A154402 -id:A154402 - cp's OEIS frontend

A147645 Number of distinct Mersenne primes dividing n.

0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 1, 1, 0

Offset: 1

Omar E. Pol, Nov 09 2008

a(n) = m first occurs at n = A098918(m). - Robert Israel, Feb 03 2020

			a(21)=2 because 1, 3, 7 and 21 are divisors of 21. Then 21 has two divisors that are Mersenne primes (A000668): 3 and 7.

Antti Karttunen, Table of n, a(n) for n = 1..131072 (terms 1..10000 from Robert Israel)

Cf. A000265, A000668, A001221, A080225, A098918, A154402, A173898, A353786.

Coincides with A331410 on A054784.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for i from 1 do
m:= numtheory:-mersenne([i]);
if m > N then break fi;
for j from m by m to N do
    V[j]:= V[j]+1
od od:
convert(V,list); # Robert Israel, Feb 03 2020

A147645(n) = { my(m=3,s=0); while(m<=n, s += (isprime(m)*!(n%m)); m += (m+1)); (s); }; \\ Antti Karttunen, May 12 2022

From Antti Karttunen, May 12 2022: (Start)
a(n) = A154402(n) - A353786(n)
a(n) = a(2*n) = a(A000265(n)).
a(n) <= A331410(n). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A173898 = 0.516454... . - Amiram Eldar, Dec 31 2023

A161790 The positive integer n is included if 1 is the largest integer of the form {2^k - 1} to divide n.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 26, 29, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 50, 52, 53, 55, 58, 59, 61, 64, 65, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 88, 89, 92, 94, 95, 97, 100, 101, 103, 104, 106, 107, 109, 110, 113, 115, 116, 118, 121, 122, 125, 128, 130, 131, 134, 136, 137, 139, 142, 143

Offset: 1

Leroy Quet, Jun 19 2009

nonn

Numbers which are not multiple of 2^k-1, k > 1. Because 2^k-1 = 1+2+...+2^(k-1), these numbers are also not the sum of positive integers in a geometric progression with common ratio 2 (cf. the primes A000040 which satisfy a similar property with arithmetic progressions with common difference 2). - Jean-Christophe Hervé, Jun 19 2014

The asymptotic density of this sequence is 1 - Sum_{s subset of A000225 \ {0, 1}} (-1)^(card(s)+1)/LCM(s) = 0.54830... - Amiram Eldar, Jun 30 2025

Diana L. Mecum, Table of n, a(n) for n = 1..2000

Cf. A000225, A382875 (complement).

Positions of ones in A154402, A161788 and A161789.

DivisorList=Drop[Table[2^k-1,{k,1,20}],1]
A161790=Union[Table[If[Length[Join[DivisorList,Drop[Divisors[n],1]]]==Length[Union[DivisorList,Drop[Divisors[n],1]]],n,],{n,1,5000}]]
(* Second program: *)
Position[Table[DivisorSum[n, 1 &, IntegerQ@ Log2[# + 1] &], {n, 143}], 1][[All, 1]] (* Michael De Vlieger, Jun 11 2018 *)

isok(k) = forstep(i = logint(k+1, 2), 2, -1, if(k % (2^i-1) == 0, return(0))); 1; \\ Amiram Eldar, Jun 30 2025

A161788(a(n)) = A161789(a(n)) = 1.

Also A154402(a(n)) = 1. - Antti Karttunen, Jun 11 2018

A342339 Heinz numbers of the integer partitions counted by A342337, which have all adjacent parts (x, y) satisfying either x = y or x = 2y.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 36, 37, 41, 42, 43, 47, 48, 49, 53, 54, 59, 61, 63, 64, 65, 67, 71, 72, 73, 79, 81, 83, 84, 89, 96, 97, 101, 103, 107, 108, 109, 113, 121, 125, 126, 127, 128, 131, 133, 137

Offset: 1

Gus Wiseman, Mar 11 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}            19: {8}             48: {1,1,1,1,2}
      2: {1}           21: {2,4}           49: {4,4}
      3: {2}           23: {9}             53: {16}
      4: {1,1}         24: {1,1,1,2}       54: {1,2,2,2}
      5: {3}           25: {3,3}           59: {17}
      6: {1,2}         27: {2,2,2}         61: {18}
      7: {4}           29: {10}            63: {2,2,4}
      8: {1,1,1}       31: {11}            64: {1,1,1,1,1,1}
      9: {2,2}         32: {1,1,1,1,1}     65: {3,6}
     11: {5}           36: {1,1,2,2}       67: {19}
     12: {1,1,2}       37: {12}            71: {20}
     13: {6}           41: {13}            72: {1,1,1,2,2}
     16: {1,1,1,1}     42: {1,2,4}         73: {21}
     17: {7}           43: {14}            79: {22}
     18: {1,2,2}       47: {15}            81: {2,2,2,2}

The first condition alone gives A000961 (perfect powers).
The second condition alone is counted by A154402.
These partitions are counted by A342337.
A018819 counts partitions into powers of 2.
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A045690 counts sets with maximum n in with adjacent elements y < 2x.
A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342334 counts compositions with adjacent parts x >= 2y or y > 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
A342342 counts strict compositions with adjacent parts x <= 2y and y <= 2x.
Cf. A003114, A003242, A034296, A040039, A167606. A342083, A342084, A342087, A342191, A342336, A342339, A342340.

Select[Range[100],With[{y=PrimePi/@First/@FactorInteger[#]},And@@Table[y[[i]]==y[[i-1]]||y[[i]]==2*y[[i-1]],{i,2,Length[y]}]]&]

A350330 Lexicographically earliest sequence of positive integers such that the Hankel matrix of any odd number of consecutive terms is invertible.

Original entry on oeis.org

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

Offset: 1

Pontus von Brömssen, Dec 25 2021

nonn

No linear relation of the form c_1*a(j) + ... + c_k*a(j+k-1) = 0, with at least one c_i nonzero, holds for k consecutive values of j.
Is a(n) <= 3 for all n? (It is true for n <= 400.) If not, what is the largest term? Or is the sequence unbounded?
There seems to be some regularity in the sequence of values of n for which a(n) > 2: 15, 29, 36, 51, 65, 71, 86, 100, ... . The first differences of these are: 14, 7, 15, 14, 6, 15, 14, 5, 15, 14, 3, 15, 14, 1, 15, 13, 11, 15, 14, 7, 15, 14, 5, 15, 14, 3, 15, 14, 1, ... . The differences are all less than or equal to 15, because A350364(15,2) = 0.
Agrees with A154402 for the first 20 terms, but differs on the 21st.

			a(15) = 3, because the Hankel matrix of (a(11), ..., a(15)) is
  [1  2   1  ]
  [2  1   2  ]
  [1  2 a(15)],
which is singular if a(15) = 1, and the Hankel matrix of (a(5), ..., a(15)) is
  [1  2  2  1  2   1  ]
  [2  2  1  2  1   1  ]
  [2  1  2  1  1   2  ]
  [1  2  1  1  2   1  ]
  [2  1  1  2  1   2  ]
  [1  1  2  1  2 a(15)],
which is singular if a(15) = 2, but if a(15) = 3 the Hankel matrix of (a(k), ..., a(15)) is invertible for all odd k <= 15.

Robert Israel, Table of n, a(n) for n = 1..1000 (1..400 from _Pontus von Brömssen)
Wikipedia, Hankel matrix

Cf. A350351, A350348, A350349, A350350, A350364, A154402.

from sympy import Matrix
from itertools import count
def A350330_list(nmax):
    a=[]
    for n in range(nmax):
        a.append(next(k for k in count(1) if all(Matrix((n-r)//2+1,(n-r)//2+1,lambda i,j:(a[r:]+[k])[i+j]).det()!=0 for r in range(n-2,-1,-2))))
    return a

# Faster version using numpy instead of sympy.
# Due to floating point errors, the results may be inaccurate for large n. Correctness verified up to n=400 for numpy 1.20.2.
from numpy import array
from numpy.linalg import det
from itertools import count
def A350330_list(nmax):
    a=[]
    for n in range(nmax):
        a.append(next(k for k in count(1) if all(abs(det(array([[(a[r:]+[k])[i+j] for j in range((n-r)//2+1)] for i in range((n-r)//2+1)])))>0.5 for r in range(n-2,-1,-2))))
    return a

A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 144, 147, 150, 156, 162, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258, 260, 264, 266, 270

Offset: 1

Gus Wiseman, Jan 20 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at least two adjacent prime indices of quotient 1/2.

			The terms and corresponding partitions begin:
   6: (2,1)
  12: (2,1,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  30: (3,2,1)
  36: (2,2,1,1)
  42: (4,2,1)
  48: (2,1,1,1,1)
  54: (2,2,2,1)
  60: (3,2,1,1)
  63: (4,2,2)
  65: (6,3)
  66: (5,2,1)
  72: (2,2,1,1,1)
  78: (6,2,1)
  84: (4,2,1,1)
  90: (3,2,2,1)
  96: (2,1,1,1,1,1)

The complement is A350838, counted by A350837.
The strict complement is counted by A350840.
These partitions are counted by A350846.
A000041 = integer partitions.
A000045 = sets containing n with all differences > 2.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free integer partitions, ranked by A320340.
A325160 ranks strict partitions with no successions, counted by A003114.
A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],MemberQ[Divide@@@Partition[primeptn[#],2,1],2]&]

A350846 Number of integer partitions of n with at least two adjacent parts of quotient 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 8, 12, 18, 25, 36, 48, 65, 89, 119, 157, 207, 269, 350, 448, 574, 729, 927, 1166, 1465, 1830, 2282, 2827, 3501, 4309, 5300, 6483, 7923, 9641, 11718, 14187, 17155, 20674, 24885, 29860, 35787, 42772, 51054, 60791, 72289, 85772, 101641

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(3) = 1 through a(9) = 12 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (521)      (621)
                       (2211)   (3211)    (3221)     (3321)
                       (21111)  (22111)   (4211)     (4221)
                                (211111)  (22211)    (5211)
                                          (32111)    (22221)
                                          (221111)   (32211)
                                          (2111111)  (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)

The complement is counted by A350837, strict A350840.
The complimentary additive version is A350842, strict A350844.
These partitions are ranked by A350845, complement A350838.
A000041 = integer partitions.
A323092 = double-free integer partitions, ranked by A320340.
Cf. A000929, A003000, A003114, A018819, A045690, A045691, A116931, A120641, A154402, A323093, A342094, A342095, A342096, A342098.

Table[Length[Select[IntegerPartitions[n], MemberQ[Divide@@@Partition[#,2,1],2]&]],{n,0,30}]

A364675 Number of integer partitions of n whose nonzero first differences are a submultiset of the parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 7, 10, 12, 15, 15, 26, 25, 35, 45, 55, 60, 86, 94, 126, 150, 186, 216, 288, 328, 407, 493, 610, 699, 896, 1030, 1269, 1500, 1816, 2130, 2620, 3029, 3654, 4300, 5165, 5984, 7222, 8368, 9976, 11637, 13771, 15960, 18978, 21896, 25815, 29915

Offset: 0

Gus Wiseman, Aug 04 2023

nonn

Conjecture: For subsets of {1..n} instead of partitions of n we have A101925.

Conjecture: The strict version is A154402.

			The partition y = (3,2,1,1) has first differences (1,1,0), and (1,1) is a submultiset of y, so y is counted under a(7).
The a(1) = 1 through a(8) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (221)    (33)      (421)      (44)
             (111)  (211)   (2111)   (42)      (2221)     (422)
                    (1111)  (11111)  (222)     (3211)     (2222)
                                     (2211)    (22111)    (4211)
                                     (21111)   (211111)   (22211)
                                     (111111)  (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

For subsets of {1..n} we appear to have A101925, A364671, A364672.
The strict case (no differences of 0) appears to be A154402.
Starting with the distinct parts gives A342337.
For disjoint multisets: A363260, subsets A364463, strict A364464.
For overlapping multisets: A364467, ranks A364537, strict A364536.
For subsets instead of submultisets we have A364673.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions, complement A237113.
A325325 counts partitions with distinct first differences.
Cf. A002865, A007862, A108917, A229816, A237667, A237668, A320347, A363225, A364272, A364345, A364466.

submultQ[cap_,fat_] := And@@Function[i,Count[fat,i] >= Count[cap,i]] /@ Union[List@@cap];
Table[Length[Select[IntegerPartitions[n], submultQ[Differences[Union[#]],#]&]], {n,0,30}]

A305436 Number of divisors of n of the form 2^k + 1 for k >= 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

a(n) is the number of terms of A000051 that divide n.

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

Cf. A000051, A209229, A292315 (positions of zeros), A305435, A323482.

Cf. also A154402.

Table[DivisorSum[n, 1 &, IntegerQ@ Log2[# - 1] &], {n, 105}] (* Michael De Vlieger, Jun 11 2018 *)

A209229(n) = (n && !bitand(n,n-1));
A305436(n) = sumdiv(n,d,A209229(d-1));

a(n) = Sum_{d|n} A209229(d-1).
a(n) = A305435(n) + A209229(n-1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A323482 = 1.264499... . - Amiram Eldar, Dec 31 2023

A305426 Number of proper divisors of n of the form 2^k - 1 for k >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

a(n) is the number of terms of A000225 less than n that divide n.

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

Cf. A000225, A036987, A154402.

Cf. also A305435.

Table[DivisorSum[n, 1 &, And[IntegerQ@ Log2[# + 1], # < n] &], {n, 105}] (* Michael De Vlieger, Jun 11 2018 *)

A209229(n) = (n && !bitand(n,n-1));
A036987(n) = A209229(1+n);
A305426(n) = sumdiv(n,d,(dA036987(d));

a(n) = Sum_{d|n, dA036987(d).

a(n) = A154402(n) - A036987(n).

A353786 Number of distinct nonprime numbers of the form 2^k - 1 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Antti Karttunen, May 12 2022

nonn

			Divisors of 255 are [1, 3, 5, 15, 17, 51, 85, 255], of these of the form 2^k - 1 (A000225) are 1, 3, 15 and 255, but only three of them are counted (because 3 is a prime), therefore a(255) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000225, A000265, A147645, A154402.

Cf. A065442, A135972, A173898.

a[n_] := DivisorSum[n, 1 &, !PrimeQ[#] && # + 1 == 2^IntegerExponent[# + 1, 2] &]; Array[a, 120] (* Amiram Eldar, May 12 2022 *)

A353786(n) = { my(m=1,s=0); while(m<=n, s += (!isprime(m))*!(n%m); m += (m+1)); (s); };

a(n) = A154402(n) - A147645(n).
a(n) = a(2*n) = a(A000265(n)).
For all primes p, a(p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A135972(n) = A065442 - A173898 = 1.0902409734... . - Amiram Eldar, Dec 31 2023

cp's OEIS Frontend

A147645 Number of distinct Mersenne primes dividing n.

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A161790 The positive integer n is included if 1 is the largest integer of the form {2^k - 1} to divide n.

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A342339 Heinz numbers of the integer partitions counted by A342337, which have all adjacent parts (x, y) satisfying either x = y or x = 2y.

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A350330 Lexicographically earliest sequence of positive integers such that the Hankel matrix of any odd number of consecutive terms is invertible.

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A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

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A350846 Number of integer partitions of n with at least two adjacent parts of quotient 2.

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A364675 Number of integer partitions of n whose nonzero first differences are a submultiset of the parts.

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A305436 Number of divisors of n of the form 2^k + 1 for k >= 0.

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