A364531
Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77
Offset: 1
A299701 counts distinct subset-sums of prime indices.
A363260 counts partitions disjoint from differences, complement
A364467.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]=={}&]
A364464
Number of strict integer partitions of n where no part is the difference of two consecutive parts.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 2, 4, 4, 6, 5, 8, 9, 12, 13, 16, 16, 21, 23, 29, 34, 38, 41, 49, 57, 64, 73, 86, 95, 110, 120, 135, 160, 171, 197, 219, 247, 277, 312, 342, 386, 431, 476, 527, 598, 640, 727, 796, 893, 966, 1097, 1178, 1327, 1435, 1602, 1740, 1945, 2084, 2337
Offset: 0
The strict partition y = (9,5,3,1) has differences (4,2,2), and these are disjoint from the parts, so y is counted under a(18).
The a(1) = 1 through a(9) = 6 strict partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(3,1) (3,2) (5,1) (4,3) (5,3) (5,4)
(4,1) (5,2) (6,2) (7,2)
(6,1) (7,1) (8,1)
(4,3,2)
(5,3,1)
For length instead of differences we have
A240861, non-strict
A229816.
For all differences of pairs of elements we have
A364346, for subsets
A007865.
For subsets instead of strict partitions we have
A364463, complement
A364466.
A320347 counts strict partitions w/ distinct differences, non-strict
A325325.
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Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]=={}&]],{n,0,15}]
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from collections import Counter
from sympy.utilities.iterables import partitions
def A364464(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), filter(lambda p:max(p[1].values(),default=1)==1,partitions(n,size=True))) if set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023
A364466
Number of subsets of {1..n} where some element is a difference of two consecutive elements.
Original entry on oeis.org
0, 0, 1, 2, 6, 14, 34, 74, 164, 345, 734, 1523, 3161, 6488, 13302, 27104, 55150, 111823, 226443, 457586, 923721, 1862183, 3751130, 7549354, 15184291, 30521675, 61322711, 123151315, 247230601, 496158486, 995447739, 1996668494, 4004044396, 8027966324, 16092990132, 32255168125
Offset: 0
The a(0) = 0 through a(5) = 14 subsets:
. . {1,2} {1,2} {1,2} {1,2}
{1,2,3} {2,4} {2,4}
{1,2,3} {1,2,3}
{1,2,4} {1,2,4}
{1,3,4} {1,2,5}
{1,2,3,4} {1,3,4}
{1,4,5}
{2,3,5}
{2,4,5}
{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{1,2,3,4,5}
The complement is counted by
A364463.
For subset-sums instead of differences we have
A364534, complement
A325864.
A325325 counts partitions with all distinct differences, strict
A320347.
-
Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]!={}&]],{n,0,10}]
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from itertools import combinations
def A364466(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if not set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023
A364536
Number of strict integer partitions of n where some part is a difference of two consecutive parts.
Original entry on oeis.org
0, 0, 0, 1, 0, 0, 2, 1, 2, 2, 5, 4, 6, 6, 9, 11, 16, 17, 23, 25, 30, 38, 48, 55, 65, 78, 92, 106, 127, 146, 176, 205, 230, 277, 315, 366, 421, 483, 552, 640, 727, 829, 950, 1083, 1218, 1408, 1577, 1794, 2017, 2298, 2561, 2919, 3255, 3685, 4116, 4638, 5163
Offset: 0
The a(3) = 1 through a(15) = 11 partitions (A = 10, B = 11, C = 12):
21 . . 42 421 431 63 532 542 84 742 743 A5
321 521 621 541 632 642 841 752 843
631 821 651 A21 761 942
721 5321 921 5431 842 C21
4321 5421 6421 B21 6432
6321 7321 6431 6531
6521 7431
7421 7521
8321 8421
9321
54321
A325325 counts partitions with distinct first-differences, strict
A320347.
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Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]!={}&]],{n,0,30}]
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from collections import Counter
from sympy.utilities.iterables import partitions
def A364536(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), filter(lambda p:max(p[1].values(),default=1)==1,partitions(n,size=True))) if not set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023
A364537
Heinz numbers of integer partitions where some part is the difference of two consecutive parts.
Original entry on oeis.org
6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 65, 66, 70, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 130, 132, 133, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 195, 198, 204, 210, 216, 222, 228, 231, 234, 240, 246, 252, 258
Offset: 1
The partition {3,4,5,7} with Heinz number 6545 has first differences (1,1,2) so is not in the sequence.
The terms together with their prime indices begin:
6: {1,2}
12: {1,1,2}
18: {1,2,2}
21: {2,4}
24: {1,1,1,2}
30: {1,2,3}
36: {1,1,2,2}
42: {1,2,4}
48: {1,1,1,1,2}
54: {1,2,2,2}
60: {1,1,2,3}
63: {2,2,4}
65: {3,6}
66: {1,2,5}
70: {1,3,4}
72: {1,1,1,2,2}
78: {1,2,6}
84: {1,1,2,4}
90: {1,2,2,3}
96: {1,1,1,1,1,2}
For all differences of pairs the complement is
A364347, counted by
A364345.
Subsets of {1..n} of this type are counted by
A364466, complement
A364463.
A325325 counts partitions with distinct first differences.
Cf.
A002865,
A025065,
A093971,
A108917,
A196723,
A229816,
A236912,
A237113,
A237667,
A320347,
A326083.
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prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#],Differences[prix[#]]]!={}&]
A364673
Number of (necessarily strict) integer partitions of n containing all of their own first differences.
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 2, 5, 2, 2, 4, 2, 3, 6, 4, 4, 8, 4, 4, 10, 8, 7, 8, 13, 9, 15, 12, 13, 17, 20, 15, 31, 24, 27, 32, 33, 32, 50, 42, 45, 53, 61, 61, 85, 76, 86, 101, 108, 118, 137, 141, 147, 179, 184, 196, 222, 244, 257, 295, 324, 348, 380, 433
Offset: 0
The partition y = (12,6,3,2,1) has differences (6,3,1,1), and {1,3,6} is a subset of {1,2,3,6,12}, so y is counted under a(24).
The a(n) partitions for n = 1, 3, 6, 12, 15, 18, 21:
(1) (3) (6) (12) (15) (18) (21)
(2,1) (4,2) (8,4) (10,5) (12,6) (14,7)
(3,2,1) (6,4,2) (8,4,2,1) (9,6,3) (12,6,3)
(5,4,2,1) (5,4,3,2,1) (6,5,4,2,1) (8,6,4,2,1)
(6,3,2,1) (7,5,3,2,1) (9,5,4,2,1)
(8,4,3,2,1) (9,6,3,2,1)
(10,5,3,2,1)
(6,5,4,3,2,1)
Containing all differences:
A007862.
For submultisets instead of subsets we have
A364675.
A236912 counts sum-free partitions w/o re-used parts, complement
A237113.
A325325 counts partitions with distinct first differences.
Cf.
A002865,
A025065,
A196723,
A229816,
A237667,
A320347,
A363225,
A364272,
A364345,
A364463,
A364537,
A370386.
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Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SubsetQ[#,-Differences[#]]&]],{n,0,30}]
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from collections import Counter
def A364673_list(maxn):
count = Counter()
for i in range(maxn//3):
A,f,i = [[(i+1, )]],0,0
while f == 0:
A.append([])
for j in A[i]:
for k in j:
x = j + (j[-1] + k, )
y = sum(x)
if y <= maxn:
A[i+1].append(x)
count.update({y})
if len(A[i+1]) < 1: f += 1
i += 1
return [count[z]+1 for z in range(maxn+1)] # John Tyler Rascoe, Mar 09 2024
A364671
Number of subsets of {1..n} containing all of their own first differences.
Original entry on oeis.org
1, 2, 4, 6, 10, 14, 23, 34, 58, 96, 171, 302, 565, 1041, 1969, 3719, 7105, 13544, 25999, 49852, 95949, 184658, 356129, 687068, 1327540, 2566295, 4966449, 9617306, 18640098, 36150918, 70166056, 136272548, 264844111, 515036040, 1002211421, 1951345157, 3801569113
Offset: 0
The subset {1,2,4,5,10,14} has differences (1,2,1,5,4) so is counted under a(14).
The a(0) = 1 through a(5) = 14 subsets:
{} {} {} {} {} {}
{1} {1} {1} {1} {1}
{2} {2} {2} {2}
{1,2} {3} {3} {3}
{1,2} {4} {4}
{1,2,3} {1,2} {5}
{2,4} {1,2}
{1,2,3} {2,4}
{1,2,4} {1,2,3}
{1,2,3,4} {1,2,4}
{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,2,3,4,5}
For differences of all strict pairs we have
A054519, for partitions
A007862.
For "disjoint" instead of "subset" we have
A364463, partitions
A363260.
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Table[Length[Select[Subsets[Range[n]], SubsetQ[#,Differences[#]]&]], {n,0,10}]
A364672
Number of subsets of {1..n} not containing all of their own first differences.
Original entry on oeis.org
0, 0, 0, 2, 6, 18, 41, 94, 198, 416, 853, 1746, 3531, 7151, 14415, 29049, 58431, 117528, 236145, 474436, 952627, 1912494, 3838175, 7701540, 15449676, 30988137, 62142415, 124600422, 249795358, 500719994, 1003575768, 2011211100, 4030123185, 8074898552, 16177657763, 32408393211, 64917907623
Offset: 0
The a(0) = 0 through a(5) = 18 subsets:
. . . {1,3} {1,3} {1,3}
{2,3} {1,4} {1,4}
{2,3} {1,5}
{3,4} {2,3}
{1,3,4} {2,5}
{2,3,4} {3,4}
{3,5}
{4,5}
{1,2,5}
{1,3,4}
{1,3,5}
{1,4,5}
{2,3,4}
{2,3,5}
{2,4,5}
{3,4,5}
{1,3,4,5}
{2,3,4,5}
For disjunction instead of containment we have
A364463, partitions
A363260.
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Table[Length[Select[Subsets[Range[n]],!SubsetQ[#,Differences[#]]&]],{n,0,10}]
A364675
Number of integer partitions of n whose nonzero first differences are a submultiset of the parts.
Original entry on oeis.org
1, 1, 2, 3, 4, 4, 7, 7, 10, 12, 15, 15, 26, 25, 35, 45, 55, 60, 86, 94, 126, 150, 186, 216, 288, 328, 407, 493, 610, 699, 896, 1030, 1269, 1500, 1816, 2130, 2620, 3029, 3654, 4300, 5165, 5984, 7222, 8368, 9976, 11637, 13771, 15960, 18978, 21896, 25815, 29915
Offset: 0
The partition y = (3,2,1,1) has first differences (1,1,0), and (1,1) is a submultiset of y, so y is counted under a(7).
The a(1) = 1 through a(8) = 10 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (221) (33) (421) (44)
(111) (211) (2111) (42) (2221) (422)
(1111) (11111) (222) (3211) (2222)
(2211) (22111) (4211)
(21111) (211111) (22211)
(111111) (1111111) (32111)
(221111)
(2111111)
(11111111)
The strict case (no differences of 0) appears to be
A154402.
Starting with the distinct parts gives
A342337.
For subsets instead of submultisets we have
A364673.
A325325 counts partitions with distinct first differences.
Cf.
A002865,
A007862,
A108917,
A229816,
A237667,
A237668,
A320347,
A363225,
A364272,
A364345,
A364466.
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submultQ[cap_,fat_] := And@@Function[i,Count[fat,i] >= Count[cap,i]] /@ Union[List@@cap];
Table[Length[Select[IntegerPartitions[n], submultQ[Differences[Union[#]],#]&]], {n,0,30}]
A364674
Number of integer partitions of n containing all of their own nonzero first differences.
Original entry on oeis.org
1, 1, 2, 3, 4, 4, 8, 7, 11, 13, 17, 18, 32, 30, 44, 54, 70, 78, 114, 125, 171, 205, 257, 302, 408, 464, 592, 711, 892, 1042, 1330, 1543, 1925, 2279, 2787, 3291, 4061, 4727, 5753, 6792, 8197, 9583, 11593, 13505, 16198, 18965, 22548, 26290, 31340, 36363, 43046
Offset: 0
The partition (10,5,3,3,2,1) has nonzero differences (5,2,1,1) so is counted under a(24).
The a(1) = 1 through a(9) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(11) (21) (22) (221) (33) (421) (44) (63)
(111) (211) (2111) (42) (2221) (422) (333)
(1111) (11111) (222) (3211) (2222) (3321)
(321) (22111) (3221) (4221)
(2211) (211111) (4211) (22221)
(21111) (1111111) (22211) (32211)
(111111) (32111) (42111)
(221111) (222111)
(2111111) (321111)
(11111111) (2211111)
(21111111)
(111111111)
For subsets instead of partitions we have
A364671, complement
A364672.
The strict case (no differences of 0) is counted by
A364673.
For submultisets instead of subsets we have
A364675.
A236912 counts sum-free partitions w/o re-used parts, complement
A237113.
A325325 counts partitions with distinct first differences.
-
Table[Length[Select[IntegerPartitions[n], SubsetQ[#,Differences[Union[#]]]&]],{n,0,30}]
Showing 1-10 of 11 results.
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