A379027 -id:A379027 - cp's OEIS frontend

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).

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

A220218 Numbers where all exponents in its prime factorization are one less than a prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Charles R Greathouse IV, Dec 07 2012

nonn

Sequence has positive density, between 0.83 and 0.89; probably about 0.87951.
The numbers of terms not exceeding 10^k, for k=1,2,..., are 9, 90, 880, 8796, 87956, 879518, 8795126, 87951173, 879511794, ... The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + Sum_{q prime >= 5} (p-1)/p^q) = 0.87951176583716527413... - Amiram Eldar, Mar 20 2021
Numbers whose sets of unitary divisors (A077610) and modified exponential divisors (A379027) coincide. - Amiram Eldar, Dec 14 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Erdős and Leon Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc., Vol. s3-2, No. 1 (1952), pp. 257-271; alternative link.

Apart from the first term, a subsequence of A096432.

Cf. A010051, A124010, A077610, A379027.

a220218 n = a220218_list !! (n-1)
a220218_list = 1 : filter
               (all (== 1) . map (a010051' . (+ 1)) . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 30 2015

Select[Range[100],AllTrue[Transpose[FactorInteger[#]][[2]]+1,PrimeQ]&] (* Harvey P. Dale, Sep 29 2014 *)

is(n)=vecmin(apply(n->isprime(n+1),factor(max(n,2))[,2])) \\ Charles R Greathouse IV, Dec 07 2012

A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 120, 138, 150, 168, 174, 186, 210, 222, 246, 258, 270, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822

Offset: 1

Amiram Eldar, Dec 14 2024

All the squarefree abundant numbers (A087248) are terms since A241405(k) = A000203(k) for a squarefree number k.
If k is a term and m is coprime to k them k*m is also a term.
The numbers of terms that do no exceed 10^k, for k = 2, 3, ..., are 5, 67, 767, 7595, 76581, 764321, 7644328, 76468851, 764630276, ... . Apparently, the asymptotic density of this sequence exists and equals 0.07646... .

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

Cf. A000203, A241405, A323757, A323758, A323759, A379027.
Subsequence of A005101.
Subsequences: A034683, A087248, A379030, A379031.
Similar sequences: A064597, A129575, A129656, A292982, A348274, A348604.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; meAbQ[n_] := mesigma[n] > 2*n; Select[Range[1000], meAbQ]

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

A379028 The number of modified exponential divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 4, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

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 sum of these divisors is A241405(n).

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

Row lengths of A379027.
Cf. A241405.
Similar sequences: A000005 (all divisors), A049419 (exponential), A037445 (infinitary), A034444 (unitary), A286324 (bi-unitary).

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

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

Multiplicative with a(p^e) = d(e+1), where d(k) is the number of divisors of k (A000005).

cp's OEIS Frontend

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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A220218 Numbers where all exponents in its prime factorization are one less than a prime.

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A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

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A379028 The number of modified exponential divisors of n.

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