A049599 -id:A049599 - cp's OEIS frontend

A049603 (1+e)-sigma perfect numbers: (1+e) - sigma(x) = 2*x.

6, 28, 264, 1104, 3360, 75840, 6499584, 151062912, 2171581440, 4686409728, 316023611904

Offset: 1

The function (1+e) - sigma(n) is defined in A051378. This sequence lists solutions to A051378(n) = 2*n.

a(12) > 10^12. - Giovanni Resta, Jun 12 2016

			Factorizations: 2*3, 2^2*7, 2^3*3*11, 2^4*3*23, 2^5*3*5*7, 2^6*3*5*79, 2^8*3^2*7*13*31, 2^7*3^2*7*11*13*131, 2^10*3*5*7*19*1063.

Cf. A049599, A051378.

(* Assuming all terms greater than 28 are multiple of 24 *) ok[n_] := 2*n == Times @@ (1 + Sum[First[#]^s, {s, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Reap[For[n = 2, n <= 2171581440, n = n + If[n < 48, 1, 24], If[ok[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jun 26 2012 *)

151062912 inserted by Jean-François Alcover, Jun 26 2012

a(10)-a(11) from Giovanni Resta, Jun 12 2016

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).

A053784 Harmonic means of (1+e)-divisors of (1+e)-harmonic numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 9, 10, 11, 15, 15, 14, 8, 9, 17, 17, 12, 21, 19, 16, 14, 18, 29, 26, 29, 21, 20, 17, 24, 28, 22, 27, 39, 24, 30, 42, 23, 42, 48, 33, 26, 54, 41, 35, 37, 36, 34, 39, 31, 44, 40, 36, 38, 46, 51, 55, 77, 41, 60, 77, 54, 57, 88, 47, 43, 45, 46, 99

Offset: 1

Naohiro Nomoto, Apr 14 2001

nonn

If n = Product p(i)^r(i), d = Product p(i)^s(i) and s(i) = 0 or s(i) divides r(i), then d is a (1+e)-divisor of n.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A049599, A051378, A053783.

f[p_, e_] := (DivisorSigma[0, e] + 1)/(p^e + DivisorSum[e, p^(e - #) &]); h[n_] := n*Times @@ (f @@@ FactorInteger[n]); Select[h /@ Range[10^5], IntegerQ] (* Amiram Eldar, Sep 07 2019*)

More terms from Amiram Eldar, Sep 07 2019

A349281 a(n) is the number of prime powers (not including 1) that are (1+e)-divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 13 2021

(1+e)-divisors are defined in A049599.
First differs from A106490 at n = 64.
The total number of prime powers (not including 1) that divide n is A001222(n).
If p|n and p^e is the highest power of p that divides n, then the powers of p that are (1+e)-divisors of n are of the form p^d where d|e.

			8 has 3 (1+e)-divisors, 1, 2 and 8. Two of these divisors, 2 and 8 = 2^3 are prime powers. Therefore, a(8) = 2.

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

Cf. A000005, A000961, A001222, A046099, A049599, A077761, A106490, A246655, A349258.

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

A349281(n) = vecsum(apply(e->numdiv(e),factor(n)[,2])); \\ Antti Karttunen, Nov 13 2021

Additive with a(p^e) = A000005(e).
a(n) <= A001222(n), with equality if and only if n is cubefree (A046099).
a(n) <= A049599(n)-1, with equality if and only if n is a prime power (including 1, A000961).
Sum_{k=1..n} a(n) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.51780076119050171903..., where f(x) = -x + (1-x) * Sum_{k>=1} x^k/(1-x^k). - Amiram Eldar, Sep 29 2023

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.

A053783 (1+e)-harmonic numbers: harmonic mean of (1+e)-divisors is an integer.

Original entry on oeis.org

1, 6, 28, 140, 728, 1638, 2184, 3640, 8008, 8190, 10920, 18620, 23808, 23895, 27846, 37128, 47790, 55860, 69160, 148960, 166656, 189810, 237510, 242060, 316680, 334530, 359600, 406215, 446880, 484120, 525690, 669060, 726180, 1029952, 1078800, 1089270, 1099170

Offset: 1

Naohiro Nomoto, Apr 14 2001

nonn

If k = Product p(i)^r(i), d = Product p(i)^s(i) and s(i) = 0 or s(i) divides r(i), then d is a (1+e)-divisor of k.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A001599, A049599, A051378, A053784.

f[p_, e_] := (DivisorSigma[0, e] + 1)/(p^e + DivisorSum[e, p^(e - #) &]); aQ[n_] := IntegerQ[n * Times @@ (f @@@ FactorInteger[n])]; Select[Range[10^5], aQ] (* Amiram Eldar, Sep 07 2019 *)

More terms from Amiram Eldar, Sep 07 2019

A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

Original entry on oeis.org

6, 28, 264, 1104, 3360, 75840, 151062912, 606557952, 2171581440

Offset: 1

Amiram Eldar, Apr 10 2020

Since a recursive divisor is also a (1+e)-divisor (see A049599), then the first 6 terms and other terms of this sequence coincide with those of A049603.

			264 is a term since the sum of its recursive divisors is 1 + 2 + 3 + 6 + 8 + 11 + 22 + 24 + 33 + 66 + 88 + 264 = 528 = 2 * 264.

Cf. A049603, A282446, A333926.

Analogous sequences: A000396, A002827 (unitary), A007357 (infinitary), A054979 (exponential), A064591 (nonunitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], recDivSum[#] == 2*# &]

A349284 Numbers k such that A051378(k) > 2k and A333926(k) <= 2k.

Original entry on oeis.org

126720, 134400, 149760, 188160, 195840, 456960, 510720, 549120, 618240, 718080, 748800, 779520, 802560, 833280, 940800, 979200, 994560, 1094400, 1102080, 1155840, 1263360, 1324800, 1393920, 1424640, 1585920, 1639680, 1670400, 1785600, 1800960, 1908480, 1946880

Offset: 1

Amiram Eldar, Nov 13 2021

nonn

(1+e)-abundant numbers are numbers k such that A051378(k) > 2*k, i.e., numbers k whose sum of (1+e)-divisors exceeds 2*k.

Since all the recursive divisors (see A282446) of a number are also its (1+e)-divisors, the sequence of (1+e)-abundant numbers includes all the recursive abundant numbers (A333928). The first 21387 (1+e)-abundant numbers are also recursive abundant numbers. Therefore, this sequence includes only the (1+e)-abundant numbers that are not recursive abundant numbers.

			126720 is a term since A051378(126720) = 261144 > 2*126720 = 253440 and A333926(126720) = 246168 < 253440.

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

Cf. A049599, A049603, A051378, A282446, A333926, A333928.

oesigma[1] = 1; oesigma[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recsigma[1] = 1; recsigma[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^6], oesigma[#] > 2*# && recsigma[#] <= 2*# &]

A365551 The number of exponentially odd divisors of the smallest exponentially odd number divisible by n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 08 2023

First differs from A049599 and A282446 at n = 32, and from A353898 at n = 64.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms).

Cf. A322483, A356191.

Cf. A049599, A282446, A353898.

f[p_, e_] := Ceiling[(e + 3)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> ceil((x+3)/2), factor(n)[, 2]));

a(n) = A322483(A356191(n)).
Multiplicative with a(p^e) = ceiling((e+3)/2).
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * f(s).
Sum_{k=1..n} a(k) ~ (Pi^2 * f(1) * n / 6) * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288119...,
f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.452592479445159559037143959382854734148246511441192913672347667991...
and gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

A049603 (1+e)-sigma perfect numbers: (1+e) - sigma(x) = 2*x.

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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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A053784 Harmonic means of (1+e)-divisors of (1+e)-harmonic numbers.

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A349281 a(n) is the number of prime powers (not including 1) that are (1+e)-divisors of n.

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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.

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A053783 (1+e)-harmonic numbers: harmonic mean of (1+e)-divisors is an integer.

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A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

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A349284 Numbers k such that A051378(k) > 2*k and A333926(k) <= 2*k.

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A349284 Numbers k such that A051378(k) > 2k and A333926(k) <= 2k.