A051377 -id:A051377 - cp's OEIS frontend

A322791 Irregular triangle read by rows in which the n-th row lists the exponential divisors (or e-divisors) of n.

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 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, 12, 48, 7, 49

Offset: 1

Amiram Eldar, Dec 26 2018

			The table starts
  1
  2
  3
  2, 4
  5
  6
  7
  2, 8
  3, 9
  10

Xiaodong Cao and Wenguang Zhai, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13, No. 2 (2010), Article 10.3.7.
E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Mathematical Journal, Vol. 41, No. 2 (1974), pp. 465-471, alternative link.
Eric Weisstein's World of Mathematics, e-Divisor

Cf. A049419 (row lengths), A051377 (row sums).

Cf. A027750 (all divisors), A077609 (infinitary), A077610 (unitary), A222266 (bi-unitary).

A322791 := proc(n)
    local expundivs ,d,isue,p,ai,bi;
    expudvs := {} ;
    for d in numtheory[divisors](n) do
        isue := true ;
        for p in numtheory[factorset](n) do
            ai := padic[ordp](n,p) ;
            bi := padic[ordp](d,p) ;
            if bi > 0 then
                if modp(ai,bi) <>0 then
                    isue := false;
                end if;
            else
                isue := false ;
            end if;
        end do;
        if isue then
            expudvs := expudvs union {d} ;
        end if;
    end do:
    sort(expudvs) ;
end proc:
seq(op(A322791(n)),n=1..40) ; # R. J. Mathar, Mar 06 2023

divQ[n_, m_] := (n > 0 && m>0 && Divisible[n, m]); expDivQ[n_, d_] := Module[ {f=FactorInteger[n]}, And@@MapThread[divQ, {f[[;; , 2]], IntegerExponent[ d, f[[;; , 1]]]} ]]; expDivs[1]={1}; expDivs[n_] := Module[ {d=Rest[Divisors[n]]}, Select[ d, expDivQ[n, #]&] ]; Table[expDivs[n], {n, 1, 50}] // Flatten

isexpdiv(f, d) = { my(e); for (i=1, #f~, e = valuation(d, f[i, 1]); if(!e || (e && f[i, 2] % e), return(0))); 1; }
row(n) = {my(d = divisors(n), f = factor(n), ediv = []); if(n == 1, return([1])); for(i=2, #d, if(isexpdiv(f, d[i]), ediv = concat(ediv, d[i]))); ediv; } \\ Amiram Eldar, Mar 27 2023

A051378 Sum of (1+e)-divisors of n. Let n = Product_i p(i)^r(i) then (1+e)-sigma(n) = Product_i (1 + Sum_{s|r(i)} p(i)^s).

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, 68, 126, 96

Offset: 1

Yasutoshi Kohmoto

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

Cf. A051377, A049599.

Cf. A027748, A124010, A027750, A069915, A107758, A107759.

a051378 n = product $ zipWith sum_1e (a027748_row n) (a124010_row n)
   where sum_1e p e = 1 + sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012

A051378 := proc(n)
    local a,d,p,e,sp;
    a := 1;
    for d in ifactors(n)[2] do
        p := op(1,d) ;
        e := op(2,d) ;
        sp := 1;
        for s in numtheory[divisors](e) do
            sp := sp+p^s ;
        end do:
        a := a*sp ;
    end do:
    a;
end proc: # R. J. Mathar, Oct 26 2015

a[1] = 1; a[p_?PrimeQ] = p+1; a[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, May 04 2012 *)

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

Multiplicative with a(p^e) = 1 + Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002
a(n) = Sum_{d|n, gcd(d, n/d) = 1} A051377(d). - Daniel Suteu, Nov 01 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 + (1-1/p)*Sum_{k>=2} p^k/(p^(2*k)-1)) = 0.76636964336546210751... . - Amiram Eldar, Oct 31 2023

Corrected and extended by Naohiro Nomoto, Apr 12 2001

A054980 Primitive e-perfect numbers: primitive elements of the e-perfect numbers (A054979).

Original entry on oeis.org

36, 1800, 2700, 17424, 1306800, 4769856, 238492800, 357739200, 54531590400

Offset: 1

Jud McCranie, May 29 2000

The nonprimitive e-perfect numbers are obtained from the primitive ones by multiplying by m, if m is squarefree and relatively prime to the primitive e-perfect number.
a(10) > 10^15. - Donovan Johnson, Nov 22 2011
The following numbers also belong to this sequence; however, their actual positions are unknown: 168136940595306022660197936246988800, 11712310558743727210993873194516480000, 1307484087615221689700651798824550400000. - Andrew Lelechenko, Apr 01 2014
The number of terms with a given number of distinct prime divisors is finite (Straus and Subbarao, 1974). - Amiram Eldar, Mar 04 2021

			180 = 36*5 (nonprimitive). 252 = 36*7 (nonprimitive). 1260 = 36*5*7 (nonprimitive). 1800 = 36*5^2 (primitive, 5^2 not squarefree and coprime to 36).

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B17, pp. 110-111.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 116-117.

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.
Jan Munch Pedersen, Exponential Perfect Numbers.
E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Math. J., Vol. 41 (1974), pp. 465-471
Eric Weisstein's World of Mathematics, e-Perfect Number.

Cf. A051377, A054979, A160134 (complement).

eperfect(n)=my(f=factor(n));prod(i=1,#f[,1],sumdiv(f[i,2],d, f[i,1]^d))==2*n
is(n)=if(!eperfect(n),0,my(f=factor(n));for(i=1,#f[,1],if(f[i,2]==1&&eperfect(n/f[i,1]),return(0)));1) \\ Charles R Greathouse IV, Nov 22 2011

a(9) from Donovan Johnson, Nov 22 2011

A126164 Sum of the proper exponential divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 6, 0, 0, 0, 6, 0, 6, 0, 10, 0, 0, 0, 6, 5, 0, 3, 14, 0, 0, 0, 2, 0, 0, 0, 36, 0, 0, 0, 10, 0, 0, 0, 22, 15, 0, 0, 18, 7, 10, 0, 26, 0, 6, 0, 14, 0, 0, 0, 30, 0, 0, 21, 14, 0, 0, 0, 34, 0, 0, 0, 48, 0, 0, 15

Offset: 1

Ant King, Dec 21 2006

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.

			The exponential divisors of 240 are 30, 60 and 240, so a(240) = 30+60 = 90.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ant King, Mathematica programs for A126164 - A126166.
Eric Weisstein's World of Mathematics, e-Divisor.

Cf. A051377, A049419, A054979, A054980, A126168.

f[p_, e_] := DivisorSum[e, p^# &]; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

A051377(n) = { my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)); }; \\ This function from Charles R Greathouse IV, Nov 22 2011
A126164(n) = (A051377(n) - n); \\ Antti Karttunen, Oct 04 2017, after the given formula

a(n) = esigma(n) - n = A051377(n) - n.

A348271 a(n) is the sum of noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 0, 5, 0, 0, 16, 0, 0, 0, 12, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 24, 42, 0, 0, 0, 36, 0, 0, 0, 45, 0, 0, 20, 40, 0, 0, 0, 84, 39

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

			a(12) = 8 since 12 has 2 noninfinitary divisors, 2 and 6, and 2 + 6 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A000203, A036537, A049417, A077609, A162643, A327633.

Similar sequences: A034448, A048146, A051377, A188999.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; a[n_]:= DivisorSigma[1,n] - isigma[n]; Array[a, 100]

a(n) = A000203(n) - A049417(n).
a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).
a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).

A126166 Larger member of each exponential amicable pair.

Original entry on oeis.org

100548, 502740, 968436, 1106028, 1307124, 1709316, 2312604, 2915892, 3116988, 3720276, 4122468, 4323564, 4725756, 5027400, 4842180, 5329044, 5530140, 5932332, 6133428, 6535620, 6736716, 7138908, 7340004, 7943292, 8345484, 8546580, 8948772, 9753156, 10155348

Offset: 1

Ant King, Dec 21 2006

nonn

This sequence includes the largest member of all exponential amicable pairs and does not discriminate between primitive and nonprimitive pairs.

			a(3)= 968436 because (937692,968436) is the third exponential amicable pair

Hagis, Peter Jr.; Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr., Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-349.
Ant King, Mathematica programs for A126164 - A126166
David Moews, Perfect, amicable and sociable numbers.
J. M. Pedersen, Known exponential amicable pairs.

Cf. A126165, A051377, A054979, A049419, A054980, A126164.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; s = {}; Do[m = esigma[n] - n; If[m > n && esigma[m] - m == n, AppendTo[s, m]], {n, 1, 10^7}]; s (* Amiram Eldar, May 09 2019 *)

The values of n for which esigma(m)=esigma(n)=m+n and mA051377

Link corrected and reference added by Andrew Lelechenko, Dec 04 2011

More terms from Amiram Eldar, May 09 2019

A160135 Sum of non-exponential divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 5, 1, 8, 1, 10, 1, 10, 9, 9, 1, 15, 1, 12, 11, 14, 1, 30, 1, 16, 10, 14, 1, 42, 1, 29, 15, 20, 13, 19, 1, 22, 17, 40, 1, 54, 1, 18, 18, 26, 1, 58, 1, 33, 21, 20, 1, 60, 17, 50, 23, 32, 1, 78, 1, 34, 20, 49, 19, 78, 1, 24, 27, 74, 1, 75, 1, 40, 34, 26, 19, 90, 1, 76, 28

Offset: 1

Jaroslav Krizek, May 02 2009

nonn

The non-exponential divisors d|n of a number n = p(i)^e(i) are divisors d not of the form p(i)^s(i), s(i)|e(i) for all i.

			a(8) = A000203(8) - A051377(8) = 15 - 10 = 5. a(8) = a(2^3) = (2^4-1)/(2-1) - (2^1+2^3) = 5.

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

Cf. A000203, A051377, A000040, A006881, A120944, A000961, A053810.

lpowp := proc(n,p) local e; for e from 0 do if n mod p^(e+1) <> 0 then RETURN(e) ; fi; od: end:
expdvs := proc(n) local a,d,nfcts,b,f,iseDiv ; a := {} ; nfcts := ifactors(n)[2] ; for d in ( numtheory[divisors](n) minus {1} ) do iseDiv := true; for f in nfcts do b := lpowp(d,op(1,f) ) ; if b = 0 or op(2,f) mod b <> 0 then iseDiv := false; fi; od: if iseDiv then a := a union {d} ; fi; od: a ; end proc:
A051377 := proc(n) local k ; add( k, k = expdvs(n)) ; end: A160135 := proc(n) if n = 1 then 1; else numtheory[sigma][1](n)-A051377(n) ; fi; end: seq(A160135(n),n=1..120) ; # R. J. Mathar, May 08 2009

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; a[1] = 1; a[n_] := DivisorSigma[1, n] - esigma[n]; Array[a, 100] (* Amiram Eldar, Oct 26 2021 after Jean-François Alcover at A051377 *)

A051377(n) = { my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d)); }; \\ From A051377
A160135(n) = if(1==n,n, sigma(n) - A051377(n)); \\ Antti Karttunen, Mar 04 2018

a(n) = A000203(n) - A051377(n) for n >= 2.
a(1) = 1, a(p) = 1, a(p*q) = 1 + p + q, a(p*q*...*z) = (p + 1)*(q + 1)*...*(z + 1) - p*q*...*z, for p, q,..,z = primes (A000040), p*q = product of two distinct primes (A006881), p*q*...*z = product of k (k > 0) distinct primes (A120944).
a(p^k) = (p^(k+1)-1)/(p-1)- Sum_{d|k} p^d for p = primes (A000040), p^k = prime powers A000961(n>1), k = natural numbers (A000027)>
a(p^q) = 1+(p^1-p^1)+p^2+p^3+...+p^(q-1), for p, q = primes (A000040), p^q = prime powers of primes (A053810).

Edited by R. J. Mathar, May 08 2009

A126165 Smaller member of each exponential amicable pair.

Original entry on oeis.org

90972, 454860, 937692, 1000692, 1182636, 1546524, 2092356, 2638188, 2820132, 3365964, 3729852, 3911796, 4275684, 4548600, 4688460, 4821516, 5003460, 5367348, 5549292, 5913180, 6095124, 6459012, 6640956, 7186788, 7550676, 7732620, 8096508, 8824284, 9188172, 9370116

Offset: 1

Ant King, Dec 21 2006

nonn

This sequence includes the smallest member of all exponential amicable pairs and does not discriminate between primitive and nonprimitive pairs.

			a(3)=937692 because (937692,968436) is the third exponential amicable pair

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr., Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-349.
Ant King, Mathematica programs for A126164 - A126166
David Moews, Perfect, amicable and sociable numbers.
J. M. Pedersen, Known exponential amicable pairs.

Cf. A051377, A054979, A049419, A054980, A126164.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; s = {}; Do[m = esigma[n] - n; If[m > n && esigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^7}]; s (* Amiram Eldar, May 09 2019 *)

The values of m for which esigma(m)=esigma(n)=m+n and mA051377.

More terms from Amiram Eldar, May 09 2019

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

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A322791 Irregular triangle read by rows in which the n-th row lists the exponential divisors (or e-divisors) of n.

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A051378 Sum of (1+e)-divisors of n. Let n = Product_i p(i)^r(i) then (1+e)-sigma(n) = Product_i (1 + Sum_{s|r(i)} p(i)^s).

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A054980 Primitive e-perfect numbers: primitive elements of the e-perfect numbers (A054979).

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A126164 Sum of the proper exponential divisors of n.

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A348271 a(n) is the sum of noninfinitary divisors of n.

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A126166 Larger member of each exponential amicable pair.

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A160135 Sum of non-exponential divisors of n.

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