A000688 -id:A000688 - cp's OEIS frontend

A275980 Decimal expansion of 2^11213 - 1, the 23rd Mersenne prime A000668(23).

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

Offset: 3376

Arkadiusz Wesolowski, Aug 15 2016

			28141120136973731333931529758425841918186623820136007878924193493455151...

Arkadiusz Wesolowski, Table of n, a(n) for n = 3376..6751
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275981 = A000668(24), A275982 = A000668(25), A275983 = A000668(26), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^11213-1))[1..105];
```

First@RealDigits@N[2^11213 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^11213-1)))[1..105]
```

2^A000043(23) - 1.

A275981 Decimal expansion of 2^19937 - 1, the 24th Mersenne prime A000668(24).

Original entry on oeis.org

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

Offset: 6002

Arkadiusz Wesolowski, Aug 15 2016

			43154247973881626480552355163379198390539350432267115051652505414033306...

Arkadiusz Wesolowski, Table of n, a(n) for n = 6002..12003
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275980 = A000668(23), A275982 = A000668(25), A275983 = A000668(26), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^19937-1))[1..105];
```

First@RealDigits@N[2^19937 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^19937-1)))[1..105]
```

2^A000043(24) - 1.

A275982 Decimal expansion of 2^21701 - 1, the 25th Mersenne prime A000668(25).

Original entry on oeis.org

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

Offset: 6533

Arkadiusz Wesolowski, Aug 15 2016

			44867916611904333479495141036159177872720902372938861301036480447512785...

Arkadiusz Wesolowski, Table of n, a(n) for n = 6533..13065
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275980 = A000668(23), A275981 = A000668(24), A275983 = A000668(26), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^21701-1))[1..105];
```

First@RealDigits@N[2^21701 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^21701-1)))[1..105]
```

2^A000043(25) - 1.

A275983 Decimal expansion of 2^23209 - 1, the 26th Mersenne prime A000668(26).

Original entry on oeis.org

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

Offset: 6987

Arkadiusz Wesolowski, Aug 15 2016

			40287411577898877818187332907159176772243850689162242004102996357869459...

Arkadiusz Wesolowski, Table of n, a(n) for n = 6987..13973
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275980 = A000668(23), A275981 = A000668(24), A275982 = A000668(25), A275984 = A000668(27).

Magma
```
Reverse(Intseq(2^23209-1))[1..105];
```

First@RealDigits@N[2^23209 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^23209-1)))[1..105]
```

2^A000043(26) - 1.

A275984 Decimal expansion of 2^44497 - 1, the 27th Mersenne prime A000668(27).

Original entry on oeis.org

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

Offset: 13395

Arkadiusz Wesolowski, Aug 15 2016

			85450982430363380319330070531840303650990159130402105834326925828229006...

Arkadiusz Wesolowski, Table of n, a(n) for n = 13395..26789
Wikipedia, Mersenne prime

Cf. A169684 = A000668(11), A169681 = A000668(12), A169685 = A000668(13), A204063 = A000668(14), A248931 = A000668(15), A248932 = A000688(16), A248933 = A000668(17), A248934 = A000668(18), A248935 = A000668(19), A248936 = A000668(20), A275977 = A000668(21), A275979 = A000668(22), A275980 = A000668(23), A275981 = A000668(24), A275982 = A000668(25), A275983 = A000668(26).

Magma
```
Reverse(Intseq(2^44497-1))[1..105];
```

First@RealDigits@N[2^44497 - 1, 100] (* G. C. Greubel, Aug 15 2016 *)

PARI
```
eval(Vec(Str(2^44497-1)))[1..105]
```

2^A000043(27) - 1.

A322435 Number of pairs of factorizations of n into factors > 1 where no factor of the second divides any factor of the first.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 5, 0, 3, 0, 3, 1, 1, 0, 7, 1, 1, 2, 3, 0, 4, 0, 7, 1, 1, 1, 15, 0, 1, 1, 7, 0, 4, 0, 3, 3, 1, 0, 16, 1, 3, 1, 3, 0, 7, 1, 7, 1, 1, 0, 18, 0, 1, 3, 16, 1, 4, 0, 3, 1, 4, 0, 32, 0, 1, 3, 3, 1, 4, 0, 16, 5, 1, 0, 18, 1

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

			The a(36) = 15 pairs of factorizations:
  (2*2*3*3)|(4*9)
  (2*2*3*3)|(6*6)
  (2*2*3*3)|(36)
    (2*2*9)|(6*6)
    (2*2*9)|(36)
    (2*3*6)|(4*9)
    (2*3*6)|(36)
     (2*18)|(36)
    (3*3*4)|(6*6)
    (3*3*4)|(36)
     (3*12)|(36)
      (4*9)|(6*6)
      (4*9)|(36)
      (6*6)|(4*9)
      (6*6)|(36)

Cf. A000688, A001055, A050336, A265947, A294068, A305149, A305150, A305193, A317144, A322436, A322437, A322438, A322439, A322440, A322441, A322442.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[Tuples[facs[n],2],!Or@@Divisible@@@Tuples[#]&]],{n,100}]

A322437 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one divides any factor of the other.

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

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

First differs from A322438 at a(144) = 3, A322438(144) = 4.
From Antti Karttunen, Dec 11 2020: (Start)
Zeros occur on numbers that are either of the form p^k, or q * p^k, or p*q*r, for some primes p, q, r, and exponent k >= 0. [Note also that in all these cases, when x > 1, A307408(x) = 2+A307409(x) = 2 + (A001222(x) - 1)*A001221(x) = A000005(x)].
Proof:
  It is easy to see that for such numbers it is not possible to obtain two such distinct factorizations, that no factor of the other would not divide some factor of the other.
  Conversely, the complement set of above is formed of such composites n that have at least one unitary divisor that is either of the form
  (1) p^x * q^y, with x, y >= 2,
  or
  (2) p^x * q^y * r^z, with x >= 2, and y, z >= 1,
  or
  (3) p^x * q^y * r^z * s^w, with x, y, z, w >= 1,
  where p, q, r, s are distinct primes. Let's indicate with C the remaining portion of k coprime to p, q, r and s (which could be also 1). Then in case (1) we can construct two factorizations, the first having factors (p*q*C) and (p^(x-1) * q^(y-1)), and the second having factors (p^x * C) and (q^y) that are guaranteed to satisfy the condition that no factor in the other factorization divides any of the factors of the other factorization. For case (2) pairs like {(p * q^y * C), (p^(x-1) * r^z)} and {(p^x * C), (q^y * r^z)}, and for case (3) pairs like {(p^x * q^y * C), (r^z * s^w)} and {(p^x * r^z * C), (q^y * s^w)} offer similar examples, therefore a(n) > 0 for all such cases.
(End)

			The a(120) = 2 pairs of such factorizations:
   (6*20)|(8*15)
   (8*15)|(10*12)
The a(144) = 3 pairs of factorizations:
   (6*24)|(9,16)
   (8*18)|(12*12)
   (9*16)|(12*12)
The a(210) = 3 pairs of factorizations:
   (6*35)|(10*21)
   (6*35)|(14*15)
  (10*21)|(14*15)
[Note that 210 is the first squarefree number obtaining nonzero value]
The a(240) = 4 pairs of factorizations:
   (6*40)|(15*16)
   (8*30)|(12*20)
  (10*24)|(15*16)
  (12*20)|(15*16)
The a(1728) = 14 pairs of factorizations:
    (6*6*48)|(27*64)
   (6*12*24)|(27*64)
     (6*288)|(27*64)
    (8*8*27)|(12*12*12)
  (12*12*12)|(27*64)
  (12*12*12)|(32*54)
    (12*144)|(27*64)
    (12*144)|(32*54)
    (16*108)|(24*72)
     (18*96)|(27*64)
     (24*72)|(27*64)
     (24*72)|(32*54)
     (27*64)|(36*48)
     (32*54)|(36*48)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000688, A001055, A285572, A285573, A294068, A303362, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322441, A322442, A339626.

Cf. also comments A307408 and A307409.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[Subsets[facs[n],{2}],And[!Or@@Divisible@@@Tuples[#],!Or@@Divisible@@@Reverse/@Tuples[#]]&]],{n,100}]

factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z,Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf,d); z = factorizations(n/d, d, newf, z))); (z));
is_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]),return(0)))); (1); };
number_of_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_ndf_pair(z[i],z[j])));
A322437(n) = number_of_ndfpairs(Vec(factorizations(n))); \\ Antti Karttunen, Dec 10 2020

For n > 0, a(A002110(n)) = A322441(n)/2 = A339626(n). - Antti Karttunen, Dec 10 2020

Data section extended up to a(120) and more examples added by Antti Karttunen, Dec 10 2020

A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 30, 31, 32, 33, 34, 36, 39, 40, 41, 44, 45, 47, 48, 50, 51, 52, 54, 55, 59, 60, 62, 64, 65, 66, 67, 68, 72, 75, 78, 80, 81, 82, 83, 85, 88, 90, 93, 94, 96, 99, 100, 102, 104, 108

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   6: {{},{1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  11: {{3}}
  12: {{},{},{1}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

The initial version is A356940.
Intervals are counted by A000012, A001227, ranked by A073485.
Other types: A107742, A356936, A356937, A356938.
Other conditions: A302478, A302492, A356930, A356935, A356944, A356955.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Select[Range[100],And@@chQ/@primeMS/@primeMS[#]&]

A382080 Number of ways to partition the prime indices of n into sets with a common sum.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 20 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Also the number of factorizations of n into squarefree numbers > 1 with equal sums of prime indices.

			The prime indices of 900 are {1,1,2,2,3,3}, with the following partitions into sets with a common sum:
  {{1,2,3},{1,2,3}}
  {{3},{3},{1,2},{1,2}}
So a(900) = 2.

For just sets we have A050320, distinct A050326.
Twice-partitions of this type are counted by A279788.
For just a common sum we have A321455.
MM-numbers of these multiset partitions are A326534 /\ A302478.
For distinct instead of equal sums we have A381633.
For constant instead of strict blocks we have A381995.
Positions of 0 are A381719, counted by A381994.
A000688 counts factorizations into prime powers, distinct A050361.
A001055 counts factorizations, strict A045778.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A000961, A001222, A006171, A279784, A353866, A381635, A381871.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[mps[prix[n]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]],{n,100}]

A058162 Number of labeled Abelian groups with a fixed identity.

Original entry on oeis.org

1, 1, 1, 4, 6, 60, 120, 1920, 7560, 90720, 362880, 13305600, 39916800, 1037836800, 10897286400, 265686220800, 1307674368000, 66691392768000, 355687428096000, 20274183401472000, 202741834014720000

Offset: 1

Christian G. Bower, Nov 15 2000, Mar 12 2008

nonn

The distinction here between labeled and unlabeled Abelian groups is analogous to the distinction between unlabeled rooted trees (A000081) and labeled rooted trees (A000169).
That is, the number of Cayley tables. - Artur Jasinski, Mar 12 2008
Number of Latin squares in dimension n with first row and first column 1,2,3 ..., n which are associative and commutative (Abelian). Each of these squares is isomorphic with the Cayley table of one of the existed Abelian group in dimension n. - Artur Jasinski, Nov 02 2005. Cf. A111341.

			The 2 unlabeled Abelian groups of order 4 are C4 and C2^2. The 4 labeled Abelian groups whose identity is "0" consist of 3 of type C4 (where the nongenerator can be "2", "3", or "4") and 1 of type C2^2.

Max Alekseyev, Table of n, a(n) for n = 1..100
C. J. Hillar, D. Rhea. Automorphisms of finite Abelian groups. American Mathematical Monthly 114:10 (2007), 917-923. Preprint arXiv:math/0605185 [math.GR]
Index entries for sequences related to groups

Cf. A000688, A058160, A058161, A058163.

a(n) = A034382(n) / n. Formula for A034382 is based on the fundamental theorem of finite Abelian groups and the formula given by Hillar and Rhea (2007).

a(16) and a(21) corrected by Max Alekseyev, Sep 12 2019

cp's OEIS Frontend

A275980 Decimal expansion of 2^11213 - 1, the 23rd Mersenne prime A000668(23).

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A275981 Decimal expansion of 2^19937 - 1, the 24th Mersenne prime A000668(24).

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A275982 Decimal expansion of 2^21701 - 1, the 25th Mersenne prime A000668(25).

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A275983 Decimal expansion of 2^23209 - 1, the 26th Mersenne prime A000668(26).

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A275984 Decimal expansion of 2^44497 - 1, the 27th Mersenne prime A000668(27).

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A322435 Number of pairs of factorizations of n into factors > 1 where no factor of the second divides any factor of the first.

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A322437 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one divides any factor of the other.

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