A034891 -id:A034891 - cp's OEIS frontend

A379309 Number of strict integer partitions of n with a unique squarefree part.

0, 1, 1, 1, 0, 2, 2, 2, 0, 2, 4, 4, 1, 4, 7, 7, 2, 6, 8, 11, 4, 9, 13, 17, 7, 13, 20, 22, 13, 20, 29, 33, 21, 29, 40, 47, 27, 41, 56, 64, 42, 59, 77, 85, 60, 74, 104, 115, 83, 101, 141, 155, 113, 138, 179, 206, 156, 183, 236, 272, 212, 239, 309, 343, 282, 315

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(9) = 2 through a(15) = 7 partitions:
  (5,4)  (10)   (11)   (9,3)  (13)     (14)     (15)
  (8,1)  (6,4)  (7,4)         (8,5)    (8,6)    (8,7)
         (8,2)  (8,3)         (12,1)   (9,5)    (9,6)
         (9,1)  (9,2)         (8,4,1)  (10,4)   (11,4)
                                       (12,2)   (12,3)
                                       (8,4,2)  (8,4,3)
                                       (9,4,1)  (9,4,2)

If all parts are squarefree we have A087188, non-strict A073576 (ranks A302478).
If no parts are squarefree we have A256012, non-strict A114374 (ranks A379307).
For composite instead of squarefree we have A379303, non-strict A379302 (ranks A379301).
For prime instead of squarefree we have A379305, non-strict A379304 (ranks A331915).
The non-strict version is A379308, ranks A379316.
For old prime instead of squarefree we have A379315, non-strict A379314 (ranks A379312).
Ranked by A379316 /\ A005117 = squarefree positions of 1 in A379306.
A000041 counts integer partitions, strict A000009.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A377038 gives k-th differences of squarefree numbers.
A379310 counts nonsquarefree prime indices.
Cf. A000586, A000607, A002095, A023895, A034891, A036497, A072284, A073247, A096258, A204389, A377430.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?SquareFreeQ]==1&]],{n,0,30}]

lista(nn) = my(r=1, s=0); for(k=1, nn, if(issquarefree(k), s+=x^k, r*=1+x^k)); concat(0, Vec(r*s+O(x^(1+nn)))); \\ Jinyuan Wang, Feb 21 2025

More terms from Jinyuan Wang, Feb 21 2025

A009490 Number of distinct orders of permutations of n objects; number of nonisomorphic cyclic subgroups of symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 9, 11, 14, 16, 20, 23, 27, 31, 35, 43, 47, 55, 61, 70, 78, 88, 98, 111, 123, 136, 152, 168, 187, 204, 225, 248, 271, 296, 325, 356, 387, 418, 455, 495, 537, 581, 629, 678, 732, 787, 851, 918, 986, 1056, 1133, 1217, 1307, 1399, 1498, 1600, 1708, 1823

Offset: 0

David W. Wilson

Also number of different LCM's of partitions of n.

a(n) <= A023893(n), which counts the nonisomorphic Abelian subgroups of S_n. - M. F. Hasler, May 24 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
L. Elliott, A. Levine, and J. D. Mitchell, Counting monogenic monoids and inverse monoids, arXiv:2303.12387 [math.GR], 2023.
Index entries for sequences related to lcm's

Cf. A051613 (first differences), A000792, A000793, A034891, A051625 (all cyclic subgroups), A256067.

b:= proc(n,i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, b(n, i-1)+
      add(b(n-p^j, i-1), j=1..ilog[p](n)))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 15 2013

Table[ Length[ Union[ Apply[ LCM, Partitions[ n ], 1 ] ] ], {n, 30} ]
f[n_] := Length@ Union[LCM @@@ IntegerPartitions@ n]; Array[f, 60, 0]
(* Caution, the following is Extremely Slow and Resource Intensive *) CoefficientList[ Series[ Expand[ Product[1 + Sum[x^(Prime@ i^k), {k, 4}], {i, 10}]/(1 - x)], {x, 0, 30}], x]
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0 || i<1, 1, b[n, i-1]+Sum[b[n-p^j, i-1], {j, 1, Log[p, n]}]]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)

/* compute David W. Wilson's g.f., needs <1 sec for 1000 terms */
N=1000;  x='x+O('x^N); /* N terms */
gf=1; /* generating function */
{ forprime(p=2,N,
    sm = 1;  pp=p;  /* sum;  prime power */
    while ( ppJoerg Arndt, Jan 19 2011 */

a(n) = Sum_{k=0..n} b(k), where b(k) is the number of partitions of k into distinct prime power parts (1 excluded) (A051613). - Vladeta Jovovic

G.f.: (Product_{p prime} (1 + Sum_{k >= 1} x^(p^k))) / (1-x). - David W. Wilson, Apr 19 2000

A379312 Positive integers whose prime indices include a unique 1 or prime number.

Original entry on oeis.org

2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214

Offset: 1

Gus Wiseman, Dec 28 2024

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.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   14: {1,4}
   17: {7}
   21: {2,4}
   26: {1,6}
   31: {11}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   41: {13}
   46: {1,9}
   57: {2,8}
   58: {1,10}
   59: {17}
   65: {3,6}
   67: {19}
   69: {2,9}
   74: {1,12}
   77: {4,5}

These "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of ones in A379311, see A379313.
Partitions of this type are counted by A379314, strict A379315.
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
Cf. A038550, A073445, A087436, A173390, A379300, A379301, A379302.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]

A379316 Positive integers whose prime indices include a unique squarefree number.

Original entry on oeis.org

2, 3, 5, 11, 13, 14, 17, 21, 29, 31, 35, 38, 41, 43, 46, 47, 57, 59, 67, 69, 73, 74, 77, 79, 83, 91, 95, 98, 101, 106, 109, 111, 113, 115, 119, 122, 127, 137, 139, 142, 147, 149, 157, 159, 163, 167, 178, 179, 181, 183, 185, 191, 194, 199, 203, 206, 209, 211

Offset: 1

Gus Wiseman, Dec 29 2024

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.

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
   11: {5}
   13: {6}
   14: {1,4}
   17: {7}
   21: {2,4}
   29: {10}
   31: {11}
   35: {3,4}
   38: {1,8}
   41: {13}
   43: {14}
   46: {1,9}

For all squarefree parts we have A302478, zeros of A379310.
Positions of 1 in A379306.
For no squarefree parts we have A379307, counted by A114374, strict A256012.
Partitions of this type are counted by A379308, strict A379309.
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 prime or 1, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A072284, A087436, A112929, A377430.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],SquareFreeQ]]==1&]

A341945 Number of partitions of n into 2 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 0, 2, 1, 3, 1, 2, 0, 3, 1, 3, 1, 3, 0, 4, 1, 3, 0, 2, 0, 4, 1, 3, 1, 4, 0, 4, 0, 3, 1, 3, 0, 5, 1, 4, 1, 4, 0, 6, 1, 4, 0, 3, 0, 6, 1, 3, 0, 4, 0, 7, 1, 4, 1, 5, 0, 6, 0, 3, 1, 5, 0, 7, 1, 6, 1, 5, 0, 7, 0, 5, 1, 5, 0, 9, 1, 5, 0, 4, 0, 10

Offset: 2

Ilya Gutkovskiy, Feb 24 2021

Number of partitions of n into 2 noncomposite numbers, A008578. - Antti Karttunen, Dec 13 2021

Antti Karttunen, Table of n, a(n) for n = 2..20000

Cf. A001031, A008578, A034891, A061358, A341946, A341947, A341948, A341949, A341950, A341951.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 3)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 2):
seq(a(n), n=2..90);  # Alois P. Heinz, Feb 24 2021

a[n_] := If[2 == n, 1, Module[{s = 0}, For[p = 2, True, p = NextPrime[p], If[p > n-p, Return[s + Boole[PrimeQ[n-1]]], s += Boole[PrimeQ[n-p]]]]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Jan 03 2022, after Antti Karttunen *)

A341945(n) = if(2==n,1,my(s=0); forprime(p=2,,if(p>(n-p), return(s+isprime(n-1)), s += isprime(n-p)))); \\ Antti Karttunen, Dec 13 2021

A341947 Number of partitions of n into 4 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 5, 4, 6, 4, 7, 4, 9, 6, 10, 6, 12, 6, 14, 8, 15, 8, 18, 9, 21, 10, 20, 9, 23, 10, 26, 12, 27, 12, 31, 13, 34, 13, 33, 14, 39, 15, 42, 16, 43, 17, 48, 18, 53, 19, 52, 19, 58, 20, 61, 20, 61, 20, 68, 23, 73, 23, 73, 26, 82, 26, 84, 23, 84, 27, 92, 28, 98

Offset: 4

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A008578, A034891, A259194, A341945, A341946, A341948, A341949, A341950, A341951.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 5)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 4):
seq(a(n), n=4..76);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 5}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 4];
Table[a[n], {n, 4, 76}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)

A341948 Number of partitions of n into 5 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 5, 8, 6, 10, 7, 12, 9, 15, 10, 18, 12, 21, 14, 25, 15, 29, 18, 33, 21, 37, 20, 41, 23, 46, 26, 51, 27, 58, 31, 63, 34, 68, 33, 77, 39, 83, 42, 90, 43, 101, 48, 107, 53, 116, 52, 128, 58, 134, 61, 142, 61, 157, 68, 165, 73, 176, 73, 194, 82, 201, 84, 211

Offset: 5

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Cf. A008578, A034891, A259195, A341945, A341946, A341947, A341949, A341950, A341951.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 6)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 5):
seq(a(n), n=5..73);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 6}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 5];
Table[a[n], {n, 5, 73}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)

A341949 Number of partitions of n into 6 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 6, 9, 8, 12, 10, 16, 12, 19, 15, 24, 18, 29, 21, 35, 25, 41, 29, 49, 33, 56, 37, 63, 41, 72, 46, 82, 51, 91, 58, 105, 63, 115, 68, 128, 77, 143, 83, 158, 90, 174, 101, 193, 107, 211, 116, 231, 128, 250, 134, 273, 142, 294, 157, 321, 165, 347, 176, 374

Offset: 6

Ilya Gutkovskiy, Feb 24 2021

nonn

Cf. A008578, A034891, A259196, A341945, A341946, A341947, A341948, A341950, A341951.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 7)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 6):
seq(a(n), n=6..70);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 7}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 6];
Table[a[n], {n, 6, 70}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)

A341950 Number of partitions of n into 7 primes (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 10, 9, 14, 12, 19, 16, 23, 19, 30, 24, 37, 29, 44, 35, 55, 41, 65, 49, 75, 56, 89, 63, 102, 72, 116, 82, 134, 91, 153, 105, 171, 115, 194, 128, 220, 143, 242, 158, 273, 174, 305, 193, 334, 211, 374, 231, 412, 250, 447, 273, 494, 294, 541, 321

Offset: 7

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..10000

Cf. A008578, A034891, A259197, A341945, A341946, A341947, A341948, A341949, A341951.

b:= proc(n, i) option remember; series(`if`(n=0, 1,
     `if`(i<0, 0, (p-> `if`(p>n, 0, x*b(n-p, i)))(
     `if`(i=0, 1, ithprime(i)))+b(n, i-1))), x, 8)
    end:
a:= n-> coeff(b(n, numtheory[pi](n)), x, 7):
seq(a(n), n=7..68);  # Alois P. Heinz, Feb 24 2021

b[n_, i_] := b[n, i] = Series[If[n == 0, 1,
     If[i < 0, 0, Function[p, If[p > n, 0, x*b[n - p, i]]][
     If[i == 0, 1, Prime[i]]] + b[n, i - 1]]], {x, 0, 8}];
a[n_] := Coefficient[b[n, PrimePi[n]], x, 7];
Table[a[n], {n, 7, 68}] (* Jean-François Alcover, Feb 15 2022, after Alois P. Heinz *)

A379300 Number of prime indices of n that are composite.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 25 2024

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.

			The prime indices of 39 are {2,6}, so a(39) = 1.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 3.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Positions of first appearances are A000420.
Positions of zero are A302540, counted by A034891 (strict A036497).
Positions of one are A379301, counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A023895, A113646, A175804, A204389, A320629, A376680, A378373, A378456.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],CompositeQ]],{n,100}]

Totally additive with a(prime(k)) = A066247(k).

cp's OEIS Frontend

A379309 Number of strict integer partitions of n with a unique squarefree part.

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A009490 Number of distinct orders of permutations of n objects; number of nonisomorphic cyclic subgroups of symmetric group S_n.

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A379312 Positive integers whose prime indices include a unique 1 or prime number.

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A379316 Positive integers whose prime indices include a unique squarefree number.

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A341945 Number of partitions of n into 2 primes (counting 1 as a prime).

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A341947 Number of partitions of n into 4 primes (counting 1 as a prime).

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A341948 Number of partitions of n into 5 primes (counting 1 as a prime).

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A341949 Number of partitions of n into 6 primes (counting 1 as a prime).

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A341950 Number of partitions of n into 7 primes (counting 1 as a prime).

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