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.

Showing 1-3 of 3 results.

A338552 Non-powers of primes whose prime indices have a common divisor > 1.

Original entry on oeis.org

21, 39, 57, 63, 65, 87, 91, 111, 115, 117, 129, 133, 147, 159, 171, 183, 185, 189, 203, 213, 235, 237, 247, 259, 261, 267, 273, 299, 301, 303, 305, 319, 321, 325, 333, 339, 351, 365, 371, 377, 387, 393, 399, 417, 427, 441, 445, 453, 477, 481, 489, 497, 507
Offset: 1

Views

Author

Gus Wiseman, Nov 03 2020

Keywords

Comments

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.
Also Heinz numbers of non-constant, non-relatively prime partitions. 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 sequence of terms together with their prime indices begins:
     21: {2,4}      183: {2,18}       305: {3,18}
     39: {2,6}      185: {3,12}       319: {5,10}
     57: {2,8}      189: {2,2,2,4}    321: {2,28}
     63: {2,2,4}    203: {4,10}       325: {3,3,6}
     65: {3,6}      213: {2,20}       333: {2,2,12}
     87: {2,10}     235: {3,15}       339: {2,30}
     91: {4,6}      237: {2,22}       351: {2,2,2,6}
    111: {2,12}     247: {6,8}        365: {3,21}
    115: {3,9}      259: {4,12}       371: {4,16}
    117: {2,2,6}    261: {2,2,10}     377: {6,10}
    129: {2,14}     267: {2,24}       387: {2,2,14}
    133: {4,8}      273: {2,4,6}      393: {2,32}
    147: {2,4,4}    299: {6,9}        399: {2,4,8}
    159: {2,16}     301: {4,14}       417: {2,34}
    171: {2,2,8}    303: {2,26}       427: {4,18}

Crossrefs

A318978 allows prime powers, counted by A018783, with complement A289509.
A327685 allows nonprime prime powers.
A338330 is the coprime instead of relatively prime version.
A338554 counts the partitions with these Heinz numbers.
A338555 is the complement.
A000740 counts relatively prime compositions.
A000961 lists powers of primes, with complement A024619.
A051424 counts pairwise coprime or singleton partitions.
A108572 counts nontrivial periodic partitions, with Heinz numbers A001597.
A291166 ranks relatively prime compositions, with complement A291165.
A302696 gives the Heinz numbers of pairwise coprime partitions.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000005, A000837, A007916, A056239, A112798, A289508, A302569, A302796, A318716, A327658, A328867, A328677, A338331, A338553.

Programs

  • Mathematica
    Select[Range[100],!(#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1)&]

Formula

Equals A024619 /\ A318978.
Complement of A000961 \/ A289509.

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.

A338554 Number of non-constant integer partitions of n whose parts have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 5, 0, 9, 0, 13, 6, 18, 0, 33, 0, 40, 14, 54, 0, 87, 5, 99, 27, 133, 0, 211, 0, 226, 55, 295, 18, 443, 0, 488, 100, 637, 0, 912, 0, 1000, 198, 1253, 0, 1775, 13, 1988, 296, 2434, 0, 3358, 59, 3728, 489, 4563, 0, 6241, 0, 6840, 814
Offset: 0

Views

Author

Gus Wiseman, Nov 07 2020

Keywords

Examples

			The a(6) = 2 through a(15) = 6 partitions (empty columns indicated by dots, A = 10, B = 11, C = 12):
  (42)  .  (62)   (63)  (64)    .  (84)     .  (86)      (96)
           (422)        (82)       (93)        (A4)      (A5)
                        (442)      (A2)        (C2)      (C3)
                        (622)      (633)       (644)     (663)
                        (4222)     (642)       (662)     (933)
                                   (822)       (842)     (6333)
                                   (4422)      (A22)
                                   (6222)      (4442)
                                   (42222)     (6422)
                                               (8222)
                                               (44222)
                                               (62222)
                                               (422222)

Crossrefs

A046022 lists positions of zeros.
A082023(n) - A059841(n) is the 2-part version, n > 2.
A303280(n) - 1 is the strict case (n > 1).
A338552 lists the Heinz numbers of these partitions.
A338553 counts the complement, with Heinz numbers A338555.
A000005 counts constant partitions, with Heinz numbers A000961.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A018783 counts non-relatively prime partitions (ordered: A178472), with Heinz numbers A318978.
A282750 counts relatively prime partitions by sum and length.
Cf. A000010, A008284, A051424, A082024, A289508, A302698, A304712.

Programs

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

Formula

For n > 0, a(n) = A018783(n) - A000005(n) + 1.
Showing 1-3 of 3 results.

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