A062968 -id:A062968 - cp's OEIS frontend

A125168 a(n) = gcd(n, A032741(n)) where A032741(n) is the number of proper divisors of n.

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

Offset: 1

Mitch Cervinka (puritan(AT)toast.net), Jan 12 2007

First occurrence of k: 1, 4, 6, 16, 20, 3240000, 42, 256, 162, 18662400, 132, 5308416, 832, 784, 120, 65536, 612, 2985984, 912, 1600, 9240, 98010000, 1380, 1296, 100800, ..., (10^7). - Robert G. Wilson v, Jan 23 2007
Do all values appear? - Robert G. Wilson v, Jan 23 2007
From Bernard Schott, Oct 19 2019: (Start)
a(n) = 1 if n = p^k, p prime, k >= 0 and k <> p or,
            n = p*q, p 3 or
            n = p*q*r, p 7 or,
            n = p^2*q, p 5 or
            n = p^3*q, p 7.
a(n) = 2 if n = 2^2 or n = 2^(2*p), p prime <> 2,
a(n) = 3 if n = 3*p, p prime <> 3 or n = 3^3,
a(n) = 4 if n = 4*p^2, p prime,
a(n) = 5 if n = 5*p^2, p prime <> 5, or n = 25*p, p prime <> 5, or n = 5^5,
a(n) = 7 if n = 7*p*q with p 7 or n = 7*p^3, p prime <> 7, or n = 7^7,
a(n) = p if n = p^p, p prime. (End)

			a(6)=3 because 6 has 3 proper divisors {1,2,3} and gcd(6,3) is 3.

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

Cf. A032741, A009191, A062968.

f[n_] := GCD[n, DivisorSigma[0, n] - 1]; Array[f, 105] (* Robert G. Wilson v *)

A125168(n) = gcd(n,numdiv(n)-1); \\ Antti Karttunen, Sep 25 2018

a(n) = gcd(n, A032741(n)) = gcd(n, A062968(n)).

More terms from Robert G. Wilson v, Jan 23 2007

A059292 a(n) = n + 2 - (number of divisors of n).

Original entry on oeis.org

2, 2, 3, 3, 5, 4, 7, 6, 8, 8, 11, 8, 13, 12, 13, 13, 17, 14, 19, 16, 19, 20, 23, 18, 24, 24, 25, 24, 29, 24, 31, 28, 31, 32, 33, 29, 37, 36, 37, 34, 41, 36, 43, 40, 41, 44, 47, 40, 48, 46, 49, 48, 53, 48, 53, 50, 55, 56, 59, 50, 61, 60, 59, 59, 63, 60, 67

Offset: 1

N. J. A. Sloane, Jan 25 2001

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 121, see #17.

Harry J. Smith, Table of n, a(n) for n = 1..5000

Cf. A000005, A049820, A062968. - Omar E. Pol, Jul 16 2009

[n+2-DivisorSigma(0, n): n in [1..100]]; // Vincenzo Librandi, Jan 05 2017

Table[n + 2 - DivisorSigma[0, n], {n,1,100}] (* G. C. Greubel, Jan 04 2017 *)

a(n) = { n + 2 - numdiv(n) } \\ Harry J. Smith, Jun 25 2009

a(n) = n + 2 - A000005(n) = A049820(n) + 2. - Omar E. Pol, Jul 16 2009

A342194 Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 7, 7, 13, 11, 11, 17, 13, 15, 25, 17, 17, 29, 19, 23, 35, 25, 23, 39, 29, 29, 45, 33, 29, 55, 31, 35, 55, 39, 43, 65, 37, 43, 65, 51, 41, 77, 43, 51, 85, 53, 47, 85, 53, 65, 87, 61, 53, 99, 67, 67, 97, 67, 59, 119, 61, 71, 113, 75, 79, 123, 67, 79, 117

Offset: 0

Gus Wiseman, Apr 02 2021

nonn

			The a(1) = 1 through a(9) = 13 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)  (1,7)  (1,8)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)  (2,6)  (2,7)
                          (3,2)  (4,2)    (3,4)  (3,5)  (3,6)
                          (4,1)  (5,1)    (4,3)  (5,3)  (4,5)
                                 (1,2,3)  (5,2)  (6,2)  (5,4)
                                 (3,2,1)  (6,1)  (7,1)  (6,3)
                                                        (7,2)
                                                        (8,1)
                                                        (1,3,5)
                                                        (2,3,4)
                                                        (4,3,2)
                                                        (5,3,1)

Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Strict compositions in general are counted by A032020.
The unordered version is A049980.
The non-strict version is A175342.
A000203 adds up divisors.
A000726 counts partitions with alternating parts unequal.
A003242 counts anti-run compositions.
A224958 counts compositions with alternating parts unequal.
A342343 counts compositions with alternating parts strictly decreasing.
A342495 counts compositions with constant quotients.
A342527 counts compositions with alternating parts equal.
Cf. A000009, A001522, A002843, A049988, A062968, A070211, A114921, A325545, A325557, A342496, A342515, A342522.

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],SameQ@@Differences[#]&]],{n,0,30}]

a(n > 0) = A175342(n) - A000005(n) + 1.

a(n > 0) = 2*A049988(n) - 2*A000005(n) + 1 = 2*A049982(n) + 1.

A062969 Numbers k such that abs(d(k)-k-1) is prime, where d(k) is the number of divisors of k.

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 12, 14, 18, 22, 24, 25, 26, 28, 30, 34, 46, 49, 52, 54, 62, 66, 72, 74, 76, 78, 80, 82, 84, 86, 90, 104, 106, 108, 110, 112, 114, 134, 138, 142, 150, 160, 166, 169, 170, 172, 174, 176, 180, 186, 192, 194, 202, 204, 208, 214, 226, 230, 234, 244, 246

Offset: 1

Jason Earls, Jul 23 2001

			d(300)-300-1 = -283, absolute value of which is prime.

Cf. A062968.

j=[]; for(n=1,300, if(isprime(numdiv(n)-n-1),j=concat(j,n))); j

A366525 Irregular triangular array read by rows: T(n,k) = number of partitions p of n such that f(p) = k >= 0, where f is defined in Comments.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 3, 3, 6, 2, 6, 6, 3, 5, 9, 5, 3, 7, 9, 9, 4, 1, 7, 13, 12, 6, 4, 10, 12, 15, 12, 5, 2, 7, 16, 19, 16, 12, 5, 2, 12, 16, 24, 22, 18, 6, 3, 11, 20, 28, 29, 24, 14, 6, 3, 12, 19, 31, 34, 36, 24, 13, 4, 3, 12, 23, 36, 42, 50, 30, 25, 8, 4

Offset: 1

Clark Kimberling, Oct 12 2023

For a partition p = (p(1),p(2),...,p(k)) of n, where p(1) >= p(2) >= ... >= p(k), define r(p) by subtracting 1 from p(1) and adding 1 to p(k) and then arranging the result in nonincreasing order. Iterating r eventually results in repetition; the function f(p) is the number of iterations of r up to but not including the first repeat. For example, starting with (5,3,2,2,1,1,1,1), the r-iterates are (4,3,2,2,2,1,1,1), (3,3,2,2,2,2,1,1), (3,2,2,2,2,2,2,1), (2,2,2,2,2,2,2,2), (3,2,2,2,2,2,2,1), so that f(5,3,2,2,1,1,1,1) = 3.

			First 18 rows:
   1
   1     1
   2     1
   2     3
   4     3
   3     6     2
   6     6     3
   5     9     5     3
   7     9     9     4     1
   7    13    12     6     4
  10    12    15    12     5     2
   7    16    19    16    12     5    2
  12    16    24    22    18     6    3
  11    20    28    29    24    14    6      3
  12    19    31    34    36    24    13     4     3
  12    23    36    42    50    30    25     8     4    1
  16    23    42    54    59    45    34    15     5    4
  13    28    47    57    74    61    52    28    16    5    4
Row 6 represents 3 partitions p that are self-repeating (i.e., k = 0), 6 such that f(p) = 1, and 2 such that f(p) = 2. Specifically,
  f(p) = 0 for these partitions: [6], [2,2,1,1], [2,1,1,1].
f(p) = 1 for these: [4,2], [3,3], [3,2,1], [3,1,1,1], [2,2,2], [1,1,1,1,1,1].
f(p) = 2 for these: [5,1], [4,1,1].

John Tyler Rascoe, Rows n = 1..60, flattened

Cf. A000041 (row sums), A003479 (row lengths, after 2nd term).

Cf. A062968 (1st column).

r[list_] := If[Length[list] == 1, list, Reverse[Sort[# +
Join[{-1}, ConstantArray[0, Length[#] - 2], {1}]] &[Reverse[Sort[list]]]]];
f[list_] := NestWhileList[r, Reverse[Sort[list]], Unequal, All];
t = Table[BinCounts[#, {0, Max[#] + 1, 1}] &[Map[-1 + Length[Union[#]] &[f[#]] &, IntegerPartitions[n]]], {n, 1,20}]
Map[Length, t]; t1 = Take[t, 18]; TableForm[t1]
Flatten[t1]
(* Peter J. C. Moses, Oct 10 2023 *)

from sympy .utilities.iterables import ordered_partitions
from collections import Counter
def A366525_rowlist(row_n):
    A = []
    for i in range(1,row_n+1):
        A.append([]); p,C = list(ordered_partitions(i)),Counter()
        for j in range(0,len(p)):
            x,a1,a,b = 0,[],list(p[j]),list(p[j])
            while i:
                b[-1] -= 1; b[0] += 1
                if b[-1] == 0: b.pop(-1)
                b = sorted(b); x += 1
                if a == b or a1 == b:
                    C.update({x}); break
                else:
                    a1 = a.copy(); a = b.copy()
        for z in range(1,len(C)+1): A[i-1].append(C[z])
    return(A) # John Tyler Rascoe, Nov 09 2023

cp's OEIS Frontend

A125168 a(n) = gcd(n, A032741(n)) where A032741(n) is the number of proper divisors of n.

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A059292 a(n) = n + 2 - (number of divisors of n).

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A342194 Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.

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A062969 Numbers k such that abs(d(k)-k-1) is prime, where d(k) is the number of divisors of k.

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A366525 Irregular triangular array read by rows: T(n,k) = number of partitions p of n such that f(p) = k >= 0, where f is defined in Comments.

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