Andrew Lelechenko - 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

A189020 a(n) = Sum_{k=1..10^n} tau_4(k), where tau_4 is the number of ordered factorizations into 4 factors (A007426).

Original entry on oeis.org

1, 89, 3575, 93237, 1951526, 35270969, 578262093, 8840109380, 128217432396, 1784942188189, 24045237260214, 315312623543840, 4042957241191810, 50862246063060180, 629513636928477232, 7681900592647818929, 92587253467765253144, 1103781870246459696784, 13031388731053572679450, 152516435040764735691556, 1771079109308495896176156

Offset: 0

Andrew Lelechenko, Apr 15 2011

nonn

Using that tau_4 = tau_2 ** tau_2, where ** means Dirichlet convolution and tau_2 is (A000005), one can calculate a(n) faster than in O(10^n) operations - namely in O(10^(3n/4)) or even in O(10^(2n/3)). See links for details.

A. V. Lelechenko, The summation of the multiplicative functions (in Russian).

Cf. A057494 - partial sums up to 10^n of the divisors function tau_2 (A000005), A180361 - of the unitary divisors function tau_2* (A034444), A180365 - of the 3-divisors function tau_3 (A007425).

Also see A072692 for such sums of the sum of divisors function (A000203), A084237 for sums of Moebius function (A008683), A064018 for sums of Euler totient function (A000010).

a(n) = A061202(10^n) = Sum_{k=1..10^n} A007426(n).

a(16)-a(20) from Henri Lifchitz, Feb 05 2025

A180365 Partial sums up to 10^n of A007425.

Original entry on oeis.org

1, 53, 1471, 29425, 496623, 7518850, 106030594, 1421760251, 18362473634, 230375375227, 2824280446479, 33978264556380, 402439152166882, 4703975577244852, 54365786164534232, 622220063695532731, 7060841638683030355, 79525016215415440837

Offset: 0

Andrew Lelechenko, Jan 19 2011

nonn

Henri Lifchitz, Table of n, a(n) for n = 0..23 (first 21 terms from Hiroaki Yamanouchi)

Cf. A007425, A061201.

a(n)=sum(k=1, 10^n, numdiv(k)*floor((10^n)/k));
vector(7, n, a(n-1)) \\ Altug Alkan, Sep 24 2015

from math import isqrt
from sympy import integer_nthroot
def A180365(n): return (m:=integer_nthroot(t:=10**n,3)[0])**3+3*sum(-(s:=isqrt(r:=t//i))**2+(sum(r//k for k in range(1,s+1))<<1)-sum(t//(i*j) for j in range(1,m+1)) for i in range(1,m+1)) # Chai Wah Wu, Oct 23 2023

a(n) = A061201(10^n).

One more term (a(16)) and typo in crossrefs fixed by Andrew Lelechenko, Apr 13 2011

a(17) from Hiroaki Yamanouchi, Jul 15 2014

A180361 Sum of number of unitary divisors (A034444) from 1 to 10^n.

Original entry on oeis.org

1, 23, 359, 4987, 63869, 778581, 9185685, 105854997, 1198530315, 13385107495, 147849112851, 1618471517571, 17584519050293, 189843229312125, 2038412681323151, 21783930695524161, 231837345778656901

Offset: 0

Andrew Lelechenko, Jan 19 2011

nonn

Henri Lifchitz, Table of n, a(n) for n = 0..23

Cf. A034444, A064608

a(n)=A064608(10^n)

a(3) corrected from 4983 to 4987 by Henri Lifchitz, Nov 07 2017

a(17)-a(23) from Henri Lifchitz, Nov 07 2017

A164843 The smallest magic constant of an n X n magic square with distinct prime entries.

Original entry on oeis.org

177, 120, 233, 432, 733, 1154, 1731, 2470, 3417, 4584, 6013, 7712, 9731, 12088, 14807, 17940, 21501, 25530, 30021, 35086, 40675, 46840, 53631, 61092, 69251, 78100, 87697, 98084, 109309, 121380, 134377, 148258, 163043

Offset: 3

Andrew Lelechenko, Aug 28 2009 and Natalia Makarova, Sep 08 2009

a(n) >= m(n), where m(n) is the smallest integer of the same parity as n, which is >= (Sum_{k=1..n^2} prime(k+1))/n. For example, Sum_{k=1..5^2} prime(k+1)/5=231.8, so m(5)=233. Conjecture: for n > 4, a(n)=m(n) or a(n)=m(n)+2.

			From _Natalia Makarova_, Sep 26 2009: (Start)
Here is a 14 X 14 example:
  [  3   43   59  131  181  271  383  599  797  919  971 1039 1123 1193
  1151  433  967  211  337  491  397  691   83  523  593  773  449  613
   263  373  101 1063  877  617  419  911  787  241  151  839  739  331
   503  439  809 1051 1091  659  157 1031   71  139  379  179  743  461
   173  647 1069  389 1049   19  311  223  317 1103  283  947  499  683
   547   13 1061  353  229  853  677  751  571  983 1201   29  193  251
   643  269  887  733   23  409 1129  191  769  401   47 1109  149  953
   163  881  673  107  431  487  991  631  829  109  349  367  811  883
  1163  827  607 1171  443  653  463    5  457  577   31  293  601  421
   509 1097  313  757  167  709  761  347  857  137  619  233   89 1117
  1093 1019    7  521 1033   61   73  941 1009  859  701   11  127  257
    53  467   97  307 1153  557 1021  569  359  937  821  113  977  281
   907   17  823  641  661  929   67  719   79  587  479  563 1013  227
   541 1187  239  277   37  997  863  103  727  197 1087 1217  199   41 ]
(End)
Comment from _N. J. A. Sloane_, Sep 28 2009: this contains 192 consecutive primes, 3 to 1171, plus 1187, 1193, 1201, 1217.
For the 3 X 3 case see A024351. For the 4 X 4 magic square see the Mathworld link.

Harvey Heinz, Prime magic squares
A. Lelechenko and N. Makarova, Examples of prime magic n X n squares with minimal magic constant for n=5..13.
N. Makarova, Smallest prime magic squares, Part I (in Russian)
N. Makarova, Smallest prime magic squares, Part II (in Russian)
Mathworld, Prime magic squares
PlanetMath, Prime magic squares
Stefano Tognon, Prime Analysis

Cf. A073502, A073350, A125007.

Partially reworded by R. J. Mathar, Aug 31 2009
Edited by N. J. A. Sloane, Sep 14 2009
a(11)-a(15) from Natalia Makarova, a(16)-a(35) from Natalia Makarova and Stefano Tognon
Edited by Max Alekseyev, Feb 11 2010

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User: Andrew Lelechenko

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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A189020 a(n) = Sum_{k=1..10^n} tau_4(k), where tau_4 is the number of ordered factorizations into 4 factors (A007426).

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A180365 Partial sums up to 10^n of A007425.

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A180361 Sum of number of unitary divisors (A034444) from 1 to 10^n.

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A164843 The smallest magic constant of an n X n magic square with distinct prime entries.

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