A101417 -id:A101417 - cp's OEIS frontend

A323092 Number of double-free integer partitions of n.

1, 1, 2, 2, 4, 5, 7, 10, 14, 17, 24, 30, 40, 50, 66, 81, 104, 128, 161, 197, 246, 300, 369, 446, 546, 656, 796, 952, 1148, 1366, 1637, 1940, 2311, 2730, 3234, 3806, 4489, 5262, 6181, 7225, 8454, 9846, 11484, 13335, 15499, 17948, 20796, 24017, 27751, 31970, 36837

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

An integer partition is double-free if no part is twice any other part.

			The a(1) = 1 through a(8) = 14 double-free integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (431)
                                               (4111)     (611)
                                               (31111)    (2222)
                                               (1111111)  (3311)
                                                          (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Double-Free Set

Cf. A018819, A051424, A101417, A120641, A276431, A305148, A323093, A323094.

Table[Length[Select[IntegerPartitions[n],Intersection[#,2*#]=={}&]],{n,30}]

A320340 Heinz numbers of double-free integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Jan 07 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
An integer partition is double-free if no part is twice any other part.
Also numbers n such that if prime(m) divides n then prime(2m) does not divide n, i.e., numbers not divisible by any element of A319613.

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (41), (32), (1111), (7), (8), (311), (51), (9), (33), (61), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Double-Free Set

Cf. A018819, A056239, A087897, A101417, A112798, A120641, A276431, A305148, A319613, A323092, A323093.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[primeMS[#],2*primeMS[#]]=={}&]

A350842 Number of integer partitions of n with no difference -2.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 40, 54, 69, 89, 118, 146, 187, 239, 297, 372, 468, 575, 711, 880, 1075, 1314, 1610, 1947, 2359, 2864, 3438, 4135, 4973, 5936, 7090, 8466, 10044, 11922, 14144, 16698, 19704, 23249, 27306, 32071, 37639, 44019, 51457, 60113

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (221)    (222)     (61)
                            (2111)   (321)     (322)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (211111)
                                               (1111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Heinz number rankings are in parentheses below.
The version for no difference 0 is A000009.
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The version for all differences > -2 is A034296, strict A001227.
The opposite version is A072670.
The version for no difference -1 is A116931 (A319630), strict A003114.
The multiplicative version is A350837 (A350838), strict A350840.
The strict case is A350844.
The complement for quotients is counted by A350846 (A350845).
A000041 = integer partitions.
A027187 = partitions of even length.
A027193 = partitions of odd length (A026424).
A323092 = double-free partitions (A320340), strict A120641.
A325534 = separable partitions (A335433).
A325535 = inseparable partitions (A335448).
A350839 = partitions with a gap and conjugate gap (A350841).
Cf. A000070, A000929, A001511, A003242, A007359, A018819, A040039, A045690, A045691, A101417, A154402, A323093.

Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],-2]&]],{n,0,30}]

A120641 Number of partitions of n into distinct double-free parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 5, 5, 7, 8, 10, 12, 14, 17, 20, 24, 26, 31, 38, 45, 50, 57, 68, 77, 88, 101, 116, 132, 151, 170, 194, 222, 247, 281, 318, 356, 399, 452, 509, 567, 635, 709, 794, 885, 983, 1094, 1222, 1358, 1504, 1671, 1854, 2050, 2264, 2505, 2771, 3060, 3370

Offset: 0

Reinhard Zumkeller, Aug 17 2006

nonn

			a(10) = #{10, 9+1, 8+2, 7+3, 6+4, 5+4+1, 5+3+2} = 7;
a(11) = #{11, 10+1, 9+2, 8+3, 7+4, 7+3+1, 6+5, 6+4+1} = 8.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Double-Free Set

Cf. A000009, A050291, A101417, A303362, A320340, A323092, A323093, A323094.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,2*#]=={}&]],{n,30}] (* Gus Wiseman, Jan 07 2019 *)

a(0)=1 prepended by Alois P. Heinz, Jan 16 2019

A078135 Numbers which cannot be written as a sum of squares > 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 19, 23

Offset: 1

Reinhard Zumkeller, Nov 19 2002

Numbers such that A078134(n)=0.
"Numbers which cannot be written as sum of squares > 1" is equivalent to "Numbers which cannot be written as sum of squares of primes." Equivalently, numbers which can be written as the sum of nonzero squares can also be written as sum of the squares of primes." cf. A090677 = number of ways to partition n into sums of squares of primes. - Jonathan Vos Post, Sep 20 2006
The sequence is finite with a(12)=23 as last member. Proof: When k=a^2+b^2+..., k+4 = 2^2+a^2+b^2+... If k can be written as sum of the squares of primes, k+4 also has this property. As 24,25,26,27 have the property, by induction, all numbers > 23 can be written as sum of squares>1. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Apr 07 2007
Also, numbers which cannot be written as sum of squares of 2 and 3 (see A078137 for the proof). Explicit representation as sum of squares of primes, or rather of squares of 2 and 3, for numbers m>23: we have m=c*2^2+d*3^2, where c:=(floor(m/4) - 2*(m mod 4))>=0, d:=m mod 4. For that, the finiteness of the sequence is proved constructively. - Hieronymus Fischer, Nov 11 2007
Also numbers n such that every integer partition of n contains a squarefree number. For example, 21 does not belong to the sequence because there are integer partitions of 21 containing no squarefree numbers, namely: (12,9), (9,8,4), (9,4,4,4). - Gus Wiseman, Dec 14 2018

Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A000290, A001422, A078137, A078139, A078136, A078129.
Cf. A090677.
Cf. A078134, A078139, A090677, A078137, A134754, A134755.
Cf. A005117, A101417, A320322, A322526, A322546, A322547.

nn=100;
ser=Product[If[SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1 (* Gus Wiseman, Dec 14 2018 *)

A090677(a(n)) = 0. - Jonathan Vos Post, Sep 20 2006 [corrected by Joerg Arndt, Dec 16 2018]

A033183(a(n)) = 0. - Reinhard Zumkeller, Nov 07 2009

A351003 Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 28, 36, 42, 51, 62, 75, 88, 106, 124, 147, 173, 202, 236, 278, 320, 371, 431, 497, 572, 661, 756, 867, 993, 1132, 1291, 1474, 1672, 1898, 2155, 2439, 2756, 3117, 3512, 3957, 4458, 5008, 5624, 6316, 7072, 7919, 8862, 9899

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (411)     (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

The ordered version (compositions) is A027383.
The version for unequal instead of equal is A122135, even-length A351008.
For odd instead of even indices we have A351004, even-length A035363.
Requiring inequalities at odd positions gives A351006, even-length A351007.
The even-length case is A351012.
Cf. A000070, A018819, A088218, A101417, A122129, A350837, A351005.

Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,10}]

A323093 Number of integer partitions of n where no part is 2^k times any other part, for any k > 0.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 9, 12, 13, 18, 23, 29, 37, 49, 55, 71, 84, 104, 126, 153, 185, 221, 261, 317, 375, 446, 523, 623, 721, 854, 994, 1168, 1357, 1579, 1833, 2126, 2455, 2843, 3270, 3766, 4320, 4980, 5687, 6521, 7444, 8498, 9684, 11039, 12540, 14262

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(1) = 1 through a(8) = 12 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (311)    (51)      (52)       (53)
                    (1111)  (11111)  (222)     (61)       (62)
                                     (3111)    (322)      (71)
                                     (111111)  (331)      (332)
                                               (511)      (611)
                                               (31111)    (2222)
                                               (1111111)  (3311)
                                                          (5111)
                                                          (311111)
                                                          (11111111)

Cf. A018819, A101417, A120641, A276431, A305148, A323092, A323094.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[#,IntegerQ[Log[2,#1/#2]]&]&]],{n,30}]

A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 7, 7, 10, 9, 15, 13, 21, 19, 28, 26, 40, 35, 54, 49, 72, 64, 97, 87, 128, 115, 167, 151, 220, 195, 284, 256, 366, 328, 469, 421, 598, 537, 757, 682, 959, 859, 1204, 1085, 1507, 1354, 1880, 1694, 2338, 2104, 2892, 2609, 3574, 3218, 4394

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.

			The a(1) = 1 through a(9) = 7 partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    221    33      331      44        333
              1111  11111  222     22111    332       441
                           2211    1111111  2222      22221
                           111111           3311      33111
                                            221111    2211111
                                            11111111  111111111

The ordered version (compositions) is A016116.
The even-length case is A035363.
A reverse version is A096441, both A349060.
The version for unequal instead of equal is A122129, even-length A351008.
The version for even instead of odd indices is A351003, even-length A351012.
The strict version is A351005, opposite A351006, even-length A035457.
Cf. A000070, A018819, A027383, A088218, A067661, A101417, A122134, A122135, A350842, A351007.

Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

A350837 Number of integer partitions of n with no adjacent parts of quotient 2.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 14, 18, 24, 31, 41, 53, 70, 87, 112, 140, 178, 221, 277, 344, 428, 526, 648, 792, 971, 1180, 1436, 1738, 2103, 2533, 3049, 3660, 4387, 5242, 6259, 7450, 8860, 10511, 12453, 14723, 17387, 20489, 24121, 28343, 33269, 38982, 45632, 53327

Offset: 0

Gus Wiseman, Jan 18 2022

nonn

The first of these partitions that is not double-free (see A323092 for definition) is (4,3,2).

			The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (32)     (33)      (43)
                    (31)    (41)     (51)      (52)
                    (1111)  (311)    (222)     (61)
                            (11111)  (411)     (322)
                                     (3111)    (331)
                                     (111111)  (511)
                                               (4111)
                                               (31111)
                                               (1111111)

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version with quotients >= 2 is A000929, sets A018819.
                           <= 2 is A342094, ranked by A342191.
                           < 2 is A342096, sets A045690, strict A342097.
                           > 2 is A342098, sets A040039.
The sets version (subsets of prescribed maximum) is A045691.
These partitions are ranked by A350838.
The strict case is A350840.
A version for differences is A350842, strict A350844.
The complement is counted by A350846, ranked by A350845.
A000041 = integer partitions.
A116931 = partitions with no successions, ranked by A319630.
A116932 = partitions with differences != 1 or 2, strict A025157.
A323092 = double-free partitions, ranked by A320340.
Cf. A000070, A003000, A003114, `A003242, A051424, `A101417, A120641, A154402, A305148, A323093, A323094, A342095, A350839.

Table[Length[Select[IntegerPartitions[n], FreeQ[Divide@@@Partition[#,2,1],2]&]],{n,0,15}]

A351012 Number of even-length integer partitions y of n such that y_i = y_{i+1} for all even i.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 5, 6, 9, 10, 13, 16, 21, 24, 29, 35, 43, 50, 60, 70, 83, 97, 113, 132, 156, 178, 206, 239, 275, 316, 365, 416, 477, 545, 620, 706, 806, 912, 1034, 1173, 1326, 1496, 1691, 1902, 2141, 2410, 2704, 3034, 3406, 3808, 4261, 4765, 5317, 5932, 6617

Offset: 0

Gus Wiseman, Feb 03 2022

nonn

			The a(2) = 1 through a(8) = 9 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (3111)    (2221)    (71)
                              (111111)  (4111)    (2222)
                                        (211111)  (3221)
                                                  (5111)
                                                  (311111)
                                                  (11111111)

The ordered version (compositions) is A027383(n-2).
For odd instead of even indices we have A035363, any length A351004.
The version for unequal parts appears to be A122134, any length A122135.
This is the even-length case of A351003.
Requiring inequalities at odd positions gives A351007, any length A351006.
Cf. A000070, A018819, A035457, A088218, A101417, A122129, A350837, A351005, A351008.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]

cp's OEIS Frontend

A323092 Number of double-free integer partitions of n.

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A320340 Heinz numbers of double-free integer partitions.

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A350842 Number of integer partitions of n with no difference -2.

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A120641 Number of partitions of n into distinct double-free parts.

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A078135 Numbers which cannot be written as a sum of squares > 1.

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A351003 Number of integer partitions y of n such that y_i = y_{i+1} for all even i.

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A323093 Number of integer partitions of n where no part is 2^k times any other part, for any k > 0.

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A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

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A350837 Number of integer partitions of n with no adjacent parts of quotient 2.

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