A062319 -id:A062319 - cp's OEIS frontend

A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 135, 187, 256, 346, 463, 613, 803, 1040, 1336, 1703, 2158, 2720, 3409, 4244, 5251, 6461, 7911, 9643, 11707, 14157, 17058, 20480, 24502, 29212, 34707, 41094, 48496, 57053, 66926, 78296, 91369, 106376, 123581, 143276, 165786

Offset: 1

Gus Wiseman, May 07 2021

nonn

			The a(8) = 45 chains:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)  (1/1/1/1/1/1)
      (7)  (2/1)  (5/1/1)  (2/1/1/1)  (3/1/1/1/1)  (2/1/1/1/1/1)
           (2/2)  (5/5/1)  (2/2/1/1)  (3/3/1/1/1)  (2/2/1/1/1/1)
           (3/1)  (5/5/5)  (2/2/2/1)  (3/3/3/1/1)  (2/2/2/1/1/1)
           (3/3)           (2/2/2/2)  (3/3/3/3/1)  (2/2/2/2/1/1)
           (6/1)           (4/1/1/1)  (3/3/3/3/3)  (2/2/2/2/2/1)
           (6/2)           (4/2/1/1)               (2/2/2/2/2/2)
           (6/3)           (4/2/2/1)
           (6/6)           (4/2/2/2)
                           (4/4/1/1)
                           (4/4/2/1)           (1/1/1/1/1/1/1)
                           (4/4/2/2)
                           (4/4/4/1)
                           (4/4/4/2)
                           (4/4/4/4)

Antidiagonal sums of the array (or row sums of the triangle) A334997.
A000005 counts divisors of n.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A146291 counts divisors of n with k prime factors (with multiplicity).
A251683 counts strict length k + 1 chains of divisors from n to 1.
A253249 counts nonempty chains of divisors of n.
A334996 counts strict length k chains of divisors from n to 1.
A337255 counts strict length k chains of divisors starting with n.
Array version of A334997 has:
- column k = 2 A007425,
- transpose A077592,
- subdiagonal n = k + 1 A163767,
- strict case A343662 (row sums: A337256),
- version counting all multisets of divisors (not just chains) A343658,
- diagonal n = k A343939.
Cf. A018892, A062319, A066959, A143773, A176029, A327527, A343656.

Total/@Table[Length[Select[Tuples[Divisors[n-k],k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,0,n-1}]

A275217 Even numbers n such that A000005(n) divides A000005(n^n).

Original entry on oeis.org

4, 16, 64, 100, 196, 484, 676, 1024, 1156, 1296, 1444, 1936, 2116, 3364, 3844, 4096, 4900, 5476, 5776, 6400, 6724, 7396, 8836, 10816, 11236, 12100, 13456, 13924, 14884, 15376, 16900, 17956, 20164, 21316, 23716, 24964, 26896, 27556, 28900, 31684, 33124, 36100

Offset: 1

Altug Alkan, Jul 20 2016

nonn

This sequence is not the duplicate of A275123. See also comments section of A275123.
An even number n with prime factorization Product_i p_i^(e_i) is in this sequence iff Product_i (n*e_i+1)/(e_i+1) is an integer.
This sequence is infinite since A002110(n)^2 / 9 is always a term of this sequence for n > 1.

			4 is a term because 4 = 2^2 and (4*2+1) mod (2+1) = 0.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A062319, A275123.

is(n,f=factor(n))=f=f[,2]; n%2==0 && denominator(prod(i=1,#f,(f[i]*n+1)/(f[i]+1)))==1 \\ Charles R Greathouse IV, Jul 20 2016

A275520 Least k such that n divides d(k^k) (d = A000005, k > 0).

Original entry on oeis.org

1, 3, 2, 3, 8, 5, 6, 7, 4, 19, 10, 11, 12, 13, 14, 15, 25, 17, 9, 19, 20, 21, 22, 23, 8, 45, 26, 55, 28, 29, 30, 15, 49, 33, 34, 35, 18, 37, 38, 39, 20, 41, 42, 21, 14, 45, 46, 35, 6, 39, 25, 51, 52, 35, 54, 55, 28, 57, 58, 59, 60, 61, 62, 15, 12, 65, 66, 33, 68, 69, 70, 35, 24

Offset: 1

Altug Alkan, Jul 31 2016

If n > 1 and n-1 is squarefree, then a(n) <= n-1. # Robert Israel, Apr 11 2023

			a(5) = 8 because A000005(8^8) = 25 is divisible by 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005179, A062319.

g:= proc(k) option remember;
    local F,t;
    F:= ifactors(k)[2];
    mul(t[2]*k+1,t=F);
end proc:
f:= proc(n) local k;
  for k from 1 do if g(k) mod n = 0 then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Apr 11 2023

Table[k = 1; While[! Divisible[DivisorSigma[0, k^k], n], k++]; k, {n, 73}] (* Michael De Vlieger, Aug 02 2016 *)

a(n) = {my(k=1); while(numdiv(k^k) % n != 0, k++); k; }

A343731 Numbers k at which tau(k^k) reaches a record high, where tau is the number-of-divisors function A000005.

Original entry on oeis.org

0, 2, 3, 4, 6, 10, 12, 18, 20, 24, 30, 42, 60, 78, 84, 90, 114, 120, 140, 150, 156, 168, 180, 210, 330, 390, 420, 510, 546, 570, 630, 660, 780, 840, 990, 1020, 1050, 1092, 1140, 1170, 1260, 1530, 1540, 1560, 1680, 1848, 1890, 1980, 2100, 2280, 2310, 2730, 3570

Offset: 1

Jon E. Schoenfield, Jun 01 2021

nonn

			In the table below, asterisks indicate record high values of tau(k^k):
                        tau(k^k) =
   k  k^k = A000312(k)  A062319(k)
  --  ----------------  ----------
   0                 1           1 *
   1                 1           1
   2                 4           3 *
   3                27           4 *
   4               256           9 *
   5              3125           6
   6             46656          49 *
   7            823543           8
   8          16777216          25
   9         387420489          19
  10       10000000000         121 *
  11      285311670611          12
  12     8916100448256         325 *
.
The numbers k at which those record high values occur are 0, 2, 3, 4, 5, 6, 10, 12, ...

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 510 terms from Chai Wah Wu)

Cf. A000005, A000312, A062319.

Join[{0},DeleteDuplicates[Table[{n,DivisorSigma[0,n^n]},{n,2,3600}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]]] (* Harvey P. Dale, Jul 21 2024 *)

from functools import reduce
from operator import mul
from sympy import factorint
c, A343731_list = 0, [0]
for n in range(2,10**5):
    x = reduce(mul,(n*d+1 for d in factorint(n).values()))
    if x > c:
        c = x
        A343731_list.append(n) # Chai Wah Wu, Jun 03 2021

A343732 Numbers k at which tau(k^k) is a prime power, where tau is the number-of-divisors function A000005.

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 9, 10, 15, 22, 26, 30, 31, 36, 42, 46, 58, 66, 70, 78, 82, 102, 106, 121, 127, 130, 138, 166, 178, 190, 210, 222, 226, 238, 255, 262, 282, 310, 330, 346, 358, 366, 382, 418, 430, 438, 441, 442, 462, 466, 478, 498, 502, 511, 546, 562, 570, 586

Offset: 1

Jon E. Schoenfield, Jun 01 2021

nonn

			9^9 = (3^2)^9 = 3^18 has 19 = 19^1 divisors, so 9 is a term.
10^10 = 2^10 * 5^10 has 121 = 11^2 divisors, so 10 is a term.
11^11 has 12 = 2^2 * 3^1 divisors, so 11 is not a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A000312, A062319, A246655.

a={}; For[k=1,k<600,k++,If[PrimePowerQ[DivisorSigma[0,k^k]],AppendTo[a,k]]]; a (* Stefano Spezia, Jun 02 2021 *)
Select[Range[600],PrimePowerQ[DivisorSigma[0,#^#]]&] (* Harvey P. Dale, Oct 29 2022 *)

isok(k) = isprimepower(numdiv(k^k)); \\ Michel Marcus, Jun 02 2021

from functools import reduce
from operator import mul
from sympy import factorint
A343732_list = [n for n in range(2,10**3) if len(factorint(reduce(mul,(n*d+1 for d in factorint(n).values())))) == 1] # Chai Wah Wu, Jun 03 2021

A343936 Number of ways to choose a multiset of n divisors of n - 1.

Original entry on oeis.org

1, 2, 3, 10, 5, 56, 7, 120, 45, 220, 11, 4368, 13, 560, 680, 3876, 17, 26334, 19, 42504, 1771, 2024, 23, 2035800, 325, 3276, 3654, 201376, 29, 8347680, 31, 376992, 6545, 7140, 7770, 145008513, 37, 9880, 10660, 53524680, 41, 73629072, 43, 1712304, 1906884

Offset: 1

Gus Wiseman, May 05 2021

nonn

			The a(1) = 1 through a(5) = 5 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
      {2}  {1,3}  {1,1,2}  {1,1,1,5}
           {3,3}  {1,1,4}  {1,1,5,5}
                  {1,2,2}  {1,5,5,5}
                  {1,2,4}  {5,5,5,5}
                  {1,4,4}
                  {2,2,2}
                  {2,2,4}
                  {2,4,4}
                  {4,4,4}
The a(6) = 56 multisets:
  11111  11136  11333  12236  13366  22266  23666
  11112  11166  11336  12266  13666  22333  26666
  11113  11222  11366  12333  16666  22336  33333
  11116  11223  11666  12336  22222  22366  33336
  11122  11226  12222  12366  22223  22666  33366
  11123  11233  12223  12666  22226  23333  33666
  11126  11236  12226  13333  22233  23336  36666
  11133  11266  12233  13336  22236  23366  66666

The version for chains of divisors is A163767.
Diagonal n = k + 1 of A343658.
Choosing n divisors of n gives A343935.
A000005 counts divisors.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
- A067824 counts strict chains of divisors starting with n.
- A074206 counts strict chains of divisors from n to 1.
- A251683 counts strict length k + 1 chains of divisors from n to 1.
- A334996 counts strict length-k chains of divisors from n to 1.
- A337255 counts strict length-k chains of divisors starting with n.
- A337256 counts strict chains of divisors of n.
- A343662 counts strict length-k chains of divisors.
Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343656, A343661.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n-1],{n,50}]

a(n) = ((sigma(n - 1), n)) = binomial(sigma(n - 1) + n - 1, n) where sigma = A000005 and binomial = A007318.

A343733 Primes p at which tau(p^p) is a prime power, where tau is the number-of-divisors function A000005.

Original entry on oeis.org

2, 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Jon E. Schoenfield, Jun 01 2021

For every prime p, p^p has p+1 divisors. p=2 is a term, but for all odd primes, p+1 is even, so this sequence consists of 2 and the primes of the form 2^j - 1, i.e., 2 and the Mersenne primes (A000668).

			2^2 has 3 = 3^1 divisors, so 2 is a term.
3^3 has 4 = 2^2 divisors, so 3 is a term.
5^5 has 6 = 2*3 divisors, so 5 is not a term.

Cf. A000005, A000668, A000312, A062319.

cp's OEIS Frontend

A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

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A275217 Even numbers n such that A000005(n) divides A000005(n^n).

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PARI

A275520 Least k such that n divides d(k^k) (d = A000005, k > 0).

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A343731 Numbers k at which tau(k^k) reaches a record high, where tau is the number-of-divisors function A000005.

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A343732 Numbers k at which tau(k^k) is a prime power, where tau is the number-of-divisors function A000005.

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Python

A343936 Number of ways to choose a multiset of n divisors of n - 1.

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Formula

A343733 Primes p at which tau(p^p) is a prime power, where tau is the number-of-divisors function A000005.

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