A032741 -id:A032741 - cp's OEIS frontend

A335661 The squares visited on a square (Ulam) spiral, with a(1) = 1 and a(2) = 2, when stepping to the closest unvisited square containing a number that shares a common divisor > 1 with the number in the current square. If two or more such squares are the same distance from the current square then the one with the smallest number is chosen.

1, 2, 4, 6, 8, 22, 20, 40, 18, 39, 69, 105, 150, 104, 66, 38, 36, 63, 98, 62, 34, 14, 12, 3, 15, 5, 35, 60, 33, 30, 55, 88, 54, 87, 129, 177, 234, 299, 455, 375, 456, 374, 300, 235, 130, 90, 57, 93, 135, 186, 134, 92, 58, 32, 56, 91, 133, 182, 132, 180, 237

Offset: 1

Scott R. Shannon, Jun 17 2020

Any even number on the square spiral has 4 diagonally adjacent squares which contain an even number and thus, unless all four such squares have been previously visited, a step to one of those adjacent squares, the one containing the smallest number, will always be possible. Any visited square containing a prime number will need to step to, and be stepped to from, a square containing a multiple of that prime number.
In the first 10 million terms the longest required step is from a(97528) = 5981, a prime number which has coordinates (39,13) relative to the starting 1-square, to a(97529) = 167468 (27*5981), with coordinates (205,-18), a step of length sqrt(28517), approximately 168.9 units. This is an extremely large step length relative to the total number of steps taken up to that point - see the attached link image. It is not surpassed by any subsequent step up to 10 million steps. If the maximum step distance between adjacent terms has a finite value or is unbounded as n increases is unknown. The largest difference between terms is for a(9404208) = 8964653 to a(9404209) = 10485343, a difference of 1520690.
In the first 10 million terms the smallest unvisited square is 37, which has coordinates (-3,3) relative to the starting 1-square. It is unknown if this square, and similar unvisited squares near the origin, is eventually visited for very large values of n or is never visited. The longest run of diagonal steps in the same direction to adjacent smaller even numbers is 52, from a(3979714) = 5051162 to a(3979766) = 4594498.

			a(3) = 4 as a(2) = 2 is surrounded by eight adjacent squares with numbers 3,4,1,8,9,10,11,12. The unvisited squares 1 unit away, 3,9,11 have no common factor with 2. Of the other squares sqrt(2) units away, 4,8,10,12, all share the common factor 2 with a(2), and the smallest of those is 4.
a(10) = 39 as a(9) = 18 is surrounded by adjacent squares 5,6,19,40,39,38,17,16. The square containing 39 is 1 unit directly left of 18 and shares the common factor 3. The other squares one unit away, 5,17,19, have no common factor with 18.

Scott R. Shannon, Image of the steps from 1 to 20001. The green dot shows the starting square 1, the red dot the final square 26453, and the yellow dot the smallest unvisited square 11. The orange line shows the largest step distance, sqrt(976), from a(8538) = 233 to a(8539) = 3029. The blue line shows the longest run of adjacent diagonal steps, each of length sqrt(2), to a lower even number in the same direction, from a(3747) = 7880 and lasts for 19 steps. The pink line shows the largest change in value for a single step, from a(19032) = 15023 to a(19033) = 25159, a difference of 10136.
Scott R. Shannon, Image of the steps from 1 to 20001 with color. The color of each step is graduated across the spectrum from red to violet to show the relative visit order of the squares. Note how green colored steps, those around n = 10000, approach the origin, showing that all numbers near the origin may eventually be visited for very large values of n.
Scott R. Shannon, Image of the steps from 1 to 100000. The orange line shows the step length of sqrt(28517) units at a(97528), from 5981 to 167468. The blue line shows the new longest run of adjacent diagonal steps to lower even numbers, a series of 24 steps. The yellow dot shows the new lowest unvisited square 13, square 11 being visited at a(26321).
Scott R. Shannon, Image of the steps from 1 to 5000000 with color. Note how some violet colored steps, those around n = 4200000, approach the origin. The yellow dot shows the new lowest unvisited square 37, square 13 being visited at a(105263). Also note the visited area forms a roughly square pattern, following the largest outer numbers of the spiral. This becomes more pronounced as n increases.

Cf. A000005, A032741, A063826, A214665, A330979, A331027, A332767, A335710.

A367582 Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 1, 4, 3, 3, 2, 1, 0, 1, 3, 5, 4, 4, 3, 1, 1, 0, 1, 2, 6, 4, 8, 3, 3, 2, 1, 0, 1, 3, 7, 9, 6, 7, 4, 3, 1, 1, 0, 1, 1, 8, 7, 11, 9, 9, 4, 3, 2, 1

Offset: 0

Gus Wiseman, Nov 28 2023

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  2  1  1
  0  1  1  2  2  1
  0  1  3  3  2  1  1
  0  1  1  4  3  3  2  1
  0  1  3  5  4  4  3  1  1
  0  1  2  6  4  8  3  3  2  1
  0  1  3  7  9  6  7  4  3  1  1
  0  1  1  8  7 11  9  9  4  3  2  1
  0  1  5 10 11 13 10 11  6  5  3  1  1
  0  1  1 10 11 17 14 18 10  9  4  3  2  1
  0  1  3 12 17 19 18 22 14 12  8  4  3  1  1
  0  1  3 12 15 27 19 31 19 19 10  9  5  3  2  1
  0  1  4 15 23 27 31 33 24 26 18 12  8  4  3  1  1
  0  1  1 14 20 35 33 48 32 37 25 20 11 10  4  3  2  1
Row n = 7 counts the following partitions:
  (1111111)  (61)  (421)     (52)     (4111)  (511)  (7)
                   (2221)    (331)    (322)   (43)
                   (22111)   (31111)  (3211)
                   (211111)

Column k = 2 is A000005(n) - 1 = A032741(n).
Row sums are A000041.
The case of constant partitions is A051731, row sums A000005.
The corresponding rank statistic is A367581, row sums of A367579.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
Cf. A000837, A056239, A066328, A071625, A072774, A082090, A367580.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], Total[mmk[#]]==k&]], {n,0,10}, {k,0,n}]

A384854 The number of divisors d of n such that (-d)^d == d (mod n).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 5, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Jun 10 2025

nonn

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

Cf. A000005, A032741, A065295, A384237, A384781.

[1+#[s: s in [1..n-1] | n mod s eq 0 and Modexp((-s), s, n) eq s]: n in [1..100]];

a:= n-> add(`if`((-d)&^d-d mod n=0, 1, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 10 2025

a(n) = sumdiv(n, d, Mod(-d, n)^d == d); \\ Michel Marcus, Jun 11 2025

a(n) = 1 + number of proper divisors h of n such that (-h)^h = h (mod n).

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 20, 6, 8, 2, 48, 2, 8, 8, 84, 2, 48, 2, 48, 8, 8, 2, 320, 6, 8, 20, 48, 2, 128, 2, 264, 8, 8, 8, 864, 2, 8, 8, 320, 2, 128, 2, 48, 48, 8, 2, 2688, 6, 48, 8, 48, 2, 320, 8, 320, 8, 8, 2, 3072, 2, 8, 48, 1560, 8, 128, 2, 48, 8, 128, 2, 11520, 2, 8, 48, 48, 8, 128, 2, 2688, 84, 8, 2, 3072, 8, 8, 8, 320

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

			For n = 24, its divisors larger than one are: 2, 3, 4, 6, 8, 12, 24. Only 2 has valuation > 1, namely A286561(24,2) = 3 (as 2^3 divides 24), while the other six have valuation 1. Thus a(24) = prime(1)^6 * prime(3) = 64*5 = 320.
For n = 64, its divisors larger than one are: 2, 4, 8, 16, 32, 64. We see that 2^6 = 4^3 = 8^2 = 64, while valuation of the last three 16, 32 and 64 is 1. Thus a(64) = prime(1)^3 * prime(2) * prime(3) * prime(6) = 2^3 * 3 * 5 * 13 = 1560.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000040, A286561.

Cf. also A293515, A293524, A294876, A293442.

A293514(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(valuation(n,d)))); m; };

a(n) = Product_{d|n, d>1} A000040(A286561(n,d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A056595(n) [See A046951.]
1+A056239(a(n)) = A169594(n).
A064989(a(n)) = A293515(n).

A122934 Triangle T(n,k) = number of partitions of n into k parts, with each part size divisible by the next.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 3, 2, 4, 2, 2, 1, 1, 1, 2, 4, 2, 4, 2, 2, 1, 1, 1, 3, 4, 5, 3, 4, 2, 2, 1, 1, 1, 1, 3, 4, 5, 3, 4, 2, 2, 1, 1, 1, 5, 4, 6, 5, 6, 3, 4, 2, 2, 1, 1, 1, 1, 5, 4, 6, 5, 6, 3, 4, 2, 2, 1, 1, 1, 3, 4, 7, 6, 7, 6, 6, 3, 4, 2, 2, 1, 1

Offset: 1

Franklin T. Adams-Watters, Sep 20 2006

			Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 1, 1;
  1, 1, 2, 1, 1;
  1, 3, 2, 2, 1, 1;
  ...
T(6,3) = 2 because of the 3 partitions of 6 into 3 parts, [4,1,1] and [2,2,2] meet the definition; [3,2,1] fails because 2 does not divide 3.

Seiichi Manyama, Rows n = 1..140, flattened (Rows n = 1..50 from G. C. Greubel)
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5.

Column k=1..4 give A057427, A032741, A049822, A121895.

Row sums give A003238.

T[, 1] = 1; T[n, k_] := T[n, k] = DivisorSum[n, If[#==1, 0, T[#-1, k-1]]& ]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 30 2016 *)

T(n,1) = 1. T(n,k+1) = Sum_{d|n, d1} T(d-1,k).

A294902 Number of proper divisors of n that are in A175526.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 3, 0, 3, 0, 0, 0, 5, 0, 0, 0, 4, 0, 3, 0, 2, 2, 0, 0, 6, 0, 1, 0, 2, 0, 4, 0, 4, 0, 0, 0, 7, 0, 0, 2, 4, 0, 3, 0, 2, 0, 3, 0, 8, 0, 0, 1, 2, 0, 3, 0, 6, 2, 0, 0, 7, 0, 0, 0, 4, 0, 7, 0, 2, 0, 0, 0, 8, 0, 2, 2, 4, 0, 3, 0, 4, 3, 0, 0, 8, 0, 3, 0, 6, 0, 3, 0, 2, 2, 0, 0, 11

Offset: 1

Antti Karttunen, Nov 10 2017

Antti Karttunen, Table of n, a(n) for n = 1..25000

Cf. A000120, A032741, A175526, A292257, A294901, A294903, A294904, A294905.

Cf. also A294892.

q[n_] := DivisorSum[n, DigitCount[#, 2, 1] &] > 2 * DigitCount[n, 2, 1]; a[n_] := DivisorSum[n, 1 &, # < n && q[#] &]; Array[a, 100] (* Amiram Eldar, Jul 20 2023 *)

A292257(n) = sumdiv(n,d,(dA294905(n) = (A292257(n) <= hammingweight(n));
A294902(n) = sumdiv(n,d,(dA294905(d)));

a(n) = Sum_{d|n, dA294905(d)).
a(n) = A294904(n) + A294905(n) - 1.
a(n) + A294901(n) = A032741(n).

A294927 Number of proper divisors of n that are nondeficient (A023196).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 1, 0, 0, 0

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A023196, A032741, A071395, A294887, A294926, A294928, A294929, A294936.

a[n_] := DivisorSum[n, 1 &, # < n && DivisorSigma[1, #] >= 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

PARI
```
A294927(n) = sumdiv(n, d, (d=(2*d)));
```

a(n) = Sum_{d|n, dA294936(d).

a(n) + A294926(n) = A032741(n).

A294929 Number of proper divisors of n that are abundant (A005101).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 1

Antti Karttunen, Nov 14 2017

nonn

			The proper divisors of 24 are 1, 2, 3, 4, 6, 8, 12. Only one of these, 12, is abundant (in A005101), thus a(24) = 1.
The proper divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60. Six of these are abundant: 12, 20, 24, 30, 40, 60, thus a(120) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005101, A032741, A080224, A294889, A294926, A294927, A294928, A294930, A294937.

Cf. also A294902.

a[n_] := Count[Most[Divisors[n]], ?(DivisorSigma[1, #] > 2*# &)]; Array[a, 100] (* _Amiram Eldar, Mar 14 2024 *)

PARI
```
A294929(n) = sumdiv(n, d, (d(2*d)));
```

a(n) = Sum_{d|n, dA294937(d).
a(n) = A080224(n) - A294937(n).
a(n) + A294928(n) = A032741(n).

A051778 Triangle read by rows, where row (n) = n mod (n-1), n mod (n-2), n mod (n-3), ...n mod 2.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 2, 0, 0, 1, 2, 3, 1, 1, 1, 2, 3, 0, 2, 0, 1, 2, 3, 4, 1, 0, 1, 1, 2, 3, 4, 0, 2, 1, 0, 1, 2, 3, 4, 5, 1, 3, 2, 1, 1, 2, 3, 4, 5, 0, 2, 0, 0, 0, 1, 2, 3, 4, 5, 6, 1, 3, 1, 1, 1, 1, 2, 3, 4, 5, 6, 0, 2, 4, 2, 2, 0, 1, 2, 3, 4, 5, 6, 7, 1, 3, 0, 3, 0, 1, 1, 2, 3, 4, 5, 6, 7, 0, 2, 4, 1, 0, 1, 0

Offset: 3

Asher Auel, Dec 09 1999

Central terms: a(2*n+1,n) = n for n > 0. - Reinhard Zumkeller, Dec 03 2014

Deleting column 1 of the array at A051126 gives the array A051778 in square format (see Example). - Clark Kimberling, Feb 04 2016

			row (7) = 7 mod 6, 7 mod 5, 7 mod 4, 7 mod 3, 7 mod 2 = 1, 2, 3, 1, 1.
1;
1  0 ;
1  2  1 ;
1  2  0  0 ;
1  2  3  1  1 ;
1  2  3  0  2  0 ;
1  2  3  4  1  0  1 ;
1  2  3  4  0  2  1  0 ;
1  2  3  4  5  1  3  2  1 ;
1  2  3  4  5  0  2  0  0  0 ;
1  2  3  4  5  6  1  3  1  1  1 ;
Northwest corner of square array:
1 1 1 1 1 1 1 1 1 1 1
0 2 2 2 2 2 2 2 2 2 2
1 0 3 3 3 3 3 3 3 3 3
0 1 0 4 4 4 4 4 4 4 4
1 2 1 0 5 5 5 5 5 5 5
0 0 2 1 0 6 6 6 6 6 6
1 1 3 2 1 0 7 7 7 7 7
- _Clark Kimberling_, Feb 04 2016

Reinhard Zumkeller, Rows n=3..125 of triangle, flattened

Cf. A051777, A051127, A051126.

Cf. A004125 (row sums), A000027 (central terms), A049820 (number of nonzeros per row), A032741 (number of ones per row), A070824 (number of zeros per row).

a051778 n k = a051778_tabl !! (n-3) !! (k-1)
a051778_row n = a051778_tabl !! (n-3)
a051778_tabl = map (\xs -> map (mod (head xs + 1)) xs) $
                   iterate (\xs -> (head xs + 1) : xs) [2]
-- Reinhard Zumkeller, Dec 03 2014

Flatten[Table[Mod[n,i],{n,3,20},{i,n-1,2,-1}]] (* Harvey P. Dale, Sep 09 2012 *)
TableForm[Table[Mod[n, k], {n, 1, 12}, {k, 2, 12}]] (* square *)
(* Clark Kimberling, Feb 04 2016 *)

A336570 Number of maximal sets of proper divisors d|n, d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 3, 1, 4, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) sets for n = 36, 120, 144, 180 (ones not shown):
  {2,18}    {3,12,24}    {2,18,72}       {2,18}
  {3,12}    {5,20,40}    {3,9,18,72}     {3,12}
  {2,4,12}  {2,4,8,24}   {3,12,24,48}    {5,20}
  {3,9,18}  {2,4,8,40}   {3,12,24,72}    {5,45}
            {2,4,12,24}  {2,4,8,16,48}   {2,4,12}
            {2,4,20,40}  {2,4,8,24,48}   {2,4,20}
                         {2,4,8,24,72}   {3,9,18}
                         {2,4,12,24,48}  {3,9,45}
                         {2,4,12,24,72}

A336569 is the version for chains containing n.
A336571 is the non-maximal version.
A000005 counts divisors.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.
Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336423, A336425, A336568.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
strses[n_]:=If[n==1,{{}},Join@@Table[Append[#,d]&/@strses[d],{d,Select[Most[Divisors[n]],strsigQ]}]];
Table[Length[fasmax[strses[n]]],{n,100}]

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A367582 Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.

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Mathematica

A384854 The number of divisors d of n such that (-d)^d == d (mod n).

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Magma

Maple

PARI

Formula

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

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A122934 Triangle T(n,k) = number of partitions of n into k parts, with each part size divisible by the next.

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Mathematica

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A294902 Number of proper divisors of n that are in A175526.

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Mathematica

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Formula

A294927 Number of proper divisors of n that are nondeficient (A023196).

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Mathematica

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A294929 Number of proper divisors of n that are abundant (A005101).

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A051778 Triangle read by rows, where row (n) = n mod (n-1), n mod (n-2), n mod (n-3), ...n mod 2.