A117485 -id:A117485 - cp's OEIS frontend

A307386 Heinz numbers of integer partitions with Durfee square of length 3.

125, 175, 245, 250, 275, 325, 343, 350, 375, 385, 425, 455, 475, 490, 500, 525, 539, 550, 575, 595, 605, 625, 637, 650, 665, 686, 700, 715, 725, 735, 750, 770, 775, 805, 825, 833, 845, 847, 850, 875, 910, 925, 931, 935, 950, 975, 980, 1000, 1001, 1015, 1025

Offset: 1

Gus Wiseman, Apr 06 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The Durfee square of an integer partition is the largest square contained in its Young diagram.

			The sequence of terms together with their prime indices begins:
  125: {3,3,3}
  175: {3,3,4}
  245: {3,4,4}
  250: {1,3,3,3}
  275: {3,3,5}
  325: {3,3,6}
  343: {4,4,4}
  350: {1,3,3,4}
  375: {2,3,3,3}
  385: {3,4,5}
  425: {3,3,7}
  455: {3,4,6}
  475: {3,3,8}
  490: {1,3,4,4}
  500: {1,1,3,3,3}
  525: {2,3,3,4}
  539: {4,4,5}
  550: {1,3,3,5}
  575: {3,3,9}
  595: {3,4,7}

Gus Wiseman, Young diagrams corresponding to the first 36 terms.

Positions of 3 in A257990. The Durfee length 1 case is A093641. The Durfee length 2 case is A325164. The enumeration of Durfee length 2 partitions by sum is given by A006918, while that of Durfee length 3 partitions is given by A117485.

Cf. A056239, A112798, A115994, A252464, A325163, A325170.

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
Select[Range[100],durf[#]==3&]

A330641 a(n) is the number of partitions of n with Durfee square of size <= 3.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 230, 295, 380, 480, 607, 758, 943, 1161, 1426, 1733, 2100, 2525, 3023, 3595, 4261, 5017, 5888, 6874, 7996, 9258, 10687, 12281, 14073, 16066, 18288, 20747, 23478, 26482, 29801, 33442, 37441, 41811, 46596, 51801, 57478, 63639, 70329, 77567

Offset: 0

Omar E. Pol, Dec 22 2019

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A330640.

seq(n)={Vec((1 - x - x^2 + 2*x^4 + x^5 - 2*x^6 - x^7 + x^8 + x^9 - x^11 + x^12) / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2) + O(x*x^n))} \\ Andrew Howroyd, Dec 27 2024

a(n) = A000041(n), 0 <= n <= 15.
a(n) = A330640(n), 0 <= n <= 8.
a(n) = A330640(n) + A117485(n), n >= 9.
a(n) = n + A006918(n-3) + A117485(n), n >= 9.
From Colin Barker, Dec 31 2019: (Start)
G.f.: (1 - x - x^2 + 2*x^4 + x^5 - 2*x^6 - x^7 + x^8 + x^9 - x^11 + x^12) / ((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>12. (End)

A354234 Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 3, 1, 1, 7, 4, 2, 1, 1, 11, 7, 4, 2, 1, 1, 15, 10, 6, 3, 2, 1, 1, 22, 16, 9, 6, 3, 2, 1, 1, 30, 22, 14, 8, 5, 3, 2, 1, 1, 42, 32, 20, 13, 8, 5, 3, 2, 1, 1, 56, 44, 29, 18, 12, 7, 5, 3, 2, 1, 1, 77, 62, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1

Offset: 1

Gus Wiseman, May 22 2022

Also partitions of n with at least one part appearing k or more times. It would be interesting to have a bijective proof of this.

			Triangle begins:
   1
   2  1
   3  1  1
   5  3  1  1
   7  4  2  1  1
  11  7  4  2  1  1
  15 10  6  3  2  1  1
  22 16  9  6  3  2  1  1
  30 22 14  8  5  3  2  1  1
  42 32 20 13  8  5  3  2  1  1
  56 44 29 18 12  7  5  3  2  1  1
  77 62 41 27 17 12  7  5  3  2  1  1
For example, row n = 5 counts the following partitions:
  (5)      (32)    (32)   (41)  (5)
  (32)     (41)    (311)
  (41)     (221)
  (221)    (2111)
  (311)
  (2111)
  (11111)
At least one part appearing k or more times:
  (5)      (221)    (2111)   (11111)  (11111)
  (32)     (311)    (11111)
  (41)     (2111)
  (221)    (11111)
  (311)
  (2111)
  (11111)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

The complement is counted by A061199.
Differences of consecutive terms are A091602.
Column k = 1 is A000041.
Column k = 2 is A047967, ranked by A013929 and A324929.
Column k = 3 is A295341, ranked by A046099 and A354235.
Column k = 4 is A295342.
A000041 counts integer partitions, strict A000009.
A047966 counts uniform partitions.
Cf. A002033, A006918, A064410, A117485, A238394, A238395, A325534.

Table[Length[Select[IntegerPartitions[n],MemberQ[#/k,_?IntegerQ]&]],{n,1,15},{k,1,n}]
- or -
Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]>=k&]],{n,1,15},{k,1,n}]

\\ here P(k,n) is partitions with no part divisible by k as g.f.
P(k,n)={1/prod(i=1, n, 1 - if(i%k, x^i) + O(x*x^n))}
M(n,m=n)={my(p=P(n+1,n)); Mat(vector(m, k, Col(p-P(k,n), -n) ))}
{ my(A=M(12)); for(n=1, #A, print(A[n,1..n])) } \\ Andrew Howroyd, Jan 19 2023

A139672 Convolution of A008619 and A001400.

Original entry on oeis.org

1, 2, 5, 9, 17, 27, 44, 65, 97, 136, 191, 257, 346, 451, 587, 746, 946, 1177, 1461, 1786, 2178, 2623, 3151, 3746, 4443, 5223, 6126, 7131, 8283, 9558, 11007, 12603, 14403, 16377, 18588, 21003, 23692, 26618, 29858, 33372, 37244, 41430, 46022, 50972

Offset: 1

Alford Arnold, Apr 29 2008, May 01 2008

nonn

This is row 21 of a table of values related to Molien series. It is the product of the sequence on row 3 (A008619) with the sequence on row 7 (A001400).
This table may be constructed by moving the rows of table A008284 to prime locations and generating the composite locations by multiplication in a manner similar to the calculation illustrated in the present sequence.
Rows 1 thru 20 and 22 thru 25 are as follows:
A000012 A008619 A000027 A001399 A002620 A001400 A000217 A006918 A000601 A001401 A002623 A001402 A002621 A097701 A000292 A008630 A002624 A008631 A057524 A002622 A008632 A001752 A117485.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -3, 0, -1, 2, 2, -1, 0, -3, 1, 2, -1).

a:= proc(n) local m, r; m:= iquo (n, 12, 'r'); r:= r+1; (19+ (145+ (260+ 15* (r+9)*r+ (405+ 90*r+ 216*m) *m) *m) *m) *m/5+ [0, 1, 2, 5, 9, 17, 27, 44, 65, 97, 136, 191][r]+ [0, 16, 37, 77, 128, 208, 307, 447, 616, 840, 1105, 1441][r]*m/2+ [0, 52, 119, 213, 328, 476, 651, 865, 1112, 1404, 1735, 2117][r]*m^2/2 end: seq (a(n), n=1..50); # Alois P. Heinz, Nov 10 2008

CoefficientList[Series[x/((x^2+x+1)(x^2+1)(x+1)^3 (x-1)^6),{x,0,50}],x] (* or *) LinearRecurrence[{2,1,-3,0,-1,2,2,-1,0,-3,1,2,-1},{0,1,2,5,9,17,27,44,65,97,136,191,257},50] (* Harvey P. Dale, Feb 17 2016 *)

G.f.: x/((x^2+x+1)*(x^2+1)*(x+1)^3*(x-1)^6). - Alois P. Heinz, Nov 10 2008

a(n)= -A049347(n)/27 +(2*n+11)*(6*n^4+132*n^3+914*n^2+2068*n+1055)/69120 -(-1)^n*(51/512+n^2/256+11*n/256+A057077(n)/32 ). - R. J. Mathar, Nov 21 2008

More terms from Alois P. Heinz, Nov 10 2008

Corrected A-number in definition. Added formula. - R. J. Mathar, Nov 21 2008

A330642 a(n) is the number of partitions of n with Durfee square of size <= 4.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1957, 2434, 3005, 3708, 4545, 5568, 6779, 8245, 9974, 12046, 14478, 17372, 20747, 24732, 29360, 34782, 41045, 48337, 56716, 66410, 77498, 90247, 104763, 121366, 140181, 161590, 185755

Offset: 0

Omar E. Pol, Dec 24 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1,-4,4,4,2,0,-10,0,2,4,4,-4,-1,-2,1,2,-1).

Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A117486, A330640, A330641, A330643.

seq(n) = Vec(sum(k=0, 4, x^(k^2)/prod(j=1, k, 1 - x^j)^2) + O(x*x^n)) \\ Andrew Howroyd, Dec 27 2024

a(n) = A000041(n), 0 <= n <= 24.
a(n) = A330641(n), 0 <= n <= 15.
a(n) = A330641(n) + A117486(n-16), n >= 16.
a(n) = n + A006918(n-3) + A117485(n) + A117486(n-16), n >= 16.
From Colin Barker, Jan 01 2020: (Start)
G.f.: (1 - x - x^2 + 3*x^5 - x^7 - 2*x^8 - 2*x^9 + 3*x^10 + x^11 + x^12 - x^13 - 2*x^14 + x^15 + x^17 - x^19 + x^20) / ((1 - x)^8*(1 + x)^4*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) - 4*a(n-5) + 4*a(n-6) + 4*a(n-7) + 2*a(n-8) - 10*a(n-10) + 2*a(n-12) + 4*a(n-13) + 4*a(n-14) - 4*a(n-15) - a(n-16) - 2*a(n-17) + a(n-18) + 2*a(n-19) - a(n-20) for n>20.
(End)
G.f.: Sum_{k=0..4} x^(k^2)/(Product_{j=1..k} (1 - x^j))^2. - Andrew Howroyd, Dec 27 2024

A330643 a(n) is the number of partitions of n with Durfee square of size <= 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17976, 21635, 26010, 31175, 37318, 44547, 53109, 63153, 74996, 88850, 105113, 124078, 146256, 172032, 202056, 236844

Offset: 0

Omar E. Pol, Dec 24 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 1, -2, -1, -2, 0, 2, 6, 2, -3, -6, -5, -2, 3, 12, 3, -2, -5, -6, -3, 2, 6, 2, 0, -2, -1, -2, 1, 2, -1).

Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A117486, A117487, A330640, A330641, A330642.

seq(n) = Vec(sum(k=0, 5, x^(k^2)/prod(j=1, k, 1 - x^j)^2) + O(x*x^n)) \\ Andrew Howroyd, Dec 27 2024

a(n) = A000041(n), 0 <= n <= 35.
a(n) = A330642(n), 0 <= n <= 24.
a(n) = A330642(n) + A117487(n-24), n >= 25.
a(n) = n + A006918(n-3) + A117485(n) + A117486(n-16) + A117487(n-24), n >= 25.
G.f.: Sum_{k=0..5} x^(k^2)/(Product_{j=1..k} (1 - x^j))^2. - Andrew Howroyd, Dec 27 2024

A160647 Self-convolution of sequence A001402.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 108, 179, 284, 445, 676, 1017, 1492, 2168, 3094, 4372, 6088, 8406, 11462, 15509, 20770, 27614, 36390, 47646, 61898, 79939, 102538, 130808, 165864, 209272, 262598, 328008, 407700, 504607, 621760, 763123, 932788, 1136047

Offset: 1

Alford Arnold, May 27 2009

			a(8) = 108 because the eighth antidiagonal of the associated array is 14 11 14 15 15 14 11 14 and sums to 108.

Cf. A117566.

Sixth in a list of sequences related to numeric partitions; earlier sequences are A000027, A006918, A117485, A117486, and A117487.

A160647 := proc(n) coeftayl( convert(1/mul((1-x^j)^2,j=1..6),parfrac,x),x=0,n) ; end: seq(A160647(n),n=0..45) ; # R. J. Mathar, Jun 16 2009

More terms from R. J. Mathar, Jun 16 2009

A307370 Number of integer partitions of n with 2 distinct parts, none appearing more than twice.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 4, 6, 7, 7, 10, 10, 11, 12, 15, 13, 17, 16, 19, 18, 22, 19, 25, 22, 26, 24, 30, 25, 32, 28, 34, 30, 37, 31, 40, 34, 41, 36, 45, 37, 47, 40, 49, 42, 52, 43, 55, 46, 56, 48, 60, 49, 62, 52, 64, 54, 67, 55, 70, 58, 71, 60, 75, 61, 77, 64, 79, 66

Offset: 0

Gus Wiseman, Apr 05 2019

The Heinz numbers of these partitions appear to be given by A296205.

			The a(3) = 1 through a(10) = 10 partitions:
  (21)  (31)   (32)   (42)    (43)   (53)    (54)   (64)
        (211)  (41)   (51)    (52)   (62)    (63)   (73)
               (221)  (411)   (61)   (71)    (72)   (82)
               (311)  (2211)  (322)  (332)   (81)   (91)
                              (331)  (422)   (441)  (433)
                              (511)  (611)   (522)  (442)
                                     (3311)  (711)  (622)
                                                    (811)
                                                    (3322)
                                                    (4411)

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,2,1,0,-1,-1).

Cf. A000041, A000726, A002133, A004709, A006918, A007774, A117485, A296205, A325168.

Cf. A056594, A061347.

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2&&Max@@Length/@Split[#]<=2&]],{n,0,30}]

concat([0,0,0], Vec(x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Apr 08 2019

From Colin Barker, Apr 08 2019: (Start)
G.f.: x^3*(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 4*x^5) / ((1 - x)^2*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = -a(n-1) + a(n-3) + 2*a(n-4) + a(n-5) - a(n-7) - a(n-8) for n>8. (End)
a(n) = (27*n + 3*(n - 7)*(-1)^n - 53 - 6*A056594(n) + 8*A061347(n))/24 for n > 0. - Stefano Spezia, Feb 20 2024

A307515 Heinz numbers of integer partitions with Durfee square of length > 2.

Original entry on oeis.org

125, 175, 245, 250, 275, 325, 343, 350, 375, 385, 425, 455, 475, 490, 500, 525, 539, 550, 575, 595, 605, 625, 637, 650, 665, 686, 700, 715, 725, 735, 750, 770, 775, 805, 825, 833, 845, 847, 850, 875, 910, 925, 931, 935, 950, 975, 980, 1000, 1001, 1015, 1025

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

First differs from A307386 in having 7^4 = 2401.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The Durfee square of an integer partition is the largest square contained in its Young diagram.
The enumeration of these partitions by sum is given by A084835.

			The sequence of terms together with their prime indices begins:
  125: {3,3,3}
  175: {3,3,4}
  245: {3,4,4}
  250: {1,3,3,3}
  275: {3,3,5}
  325: {3,3,6}
  343: {4,4,4}
  350: {1,3,3,4}
  375: {2,3,3,3}
  385: {3,4,5}
  425: {3,3,7}
  455: {3,4,6}
  475: {3,3,8}
  490: {1,3,4,4}
  500: {1,1,3,3,3}
  525: {2,3,3,4}
  539: {4,4,5}
  550: {1,3,3,5}
  575: {3,3,9}
  595: {3,4,7}

Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Gus Wiseman, Table of n, a(n) for n = 1..22485
FindStat, St000183: The side length of the Durfee square of an integer partition
Wikipedia, Durfee square.

Positions of numbers > 2 in A257990.

Cf. A006918, A056239, A084835, A112798, A115994, A117485, A252464, A325163, A325170.

durf[n_]:=Length[Select[Range[PrimeOmega[n]], Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]][[#]]>=#&]];
Select[Range[100], durf[#]>2&]

A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

Original entry on oeis.org

5, 10, 13, 15, 20, 23, 25, 26, 30, 35, 37, 39, 40, 45, 46, 47, 50, 52, 55, 60, 61, 65, 69, 70, 73, 74, 75, 78, 80, 85, 89, 90, 91, 92, 94, 95, 100, 103, 104, 105, 110, 111, 113, 115, 117, 120, 122, 125, 130, 135, 137, 138, 140, 141, 143, 145, 146, 148, 150

Offset: 1

Gus Wiseman, May 23 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    5: {3}
   10: {1,3}
   13: {6}
   15: {2,3}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   26: {1,6}
   30: {1,2,3}
   35: {3,4}
   37: {12}
   39: {2,6}
   40: {1,1,1,3}
   45: {2,2,3}
   46: {1,9}
   47: {15}
   50: {1,3,3}
   52: {1,1,6}
   55: {3,5}
   60: {1,1,2,3}

For 4 instead of 3 we have A046101, counted by A295342.
This sequence ranks the partitions counted by A295341, compositions A335464.
For 2 instead of 3 we have A324929 (and A013929), counted by A047967.
A001222 counts prime factors with multiplicity, distinct A001221.
A004250 counts partitions with some part > 2, compositions A008466.
A004709 lists numbers divisible by no cube, counted by A000726.
A036966 lists 3-full numbers, counted by A100405.
A046099 lists non-cubefree numbers.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A354234 counts partitions of n with at least one part divisible by k.
Cf. A000720, A003963, A008483, A018256, A062739, A064428, A117485, A181819, A339222, A353508.

Select[Range[100],MemberQ[PrimePi/@First/@If[#==1,{},FactorInteger[#]]/3,_?IntegerQ]&]

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A307386 Heinz numbers of integer partitions with Durfee square of length 3.

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A330641 a(n) is the number of partitions of n with Durfee square of size <= 3.

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A354234 Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.

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A139672 Convolution of A008619 and A001400.

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A330642 a(n) is the number of partitions of n with Durfee square of size <= 4.

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A330643 a(n) is the number of partitions of n with Durfee square of size <= 5.

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A160647 Self-convolution of sequence A001402.

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A307370 Number of integer partitions of n with 2 distinct parts, none appearing more than twice.

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