cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A338553 Number of integer partitions of n that are either constant or relatively prime.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 20, 29, 37, 56, 68, 101, 122, 170, 213, 297, 352, 490, 587, 778, 948, 1255, 1488, 1953, 2337, 2983, 3585, 4565, 5393, 6842, 8123, 10088, 12015, 14865, 17534, 21637, 25527, 31085, 36701, 44583, 52262, 63261, 74175, 88936, 104305, 124754
Offset: 0

Views

Author

Gus Wiseman, Nov 03 2020

Keywords

Comments

The Heinz numbers of these partitions are given by A338555 = A000961 \/ A289509. 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.

Examples

			The a(1) = 1 through a(7) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (51)      (52)
                    (211)   (221)    (222)     (61)
                    (1111)  (311)    (321)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Crossrefs

A023022(n) + A059841(n) is the 2-part version.
A078374(n) + 1 is the strict case (n > 1).
A338554 counts the complement, with Heinz numbers A338552.
A338555 gives the Heinz numbers of these partitions.
A000005 counts constant partitions, with Heinz numbers A000961.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A282750 counts relatively prime partitions by sum and length.
Cf. A000010, A007360, A008284, A023023, A051424, A101271, A101391, A302698, A304712, A327516, A337664.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],SameQ@@#||GCD@@#==1&]],{n,0,30}]

Formula

For n > 0, a(n) = A000005(n) + A000837(n) - 1.

A335241 Numbers whose prime indices are not pairwise coprime, where a singleton is not coprime unless it is {1}.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 18, 19, 21, 23, 25, 27, 29, 31, 36, 37, 39, 41, 42, 43, 45, 47, 49, 50, 53, 54, 57, 59, 61, 63, 65, 67, 71, 72, 73, 75, 78, 79, 81, 83, 84, 87, 89, 90, 91, 97, 98, 99, 100, 101, 103, 105, 107, 108, 109, 111, 113, 114, 115, 117, 121
Offset: 1

Views

Author

Gus Wiseman, May 30 2020

Keywords

Comments

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (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.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}          31: {11}          61: {18}
    3: {2}         36: {1,1,2,2}     63: {2,2,4}
    5: {3}         37: {12}          65: {3,6}
    7: {4}         39: {2,6}         67: {19}
    9: {2,2}       41: {13}          71: {20}
   11: {5}         42: {1,2,4}       72: {1,1,1,2,2}
   13: {6}         43: {14}          73: {21}
   17: {7}         45: {2,2,3}       75: {2,3,3}
   18: {1,2,2}     47: {15}          78: {1,2,6}
   19: {8}         49: {4,4}         79: {22}
   21: {2,4}       50: {1,3,3}       81: {2,2,2,2}
   23: {9}         53: {16}          83: {23}
   25: {3,3}       54: {1,2,2,2}     84: {1,1,2,4}
   27: {2,2,2}     57: {2,8}         87: {2,10}
   29: {10}        59: {17}          89: {24}

Crossrefs

The complement is A302696.
The version for relatively prime instead of coprime is A318978.
The version for standard compositions is A335239.
These are the Heinz numbers of the partitions counted by A335240.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Primes and numbers with pairwise coprime prime indices are A302569.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime standard composition numbers are A333227.
Cf. A007360, A101268, A326674, A327516, A333228, A335236, A335237, A335238.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!CoprimeQ@@primeMS[#]&]

A366750 Number of strict integer partitions of n into odd parts with a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 3, 1, 0, 2, 0, 1, 3, 1, 0, 3, 2, 1, 4, 1, 1, 5, 0, 1, 5, 1, 2, 5, 1, 1, 5, 2, 2, 6, 0, 1, 9, 1, 0, 9, 0, 3, 9, 1, 1, 9, 5, 1, 11, 1, 0, 15, 1, 2, 13, 1, 5, 14, 0, 1, 18
Offset: 0

Views

Author

Gus Wiseman, Nov 01 2023

Keywords

Examples

			The a(n) partitions for n = 3, 24, 30, 42, 45, 57, 60:
  (3)  (15,9)  (21,9)  (33,9)   (45)       (57)       (51,9)
       (21,3)  (25,5)  (35,7)   (33,9,3)   (45,9,3)   (55,5)
               (27,3)  (39,3)   (21,15,9)  (27,21,9)  (57,3)
                       (27,15)  (25,15,5)  (33,15,9)  (33,27)
                                (27,15,3)  (33,21,3)  (35,25)
                                           (39,15,3)  (39,21)
                                                      (45,15)
                                                      (27,21,9,3)
                                                      (33,15,9,3)

Crossrefs

This is the case of A000700 with a common divisor.
Including evens gives A303280.
The complement is counted by A366844, non-strict version A366843.
The non-strict version is A366852, with evens A018783.
A000041 counts integer partitions, strict A000009 (also into odds).
A051424 counts pairwise coprime partitions, for odd parts A366853.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
Cf. A000837, A007360, A055922, A066208, A078374, A302697, A337485, A366842, A366845, A366848.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#>1&]], {n,0,30}]
  • Python
    from math import gcd
from sympy.utilities.iterables import partitions
def A366750(n): return sum(1 for p in partitions(n) if all(d==1 for d in p.values()) and all(d&1 for d in p) and gcd(*p)>1) # Chai Wah Wu, Nov 02 2023

Extensions

More terms from Chai Wah Wu, Nov 02 2023

A336620 Numbers that are not a product of elements of A304711.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 42, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 78, 79, 81, 83, 87, 89, 91, 97, 101, 103, 105, 107, 109, 111, 113, 114, 115, 117, 121, 125, 126, 127, 129, 130, 131, 133, 137, 139, 147, 149
Offset: 1

Views

Author

Gus Wiseman, Aug 02 2020

Keywords

Comments

A304711 lists numbers whose distinct prime indices are pairwise coprime.
The first term divisible by 4 is a(421) = 1092.

Examples

			The sequence of terms together with their prime indices begins:
      3: {2}         39: {2,6}       78: {1,2,6}
      5: {3}         41: {13}        79: {22}
      7: {4}         42: {1,2,4}     81: {2,2,2,2}
      9: {2,2}       43: {14}        83: {23}
     11: {5}         47: {15}        87: {2,10}
     13: {6}         49: {4,4}       89: {24}
     17: {7}         53: {16}        91: {4,6}
     19: {8}         57: {2,8}       97: {25}
     21: {2,4}       59: {17}       101: {26}
     23: {9}         61: {18}       103: {27}
     25: {3,3}       63: {2,2,4}    105: {2,3,4}
     27: {2,2,2}     65: {3,6}      107: {28}
     29: {10}        67: {19}       109: {29}
     31: {11}        71: {20}       111: {2,12}
     37: {12}        73: {21}       113: {30}

Links

Crossrefs

A336426 is the version for superprimorials, with complement A181818.
A336497 is the version for superfactorials, with complement A336496.
A336735 is the complement.
A000837 counts relatively prime partitions, with strict case A007360.
A001055 counts factorizations.
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A007916, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A335241, A336424, A336568, A336736.

Programs

  • Mathematica
    nn=100;
dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[nn],facsusing[dat,#]=={}&]

A336735 Products of elements of A304711.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106
Offset: 1

Views

Author

Gus Wiseman, Aug 02 2020

Keywords

Comments

A304711 lists numbers whose distinct prime indices are pairwise coprime.
First differs from A304711 in having 84.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}            28: {1,1,4}         52: {1,1,6}
      2: {1}           30: {1,2,3}         54: {1,2,2,2}
      4: {1,1}         32: {1,1,1,1,1}     55: {3,5}
      6: {1,2}         33: {2,5}           56: {1,1,1,4}
      8: {1,1,1}       34: {1,7}           58: {1,10}
     10: {1,3}         35: {3,4}           60: {1,1,2,3}
     12: {1,1,2}       36: {1,1,2,2}       62: {1,11}
     14: {1,4}         38: {1,8}           64: {1,1,1,1,1,1}
     15: {2,3}         40: {1,1,1,3}       66: {1,2,5}
     16: {1,1,1,1}     44: {1,1,5}         68: {1,1,7}
     18: {1,2,2}       45: {2,2,3}         69: {2,9}
     20: {1,1,3}       46: {1,9}           70: {1,3,4}
     22: {1,5}         48: {1,1,1,1,2}     72: {1,1,1,2,2}
     24: {1,1,1,2}     50: {1,3,3}         74: {1,12}
     26: {1,6}         51: {2,7}           75: {2,3,3}

Links

Crossrefs

A181818 is the version for superprimorials, with complement A336426.
A336496 is the version for superfactorials, with complement A336497.
A336620 is the complement.
A000837 counts relatively prime partitions, with strict case A007360.
A001055 counts factorizations.
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A001221, A007360, A007916, A056239, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A336424, A336497, A336736.

Programs

  • Mathematica
    nn=100;
dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[nn],facsusing[dat,#]!={}&]

A340267 Maximum LCM of partitions of n into pairwise coprime parts that are >= 2.

Original entry on oeis.org

2, 3, 4, 6, 6, 12, 15, 20, 30, 30, 60, 42, 84, 105, 140, 210, 210, 420, 280, 330, 360, 840, 504, 1260, 1155, 1540, 2310, 2520, 4620, 3080, 5460, 3960, 9240, 5544, 13860, 6930, 16380, 15015, 27720, 30030, 32760, 60060, 40040, 45045, 51480, 120120, 72072, 180180
Offset: 2

Views

Author

Fausto A. C. Cariboni, Jan 02 2021

Keywords

Comments

a(n) <= A123131(n).

Examples

			For n=22 we have a(22) = 360 since 22 = 5 + 8 + 9 and lcm([5, 8, 9]) = 360.
Note a(22) = 360 < A123131(22) = 420.

Links

Crossrefs

Cf. A007359, A007360, A123131.

Programs

  • PARI
    isok(p) = {for (i=1, #p, for (j=i+1, #p, if (gcd(p[i], p[j]) > 1, return(0)););); return(1);}
a(n) = {my(x=1); forpart(p=n, if ((vecmin(p)>=2) && isok(p), x = max(x, lcm(Vec(p))));); x;} \\ Michel Marcus, Jan 03 2021
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