, Exponential and infinitary divisors, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; arXiv preprint, arXiv:1405.7597 [math.NT], 2014.

Cf. A307848 (row lengths), A361175 (row sums).

Cf. A077609, A322791, A361255, A383761.

infDivQ[n_, 1] = True; infDivQ[n_, d_] := n > 0 && d > 0 && BitAnd[IntegerExponent[n, First /@ (f = FactorInteger[d])], (e = Last /@ f)] == e;
expInfDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[infDivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expInfDivs[1] = {1};
expInfDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expInfDivQ[n, #] &]];
Table[expInfDivs[n], {n, 1, 70}] // Flatten

A383863 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a unitary divisor of the p-adic valuation of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 3, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, May 12 2025

First differs from A073184 at n = 64.
First differs from A383865 at n = 256.
The number of divisors d of n such that each is a unitary divisor of an exponential unitary divisor of n (see A361255).	
Analogous to the number of (1+e)-divisors (A049599) as exponential unitary divisors (A361255, A278908) are analogous to exponential divisors (A322791, A049419).
The sum of these divisors is A383864(n).
Also, the number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a squarefree divisor of the p-adic valuation of n. The sum of these divisors is A383867(n).

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

Cf. A001221, A034444, A049419, A049599, A073184, A209061, A278908, A322791, A361255, A383864, A383865, A383867.

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

a(n) = vecprod(apply(x -> 1 + 1 << omega(x), factor(n)[, 2]));

Multiplicative with a(p^e) = 1 + 2^A001221(e) = 1 + A034444(e).

a(n) <= A049599(n), with equality if and only if n is an exponentially squarefree number (A209061).

A383865 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 3, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, May 12 2025

First differs from A383863 at n = 256.
The number of divisors d of n such that each is a unitary divisor of an exponential infinitary divisor of n (see A383760).	
Analogous to the number of (1+e)-divisors (A049599) as exponential infinitary divisors (A383760, A307848) are analogous to exponential divisors (A322791, A049419).
The sum of these divisors is A383866(n).

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

Cf. A036537, A037445, A049419, A049599, A307848, A322791, A383760, A383863, A383866.

f[p_, e_] := 2^DigitCount[e, 2, 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; ff[p_, e_] := d[e] + 1; a[1] = 1; a[n_] := Times @@ ff @@@ FactorInteger[n]; Array[a, 100]

d(n) = vecprod(apply(x -> 2^hammingweight(x), factor(n)[, 2]));
a(n) = vecprod(apply(x -> 1 + d(x), factor(n)[, 2]));

Multiplicative with a(p^e) = 1 + A037445(e).

a(n) <= A049599(n), with equality if and only if all the exponents in the prime factorization of n are in A036537.

A066990 In canonical prime factorization of n replace even exponents with 2 and odd exponents with 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 6, 25, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 6, 55, 14, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69

Offset: 1

Reinhard Zumkeller, Feb 01 2002

a(n) = n for cubefree numbers (A004709), whereas a(n) <> n for cube-full numbers (A046099).

The largest exponential divisor (A322791) of n that is cubefree (A004709). - Amiram Eldar, Jun 03 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Brahim Mittou, New properties of an arithmetic function, Mathematica Montisnigri, Vol LIII (2022), pp. 5-11.
Eric Weisstein's World of Mathematics, Cubefree.

Cf. A004709, A046099, A322791.

a066990 n = product $ zipWith (^)
           (a027748_row n) (map ((2 -) . (`mod` 2)) $ a124010_row n)
-- Reinhard Zumkeller, Dec 02 2012

fx[{a_,b_}]:={a,If[EvenQ[b],2,1]}; Table[Times@@(#[[1]]^#[[2]]&/@(fx/@ FactorInteger[n])),{n,70}] (* Harvey P. Dale, Jan 01 2012 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2 - f[i,2]%2));} \\ Amiram Eldar, Oct 28 2022

from math import prod
from sympy import factorint
def a(n): return prod(p**(2-(e&1)) for p, e in factorint(n).items())
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Jun 03 2025

Multiplicative with a(p^e) = p^(2 - e mod 2), p prime, e>0.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/30) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 0.4296463408... . - Amiram Eldar, Oct 28 2022

A157488 a(1) = 1; for n > 1, a(n) = product of exponential divisors of n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 16, 27, 10, 11, 72, 13, 14, 15, 128, 17, 108, 19, 200, 21, 22, 23, 144, 125, 26, 81, 392, 29, 30, 31, 64, 33, 34, 35, 46656, 37, 38, 39, 400, 41, 42, 43, 968, 675, 46, 47, 3456, 343, 500, 51, 1352, 53, 324, 55, 784, 57, 58, 59, 1800, 61, 62, 1323, 4096

Offset: 1

Jaroslav Krizek, Mar 01 2009

nonn

The exponential divisors of a number n = Product p(i)^e(i) are all numbers of the form Product p(i)^s(i) where s(i) divides e(i) for all i.

Not multiplicative: a(3)=3 (e-divisor 3^1), a(4)=8 (e-divisors 2^1 and 2^2), but a(12)=72 (e-divisors 3*2 and 3*2^2) <> a(3)*a(4). - R. J. Mathar, Apr 14 2011

			For n = 16 = 2^4 = the a(16) = 2^(A000203(4)) = 2^7 = 128. e-divisors of number 16 is 2, 4, 16, their product is 128.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, A note on exponential divisors and related arithmetic functions, Scientia Magna, Vol. 1, No. 1 (2005), pp. 97-101.

Cf. A000005, A000027, A000040, A000203, A006881, A000961, A049419, A051377, A120944, A322791.

Similar sequences: A007955, A061537, A274029.

[ &*[ d: d in Divisors(n) | forall(t) { p: p in P | v gt 0 and e mod v eq 0 where v is Valuation(d, p) where e is Valuation(n, p) } where P is PrimeDivisors(n) ]: n in [2..64] ]; // Klaus Brockhaus, May 26 2009

f[p_, e_] := p^(DivisorSigma[1, e]/DivisorSigma[0, e]); a[n_] :=(Times @@ (f @@@ (fct = FactorInteger[n])))^(Times @@ DivisorSigma[0, Last /@ fct]); Array[a, 100] (* Amiram Eldar, Jun 03 2020 *)

a(1) = 1, a(p) = p, a(p*q) = p*q, a(p*q...*z) = pq...z, a(p^k) = p^(A000203(k)), for p, q, ..., z distinct primes and k > 1 an integer.
From Amiram Eldar, Jun 03 2020: (Start)
If n = Product_{i} p_i^e_i then a(n) = Product_{i} p_i^(sigma(e_i) * d_exp(n) / d(e_i)), where d_exp(n) = Product_{i} d(e_i) is the number of exponential divisors of n (A049419), d(e) and sigma(e) are the number of divisors (A000005) of e and their sum (A000203).
a(n) <= A007955(n) with equality if and only if n is noncomposite. (End)

a(1) = 1 from N. J. A. Sloane, Mar 02 2009

a(60) corrected by Klaus Brockhaus, May 26 2009

A348961 Exponential harmonic (or e-harmonic) numbers of type 1: numbers k such that esigma(k) | k * d_e(k), where d_e(k) is the number of exponential divisors of k (A049419) and esigma(k) is their sum (A051377).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105

Offset: 1

Amiram Eldar, Nov 05 2021

nonn

First differs from A005117 at n = 24, from A333634 and A348499 at n = 47, and from A336223 at n = 63.
Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 1, and that an e-perfect number (A054979) is a term of this sequence if and only if at least one of the exponents in its prime factorization is not a perfect square.
Since all the e-perfect numbers are products of a primitive e-perfect number (A054980) and a coprime squarefree number, and all the known primitive e-perfect numbers have a nonsquare exponent in their prime factorizations, there is no known e-perfect number that is not in this sequence.

			3 is a term since esigma(3) = 3, 3 * d_e(3) = 3 * 1, so esigma(3) | 3 * d_e(3).

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117 and A348962 are subsequences.

Cf. A049419, A051377, A322791, A333634, A336223, A348499, A348964.

f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^# &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], ehQ]

A368980 The number of exponential divisors of n that are squares (A000290).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Jan 11 2024

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

Cf. A000005, A000037, A000290, A049419, A183063, A322791, A327837, A368979.

Similar sequences: A046951, A056624, A358260, A368978.

f[p_, e_] := If[OddQ[e], 0, DivisorSigma[0, e/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, 0, numdiv(x/2)), factor(n)[, 2]));

a(n^2) = A049419(n). [corrected by Ridouane Oudra, Nov 19 2024]
Multiplicative with a(p^e) = A183063(e), or equivalently, a(p^e) = 0 if e is odd, and A000005(e/2) if e is even.
a(n) >= 0, with equality if and only if n is not a square number (A000037).
a(n) <= A049419(n), with equality if and only if n = 1.
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1.602317... (A327837).

cp's OEIS Frontend

A335218 Exponential Zumkeller numbers: numbers whose exponential divisors can be partitioned into two disjoint subsets of equal sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A340233 a(n) is the least number with exactly n exponential divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A379027 Irregular table read by rows in which the n-th row lists the modified exponential divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A383760 Irregular triangle read by rows in which the n-th row lists the exponential infinitary divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A383863 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a unitary divisor of the p-adic valuation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383865 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or an infinitary divisor of the p-adic valuation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A066990 In canonical prime factorization of n replace even exponents with 2 and odd exponents with 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A157488 a(1) = 1; for n > 1, a(n) = product of exponential divisors of n.

Views