Displaying 1-10 of 13 results found.
Number of alternately co-strong integer partitions of n.
+10
16
1, 1, 2, 3, 5, 6, 11, 13, 19, 25, 35, 42, 61, 74, 98, 122, 161, 194, 254, 304, 388, 472, 589, 700, 878, 1044, 1278, 1525, 1851, 2182, 2651, 3113, 3735, 4389, 5231, 6106, 7278, 8464, 9995, 11631, 13680, 15831, 18602, 21463, 25068, 28927, 33654, 38671, 44942, 51514
COMMENTS
A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
Also the number of alternately strong reversed integer partitions of n.
EXAMPLE
The a(1) = 1 through a(7) = 13 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) (321) (421)
(411) (511)
(2211) (3211)
(3111) (4111)
(21111) (22111)
(111111) (31111)
(211111)
(1111111)
For example, starting with the partition y = (3,2,2,1,1) and repeatedly taking run-lengths and reversing gives (3,2,2,1,1) -> (2,2,1) -> (1,2), which is not weakly decreasing, so y is not alternately co-strong. On the other hand, we have (3,3,2,2,1,1,1) -> (3,2,2) -> (2,1) -> (1,1) -> (2) -> (1), so (3,3,2,2,1,1,1) is counted under a(13).
MATHEMATICA
tniQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[IntegerPartitions[n], tniQ]], {n, 0, 30}]
CROSSREFS
Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A317081, A317086, A317245, A317258. The Heinz numbers of these partitions are given by A317257. The total (instead of alternating) version is A332275. Dominates A332289 (the normal version). The generalization to compositions is A332338. The case of reversed partitions is (also) A332339.
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Mar 08 2020
Heinz numbers of widely totally strongly normal integer partitions.
+10
13
1, 2, 4, 6, 8, 16, 18, 30, 32, 64, 128, 210, 256, 450, 512, 1024, 2048, 2250, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 131072, 262144, 510510, 524288
COMMENTS
An integer partition is widely totally strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which are themselves a widely totally strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
This sequence is closed under A304660, so there are infinitely many terms that are not powers of 2 or primorial numbers.
EXAMPLE
The sequence of all widely totally strongly normal integer partitions together with their Heinz numbers begins:
1: ()
2: (1)
4: (1,1)
6: (2,1)
8: (1,1,1)
16: (1,1,1,1)
18: (2,2,1)
30: (3,2,1)
32: (1,1,1,1,1)
64: (1,1,1,1,1,1)
128: (1,1,1,1,1,1,1)
210: (4,3,2,1)
256: (1,1,1,1,1,1,1,1)
450: (3,3,2,2,1)
512: (1,1,1,1,1,1,1,1,1)
1024: (1,1,1,1,1,1,1,1,1,1)
2048: (1,1,1,1,1,1,1,1,1,1,1)
2250: (3,3,3,2,2,1)
2310: (5,4,3,2,1)
4096: (1,1,1,1,1,1,1,1,1,1,1,1)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], GreaterEqual@@Length/@Split[ptn], totnQ[Length/@Split[ptn]]]];
Select[Range[10000], totnQ[Reverse[primeMS[#]]]&]
CROSSREFS
The case of reversed partitions is (also) A332293. Heinz numbers of normal partitions with decreasing run-lengths are A025487. Cf. A055932, A056239, A181819, A242031, A317089, A317246, A317257, A317492, A329747, A332277, A332278, A332290, A332292, A332297, A332337.
Number of alternately co-strong compositions of n.
+10
9
1, 1, 2, 4, 7, 12, 24, 39, 72, 125, 224, 387, 697, 1205, 2141, 3736, 6598, 11516, 20331, 35526, 62507, 109436, 192200, 336533, 590582, 1034187
COMMENTS
A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
EXAMPLE
The a(1) = 1 through a(5) = 12 compositions:
(1) (2) (3) (4) (5)
(11) (12) (13) (14)
(21) (22) (23)
(111) (31) (32)
(112) (41)
(121) (113)
(1111) (131)
(212)
(221)
(1112)
(1121)
(11111)
For example, starting with the composition y = (1,6,2,2,1,1,1,1) and repeatedly taking run-lengths and reversing gives (1,6,2,2,1,1,1,1) -> (4,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2). All of these have weakly increasing run-lengths and the last is a singleton, so y is counted under a(15).
MATHEMATICA
tniQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], tniQ]], {n, 0, 10}]
CROSSREFS
The recursive (rather than alternating) version is A332274. The total (rather than alternating) version is (also) A332274. The strong version is this same sequence.
The case of reversed partitions is A332339. The normal version is A332340(n) + 1 for n > 1. Cf. A001462, A100883, A181819, A182850, A316496, A317257, A329744, A329746, A332275, A332292, A332296.
Number of alternately co-strong reversed integer partitions of n.
+10
9
1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 18, 20, 29, 28, 40, 45, 54, 59, 82, 81, 108, 118, 141, 154, 204, 204, 255, 285, 339, 363, 458, 471, 580, 632, 741, 806, 983, 1015, 1225, 1341, 1562, 1667, 2003, 2107, 2491, 2712, 3101, 3344, 3962, 4182, 4860, 5270, 6022, 6482
COMMENTS
A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
Also the number of alternately strong integer partitions of n.
EXAMPLE
The a(1) = 1 through a(8) = 12 reversed partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (12) (13) (14) (15) (16) (17)
(111) (22) (23) (24) (25) (26)
(1111) (122) (33) (34) (35)
(11111) (123) (124) (44)
(222) (133) (125)
(1122) (1222) (134)
(111111) (1111111) (233)
(1133)
(2222)
(11222)
(11111111)
For example, starting with the composition y = (1,2,3,3,4,4,4) and repeatedly taking run-lengths and reversing gives (1,2,3,3,4,4,4) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2) -> (1). All of these have weakly increasing run-lengths and the last is equal to (1), so y is counted under a(21).
MATHEMATICA
tniQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Sort/@IntegerPartitions[n], tniQ]], {n, 0, 30}]
CROSSREFS
The total (instead of alternating) version is A316496. Alternately strong partitions are A317256. The case of ordinary (not reversed) partitions is (also) A317256. The generalization to compositions is A332338.
Heinz numbers of superperiodic integer partitions.
+10
8
2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233
COMMENTS
First differs from A061345 at a(1) = 2 and next at a(98) = 441. A number n is in the sequence iff n = 2 or the prime indices of n have a common divisor > 1 and the Heinz number of the multiset of prime multiplicities of n, namely A181819(n), is already in the sequence. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
EXAMPLE
The sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (2), (3), (4), (2,2), (5), (6), (7), (8), (9), (3,3), (2,2,2), (10), (11), (12), (13), (14), (15), (4,4), (16), (17), (18), (19), (20), (21), (22), (2,2,2,2).
MATHEMATICA
supperQ[n_]:=Or[n==2, And[GCD@@PrimePi/@FactorInteger[n][[All, 1]]>1, supperQ[Times@@Prime/@FactorInteger[n][[All, 2]]]]];
Select[Range[500], supperQ]
CROSSREFS
Cf. A001597, A056239, A072774, A098859, A181819, A182850, A289509, A296150, A298748, A304464, A305732, A317246, A317257, A319149, A319152, A319157.
Heinz numbers of widely alternately co-strongly normal integer partitions.
+10
8
1, 2, 4, 6, 8, 12, 16, 30, 32, 60, 64, 128, 210, 256, 360, 512, 1024, 2048, 2310, 2520, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288
COMMENTS
An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
This sequence is closed under A181821, so there are infinitely many terms that are not powers of 2 or primorial numbers.
EXAMPLE
The sequence of all widely alternately co-strongly normal integer partitions together with their Heinz numbers begins:
1: ()
2: (1)
4: (1,1)
6: (2,1)
8: (1,1,1)
12: (2,1,1)
16: (1,1,1,1)
30: (3,2,1)
32: (1,1,1,1,1)
60: (3,2,1,1)
64: (1,1,1,1,1,1)
128: (1,1,1,1,1,1,1)
210: (4,3,2,1)
256: (1,1,1,1,1,1,1,1)
360: (3,2,2,1,1,1)
512: (1,1,1,1,1,1,1,1,1)
1024: (1,1,1,1,1,1,1,1,1,1)
2048: (1,1,1,1,1,1,1,1,1,1,1)
2310: (5,4,3,2,1)
2520: (4,3,2,2,1,1,1)
For example, starting with y = (4,3,2,2,1,1,1), which has Heinz number 2520, and repeatedly taking run-lengths and reversing gives (4,3,2,2,1,1,1) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal with weakly increasing run-lengths and the last is all 1's, so 2520 belongs to the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], LessEqual@@Length/@Split[ptn], totnQ[Reverse[Length/@Split[ptn]]]]];
Select[Range[10000], totnQ[Reverse[primeMS[#]]]&]
CROSSREFS
The non-co-strong version is A332276. The enumeration of these partitions by sum is A332289. The total (rather than alternating) version is A332293. Cf. A055932, A056239, A100883, A133808, A181819, A317089, A317090, A317246, A317257, A317492, A329747, A332292, A332340.
Heinz numbers of widely totally co-strongly normal integer partitions.
+10
7
1, 2, 4, 6, 8, 12, 16, 30, 32, 64, 128, 180, 210, 256, 360, 512, 1024, 2048, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288
COMMENTS
An integer partition is widely totally co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which are themselves a widely totally co-strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
16: {1,1,1,1}
30: {1,2,3}
32: {1,1,1,1,1}
64: {1,1,1,1,1,1}
128: {1,1,1,1,1,1,1}
180: {1,1,2,2,3}
210: {1,2,3,4}
256: {1,1,1,1,1,1,1,1}
360: {1,1,1,2,2,3}
512: {1,1,1,1,1,1,1,1,1}
1024: {1,1,1,1,1,1,1,1,1,1}
2048: {1,1,1,1,1,1,1,1,1,1,1}
2310: {1,2,3,4,5}
4096: {1,1,1,1,1,1,1,1,1,1,1,1}
8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
For example, 180 is the Heinz number of (3,2,2,1,1), with run-lengths (3,2,2,1,1) -> (1,2,2) -> (1,2) -> (1,1). These are all normal with weakly increasing multiplicities and the last is all 1's, so 180 belongs to the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
gnaQ[y_]:=Or[y=={}, Union[y]=={1}, And[normQ[y], LessEqual@@Length/@Split[y], gnaQ[Length/@Split[y]]]];
Select[Range[1000], gnaQ[Reverse[primeMS[#]]]&]
CROSSREFS
The non-co-strong version is A332276. The enumeration of these partitions by sum is A332278. The alternating version is A332290. The case of reversed partitions is (also) A332291. Cf. A000009, A056239, A133808, A182850, A304660, A317089, A317246, A317257, A317492, A329747, A332277, A332289.
Heinz numbers of totally strong integer partitions.
+10
6
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83
COMMENTS
The alternating version first differs from this sequence in having 150 and lacking 450.
An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
EXAMPLE
Starting with (3,3,2,1), which has Heinz number 150, and repeatedly taking run-lengths gives (3,3,2,1) -> (2,1,1) -> (1,2), so 150 is not in the sequence.
Starting with (3,3,2,2,1), which has Heinz number 450, and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2) -> (1), so 450 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
totstrQ[q_]:=Or[q=={}, q=={1}, And[GreaterEqual@@Length/@Split[q], totstrQ[Length/@Split[q]]]];
Select[Range[100], totstrQ[Reverse[primeMS[#]]]&]
CROSSREFS
The enumeration of these partitions by sum is A316496. The widely normal version is A332291. Partitions with weakly decreasing run-lengths are A100882.
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Mar 08 2020
Heinz numbers of integer partitions that are not totally nonincreasing.
+10
5
18, 50, 54, 75, 90, 98, 108, 126, 147, 150, 162, 180, 198, 234, 242, 245, 250, 252, 270, 294, 300, 306, 324, 338, 342, 350, 363, 375, 378, 396, 414, 450, 468, 486, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 594, 600, 605, 612, 630, 648, 650, 666, 684
COMMENTS
An integer partition is totally nonincreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly decreasing and are themselves a totally nonincreasing integer partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
Sequence of all integer partitions that are not totally nonincreasing begins: (221), (331), (2221), (332), (3221), (441), (22211), (4221), (442), (3321), (22221), (32211), (5221), (6221), (551), (443), (3331), (42211), (32221), (4421), (33211), (7221), (222211), (661), (8221), (4331), (552), (3332), (42221), (52211), (9221), (33221).
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
totincQ[q_]:=Or[Length[q]<=1, And[OrderedQ[Length/@Split[q]], totincQ[Reverse[Length/@Split[q]]]]];
Select[Range[1000], !totincQ[Reverse[primeMS[#]]]&]
CROSSREFS
Cf. A056239, A071365, A100883, A112769, A181819, A182850, A242031, A296150, A305733, A317256, A317257.
Nonprime Heinz numbers of superperiodic integer partitions.
+10
3
9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 441, 529, 625, 729, 841, 961, 1331, 1369, 1521, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3249, 3481, 3721, 4225, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7569, 7921, 8281, 9261, 9409, 10201
COMMENTS
A number n is in the sequence iff n = 2 or the prime indices of n have a common divisor > 1 and the Heinz number of the multiset of prime multiplicities of n, namely A181819(n), is already in the sequence. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
EXAMPLE
The sequence of partitions whose Heinz numbers belong to the sequence begins: (22), (33), (222), (44), (2222), (55), (333), (66), (22222), (77), (444), (88), (4422), (99), (3333), (222222).
MATHEMATICA
supperQ[n_]:=Or[n==2, And[GCD@@PrimePi/@FactorInteger[n][[All, 1]]>1, supperQ[Times@@Prime/@FactorInteger[n][[All, 2]]]]];
Select[Range[10000], And[!PrimeQ[#], supperQ[#]]&]
CROSSREFS
Cf. A001597, A056239, A072774, A181819, A182850, A289509, A296150, A298748, A304464, A305732, A317246, A317257, A319149, A319151.
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