A051378 -id:A051378 - cp's OEIS frontend

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

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*# &]

A349283 Numbers k such that A051378(k) = A051378(k+1).

Original entry on oeis.org

14, 206, 957, 1334, 1364, 1485, 1634, 2685, 2974, 4136, 4364, 14841, 20145, 24957, 33998, 36566, 42818, 61183, 64672, 74918, 79826, 79833, 84134, 86343, 92685, 104192, 109864, 111506, 122073, 138237, 147454, 159711, 162602, 166934, 187863, 190773, 193893, 201597

Offset: 1

Amiram Eldar, Nov 13 2021

nonn

First differs from A333949 at n = 18.

			14 is a term since A051378(14) = A051378(15) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A051378.

Similar sequences: A002961, A064115, A064125, A293183, A306985, A333949.

s[1] = 1; s[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[2*10^5], s[#] == s[#+1] &]

A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 22, 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, 66, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67, 102, 69, 70, 71

Offset: 1

Yasutoshi Kohmoto

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.

a(n) = n if and only if n is squarefree. - Jon Perry, Nov 13 2012

			a(8)=10 because 2 and 2^3 are e-divisors of 8 and 2+2^3=10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.
Nicussor Minculete, Concerning some arithmetic functions which use exponential divisors, Acta Universitatis Apulensis, No. 28/2011, pp. 125-133.
Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar. 77 (1997), 159-175.
Eric Weisstein's World of Mathematics, e-Divisor

Cf. A051378, A049419 (number of e-divisors).
Cf. A027748, A124010, A027750.
Row sums of A322791.
See A307042 and A275480 where the formula and constant appear.

A051377:=n->Product(List(Collected(Factors(n)), p -> Sum(DivisorsInt(p[2]),d->p[1]^d))); List([1..10^4], n -> A051377(n)); # Muniru A Asiru, Oct 29 2017

a051377 n = product $ zipWith sum_e (a027748_row n) (a124010_row n) where
   sum_e p e = sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012

A051377 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        add(p^d,d=numtheory[divisors](e)) ;
        a := a*% ;
    end do:
    a ;
end proc:
seq(A051377(n),n=1..100) ; # R. J. Mathar, Oct 05 2017

a[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Apr 06 2012 *)

a(n)=my(f=factor(n));prod(i=1,#f[,1],sumdiv(f[i,2],d,f[i,1]^d)) \\ Charles R Greathouse IV, Nov 22 2011

ediv(n,f=factor(n))=my(v=List(),D=apply(divisors,f[,2]~),t=#f~); forvec(u=vector(t,i,[1,#D[i]]), listput(v,prod(j=1,t,f[j,1]^D[j][u[j]]))); Set(v)
a(n)=vecsum(ediv(n)) \\ Charles R Greathouse IV, Oct 29 2018

Multiplicative with a(p^e) = Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002
a(n) = A126164(n)+n. - R. J. Mathar, Oct 05 2017
The average order of a(n) is Dn + O(n^e) for any e > 0, due to Fabrykowski & Subbarao, where D is about 0.568. (D >= 0.5 since a(n) >= n.) - Charles R Greathouse IV, Sep 22 2023

More terms from Jud McCranie, May 29 2000

Definition corrected by Jaroslav Krizek, Feb 27 2009

A049599 Number of (1+e)-divisors of n: 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.

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

Offset: 1

Yasutoshi Kohmoto

A divisor of n is a (1+e)-divisor if and only if it is a unitary divisor of an exponential divisor of n (see A077610 and A322791). - Amiram Eldar, Feb 26 2024

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

Cf. A049603, A051378.

Cf. A124010, A000005, A049419, A077610, A322791.

a049599 = product . map ((+ 1) . a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

a[n_] := Times @@ (DivisorSigma[0, #] + 1 &)  /@ FactorInteger[n][[All, 2]]; a[1] = 1; Table[a[n], {n, 1, 103}] (* Jean-François Alcover, Oct 10 2011 *)

a(n) = vecprod(apply(x->numdiv(x)+1, factor(n)[, 2])); \\ Amiram Eldar, Aug 13 2023

If n = Product p(i)^r(i) then a(n) = Product (tau(r(i))+1), where tau(n) = number of divisors of n, cf. A000005. - Vladeta Jovovic, Apr 29 2001

More terms from Naohiro Nomoto, Apr 12 2001

A241405 Sum of modified exponential divisors: if n = Product p_i^r_i then me-sigma(x) = Product (sum p_i^s_i such that s_i+1 divides r_i+1).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 11, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 44, 26, 42, 31, 40, 30, 72, 32, 39, 48, 54, 48, 50, 38, 60, 56, 66, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 93, 72, 88, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68, 90, 96, 144, 72, 110, 74, 114, 104, 100, 96, 168, 80

Offset: 1

Andrew Lelechenko, May 06 2014

The modified exponential divisors of a number n = product p_i^r_i are all numbers of the form product p_i^s_i such that s_i+1 divides r_i+1 for each i.

The concept of modified exponential divisors simplifies combinatorial problems on the sum of exponential divisors A051377 such as a search of e-perfect numbers. Each primitive e-perfect number A054980 corresponds to a unique me-perfect number of smaller magnitude.

Antti Karttunen, Table of n, a(n) for n = 1..16384
David Moews, Perfect, amicable and sociable numbers.
Index entries for sequences related to sums of divisors.

Cf. A007947, A049419, A051377, A054980, A051378, A379027, A379028.

f[p_, e_] := DivisorSum[e+1, p^(#-1)&]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 03 2023 *)

A241405(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)))}

a(n / A007947(n)) = A051377(n).

Multiplicative with a(p^a) = sum p^b such that b+1 divides a+1.

More terms from Antti Karttunen, Nov 23 2017

Incorrect comment removed by Amiram Eldar, Dec 14 2024

A333926 The sum of recursive divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 23, 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, 92, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset: 1

Amiram Eldar, Apr 10 2020

The definition of recursive divisors and the number of recursive divisors of n are in A282446.

First differs from A051378 at n = 256.

			The recursive divisors of 8 are 1, 2 and 8, therefore a(8) = 1 + 2 + 8 = 11.

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

Cf. A000203, A051378, A282446, A287957, A287958.

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]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100]

Multiplicative with a(p^k) = 1 + Sum_{d recursive divisor of k} p^d.

a(n) <= A051378(n) <= A000203(n).

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

Original entry on oeis.org

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

Offset: 1

Yasutoshi Kohmoto

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

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

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

A069915 Sum of (1+phi)-divisors of n (cf. A061389).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 7, 4, 18, 12, 12, 14, 24, 24, 11, 18, 12, 20, 18, 32, 36, 24, 28, 6, 42, 13, 24, 30, 72, 32, 31, 48, 54, 48, 12, 38, 60, 56, 42, 42, 96, 44, 36, 24, 72, 48, 44, 8, 18, 72, 42, 54, 39, 72, 56, 80, 90, 60, 72, 62, 96, 32, 35, 84, 144, 68, 54, 96, 144, 72

Offset: 1

Vladeta Jovovic, Apr 23 2002

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

Cf. A061389.

Cf. A027748, A124010, A038566, A051378.

a069915 n = product $ zipWith sum_1phi (a027748_row n) (a124010_row n)
   where sum_1phi p e = 1 + sum [p ^ k | k <- a038566_row e]
-- Reinhard Zumkeller, Mar 13 2012

a[1] = 1; a[p_?PrimeQ] = p + 1; a[n_] := Times @@ (1 + Sum[If[GCD[k, Last[#]] == 1, First[#]^k, 0], {k, 1, Last[#]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 04 2012 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sum(k = 1, f[i, 2], (gcd(k, f[i, 2]) == 1) * f[i, 1]^k));} \\ Amiram Eldar, Aug 15 2023

Multiplicative with a(p^e) = 1+Sum_{k=1..e, gcd(k, e)=1} p^k.

cp's OEIS Frontend

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

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A349283 Numbers k such that A051378(k) = A051378(k+1).

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A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

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A049599 Number of (1+e)-divisors of n: 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.

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A241405 Sum of modified exponential divisors: if n = Product p_i^r_i then me-sigma(x) = Product (sum p_i^s_i such that s_i+1 divides r_i+1).

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A333926 The sum of recursive divisors of n.

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

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