A361255 -id:A361255 - cp's OEIS frontend

A322857 a(1) = 1; a(n) = sum of exponential unitary divisors of n for n > 1.

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 18, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 54, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67

Offset: 1

Amiram Eldar, Dec 29 2018

The exponential unitary (or e-unitary) divisors of n = Product p(i)^a(i) are all the numbers of the form Product p(i)^b(i) where b(i) is a unitary divisor of a(i).

Ivan Neretin, Table of n, a(n) for n = 1..10000
Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.

Cf. A361255, A051377, A077610, A278908 (number of exponential unitary divisors).

f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ f @@@ FactorInteger[n]; Array[eusigma, 100]

ff(p, e) = sumdiv(e, d, if (gcd(d, e/d)==1, p^d));
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = ff(f[k,1], f[k,2]); f[k,2] = 1); factorback(f); \\ Michel Marcus, Dec 29 2018

Multiplicative with a(p^e) = Sum_{d|e, gcd(d, e/d)==1} p^d.

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; }

A383693 Exponential unitary abundant numbers: numbers k such that A322857(k) > 2*k.

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 6300, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 36900, 38700, 40572, 42300, 42588, 44100, 47700, 47916, 49284, 49500

Offset: 1

Amiram Eldar, May 05 2025

First differs from its subsequence A383697 at n = 21.
All the terms are nonsquarefree numbers (A013929), since A322857(k) = k if k is a squarefree number (A005117).
If an exponential abundant number (A129575) is exponentially squarefree (A209061), then it is in this sequence. Terms of this sequence that are not exponentially squarefree are a(21) = 22500, a(77) = 86436, a(140) = 157500, etc..
The least odd term is a(202273) = 225450225, and the least term that is coprime to 6 is a(1.002..*10^18) = 1117347505588495206025.
The asymptotic density of this sequence is Sum_{n>=1} f(A383694(n)) = 0.00089722..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).

			900 is a term since A322857(900) = 2160 > 2*900 = 1800.

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

Cf. A005117, A209061, A322857, A322858, A361255.
Subsequence of A013929 and A129575.
Subsequences: A383694, A383697, A383698.

f[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Times @@ f @@@ FactorInteger[n] > 2 n; Select[Range[50000], q]

fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(k) = {my(f = factor(k)); prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}

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).

A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential 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, 4, 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, 12, 7, 49, 10, 50

Offset: 1

Amiram Eldar, May 09 2025

Differs from A322791, A361255 and A383760 at rows 16, 48, 80, 81, 112, 144, 162, ... .

An exponential squarefree exponential divisor (or e-squarefree e-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 a squarefree 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..15331 (first 10000 rows, flattened)
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Cf. A278908 (row lengths), A361174 (row sums).

Cf. A322791, A361255, A383760.

sqfDivQ[n_, d_] := SquareFreeQ[d] && Divisible[n, d];
expSqfDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[sqfDivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expSqfDivs[1] = {1};
expSqfDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expSqfDivQ[n, #] &]];
Table[expSqfDivs[n], {n, 1, 70}] // Flatten

A383864 The sum 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, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 19, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 76, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset: 1

Amiram Eldar, May 12 2025

First differs from A383866 at n = 256.
The sum of divisors d of n such that each is a unitary divisor of an exponential unitary divisor of n (see A361255).
Analogous to the sum of (1+e)-divisors (A051378) as exponential unitary divisors (A361255, A322857) are analogous to exponential divisors (A322791, A051377).
The number of these divisors is A383863(n).

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

Cf. A051377, A209061, A322791, A322857, A361255, A383863, A383866.

f[p_, e_] := 1 + DivisorSum[e, p^# &, CoprimeQ[#, e/#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sumdiv(f[i, 2], d, if(gcd(d, f[i, 2]/d) == 1, f[i, 1]^d)));}

Multiplicative with a(p^e) = 1 + Sum_{d|e, gcd(d, e/d) = 1} p^d.
a(n) <= A051378(n), with equality if and only if n is an exponentially squarefree number (A209061).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.52168352620962354041..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d|k, gcd(d, k/d)=1} x^(2*k-d))).

A383959 The number of prime powers p^e having the property that e is a unitary divisor of the p-adic valuation of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 16 2025

First differs from A238949 at n = 64.	
First differs from A383960 at n = 256.		
Also, the number of prime powers p^e having the property that e is a squarefree divisor of the p-adic valuation of n.

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

Cf. A001221, A034444, A077761, A085548, A238949, A278908, A361255, A383863, A383960.

f[p_, e_] := 2^PrimeNu[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Additive with a(p^e) = A034444(e) = 2^A001221(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A034444(e)/p^e = 0.92341081050532387352... .

cp's OEIS Frontend

A322857 a(1) = 1; a(n) = sum of exponential unitary divisors of n for n > 1.

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

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A383693 Exponential unitary abundant numbers: numbers k such that A322857(k) > 2*k.

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

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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.

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A383761 Irregular triangle read by rows in which the n-th row lists the exponential squarefree exponential divisors of n.

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A383864 The sum 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.

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A383959 The number of prime powers p^e having the property that e is a unitary divisor of the p-adic valuation of n.

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