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Search: a326149 -id:a326149
Displaying 1-10 of 19 results found.
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A057568
Number of partitions of n where n divides the product of the parts.
+10
30
1, 1, 1, 2, 1, 2, 1, 6, 5, 5, 1, 22, 1, 11, 23, 80, 1, 113, 1, 150, 85, 45, 1, 737, 226, 84, 809, 726, 1, 1787, 1, 4261, 735, 260, 1925, 9567, 1, 437, 1877, 16402, 1, 14630, 1, 9861, 33057, 1152, 1, 102082, 19393, 57330, 10159, 30706, 1, 207706, 47927, 200652
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000 (terms n=1..73 from Antti Karttunen)
EXAMPLE
From Gus Wiseman, Jul 04 2019: (Start)
The a(1) = 1 through a(9) = 5 partitions are the following. The Heinz numbers of these partitions are given by A326149.
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(22) (321) (44) (63)
(422) (333)
(2222) (3321)
(4211) (33111)
(22211)
(End)
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0,
`if`(t=1, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, min(i, n-i), t/igcd(i, t)))))
end:
a:= n-> `if`(isprime(n), 1, b(n$3)):
seq(a(n), n=1..70); # Alois P. Heinz, Dec 20 2017
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Divisible[Times@@#, n]&]], {n, 20}] (* Gus Wiseman, Jul 04 2019 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 1, 1, 0], If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, Min[i, n - i], t/GCD[i, t]]]]];
a[n_] := If[PrimeQ[n], 1, b[n, n, n]];
Array[a, 70] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)
PROG
(Scheme)
;; This is a naive algorithm that scans over all partitions of each n. For fold_over_partitions_of see A000793.
(define (A057568 n) (let ((z (list 0))) (fold_over_partitions_of n 1 * (lambda (partprod) (if (zero? (modulo partprod n)) (set-car! z (+ 1 (car z)))))) (car z)))
;; Antti Karttunen, Dec 20 2017
CROSSREFS
Cf. A028422, A057567, A096276, A113309, A114324, A318950, A319000, A319005, A326149, A326152.
KEYWORD
nonn
AUTHOR
Leroy Quet, Oct 04 2000
EXTENSIONS
More terms from James A. Sellers, Oct 09 2000
STATUS
approved
A330953
Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by their sum of primes of parts.
+10
19
1, 2, 1, 2, 1, 3, 3, 4, 6, 3, 12, 10, 12, 14, 27, 38, 44, 52, 48, 77, 101, 106, 127, 206, 268, 377, 392, 496, 602, 671, 821, 1090, 1318, 1568, 1926, 2260, 2703, 3258, 3942, 4858, 5923, 6891, 8286, 9728, 11676, 13775, 16314, 19749, 23474, 27793, 32989, 38775
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..52.
EXAMPLE
The a(1) = 1 through a(11) = 12 partitions: (A = 10, B = 11):
1 2 3 4 5 6 7 8 9 A B
11 1111 222 3211 431 432 5311 542
321 22111 4211 3321 22111111 5411
11111111 32211 33221
321111 42221
2211111 53111
322211
431111
521111
2222111
3311111
32111111
For example, the partition (3,3,2,2,1) is counted under a(11) because 5*5*3*3*2 = 450 is divisible by 5+5+3+3+2 = 18.
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Divisible[Times@@Prime/@#, Plus@@Prime/@#]&]], {n, 30}]
CROSSREFS
The Heinz numbers of these partitions are given by A036844.
Numbers divisible by the sum of their prime indices are A324851.
Partitions whose product is divisible by their sum are A057568.
Partitions whose Heinz number is divisible by all parts are A330952.
Partitions whose Heinz number is divisible by their product are A324925.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose product is divisible by their sum of primes are A330954.
Cf. A001414, A003963, A056239, A112798, A120383, A326149, A326155, A331378, A331379, A331381, A331383, A331415, A331416, A331417.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2020
STATUS
approved
A331379
Number of integer partitions of n whose sum of primes of parts is divisible by n.
+10
17
1, 1, 1, 1, 1, 1, 2, 4, 6, 7, 7, 7, 9, 11, 18, 24, 33, 39, 44, 51, 55, 66, 83, 106, 121, 145, 167, 193, 232, 253, 300, 342, 427, 469, 557, 628, 729, 846, 936, 1088, 1195, 1450, 1601, 1895, 2097, 2482, 2782, 3220, 3592, 4073, 4641, 5202, 5911, 6494, 7443, 8294
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,7
LINKS
Table of n, a(n) for n=1..56.
EXAMPLE
The a(6) = 1 through a(11) = 7 partitions:
111111 52 53 54 64 641
1111111 62 63 541 5411
521 531 631 6311
11111111 621 5311 53111
5211 6211 62111
111111111 52111 521111
1111111111 11111111111
For example, the partition (5,4,1,1) has sum of primes 11+7+2+2 = 22, which is divisible by 5+4+1+1 = 11, so (5,4,1,1) is counted under a(11).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Divisible[Plus@@Prime/@#, n]&]], {n, 30}]
CROSSREFS
The Heinz numbers of these partitions are given by A331380.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product is equal to their sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A056239, A324850, A330954, A331378, A331382.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 17 2020
STATUS
approved
A331383
Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.
+10
17
0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 1, 4, 2, 2, 2, 4, 2, 3, 4, 1, 3, 4, 5, 0, 3, 3, 1, 6, 2, 1, 5, 4, 2, 3, 4, 2, 2, 3, 1, 5, 2, 3, 4, 6, 5, 2, 7, 1, 3, 5, 3, 4, 2, 5, 5, 4, 7, 3, 6, 4, 4, 2, 4, 4, 3, 9, 4, 3, 5, 3, 5, 4, 4, 4, 3, 7, 4, 2, 8, 2, 3
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
LINKS
Table of n, a(n) for n=1..87.
EXAMPLE
The a(n) partitions for n = 7, 9, 18, 24:
(4,3) (6,3) (12,4,1,1) (19,4,1)
(4,4,1) (11,4,1,1,1) (18,4,1,1)
(8,5,1,1,1,1,1) (9,6,1,1,1,1,1,1,1,1,1)
(4,2,2,2,1,1,1,1,1,1,1,1)
For example, (4,4,1) has sum of primes of parts 7+7+2 = 16 and product of parts 4*4*1 = 16, so is counted under a(9).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Times@@#==Plus@@Prime/@#&]], {n, 30}]
CROSSREFS
The Heinz numbers of these partitions are given by A331384.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product divides their sum of primes are A331381.
Cf. A000040, A001414, A324850, A330954, A331378, A331379, A331382, A331415, A331416.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jan 16 2020
EXTENSIONS
a(71)-a(87) from Robert Price, Apr 10 2020
STATUS
approved
A330954
Number of integer partitions of n whose product is divisible by the sum of primes of their parts.
+10
16
0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 4, 2, 3, 9, 8, 18, 15, 25, 35, 44, 50, 70, 71, 93, 141, 158, 226, 286, 337, 439, 532, 648, 789, 1013, 1261, 1454, 1776, 2176, 2701, 3258, 3823, 4606, 5521, 6613, 7810, 9202, 11074, 13145, 15498, 18413, 21818, 25774, 30481, 35718
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
LINKS
Table of n, a(n) for n=1..54.
EXAMPLE
The a(7) = 1 through a(15) = 8 partitions (empty column not shown):
43 63 541 83 552 6322 4433 5532
441 4222 3332 6411 7411 7322 6522
222211 5222 62221 44321 84111
33221 63311 333222
65111 432222
72221 3322221
433211 32222211
4322111 333111111
322211111
For example, the partition (3,3,2,2,1) has product 3 * 3 * 2 * 2 * 1 = 36 and sum of primes 5 + 5 + 3 + 3 + 2 = 18, and 36 is divisible by 18, so (3,3,2,2,1) is counted under a(11).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Divisible[Times@@#, Plus@@Prime/@#]&]], {n, 30}]
CROSSREFS
The Heinz numbers of these partitions are given by A331378.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Partitions whose sum of primes divides their product of primes are A330953.
Partitions whose sum of primes divides of their product are A331381.
Partitions whose product equals their sum of primes are A331383.
Cf. A000040, A001414, A036844, A056239, A324850, A326149, A330950, A331379, A331382, A331384, A331415, A331416.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2020
STATUS
approved
A331381
Number of integer partitions of n whose sum of primes of parts is divisible by their product of parts.
+10
16
1, 1, 1, 1, 1, 3, 1, 5, 2, 6, 6, 5, 5, 7, 4, 7, 7, 7, 10, 8, 9, 6, 10, 9, 9, 15, 7, 12, 10, 14, 10, 10, 8, 8, 15, 10, 7, 16, 13, 9, 10, 14, 12, 10, 8, 14, 11, 13, 11, 16, 15, 14, 15, 15, 10, 14, 18, 11, 12, 13, 13, 18, 21, 15, 16, 19, 16, 15, 8, 17, 17
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Table of n, a(n) for n=0..70.
EXAMPLE
The a(n) partitions for n = 1, 5, 7, 8, 9, 13, 14:
1 221 43 311111 63 7411 65111
311 511 11111111 441 721111 322211111
11111 3211 711 43111111 311111111111
22111 42111 421111111 11111111111111
1111111 2211111 3211111111
111111111 22111111111
1111111111111
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Divisible[Plus@@Prime/@#, Times@@#]&]], {n, 0, 30}]
CROSSREFS
The Heinz numbers of these partitions are given by A331382.
Numbers divisible by the sum of their prime factors are A036844.
Partitions whose product is divisible by their sum are A057568.
Numbers divisible by the sum of their prime indices are A324851.
Product of prime indices is divisible by sum of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Sum of prime factors is divisible by sum of prime indices: A331380
Partitions whose product is equal to their sum of primes are A331383.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379, A331415, A331416.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 16 2020
STATUS
approved
A331378
Numbers whose product of prime indices is divisible by their sum of prime factors.
+10
15
35, 65, 95, 98, 154, 189, 297, 324, 363, 364, 375, 450, 476, 585, 623, 702, 763, 765, 791, 812, 826, 918, 938, 994, 1036, 1064, 1106, 1144, 1148, 1162, 1197, 1225, 1287, 1288, 1300, 1305, 1309, 1449, 1470, 1484, 1517, 1566, 1593, 1665, 1708, 1710, 1736, 1769
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
The sequence of terms together with their prime indices begins:
35: {3,4}
65: {3,6}
95: {3,8}
98: {1,4,4}
154: {1,4,5}
189: {2,2,2,4}
297: {2,2,2,5}
324: {1,1,2,2,2,2}
363: {2,5,5}
364: {1,1,4,6}
375: {2,3,3,3}
450: {1,2,2,3,3}
476: {1,1,4,7}
585: {2,2,3,6}
623: {4,24}
702: {1,2,2,2,6}
763: {4,29}
765: {2,2,3,7}
791: {4,30}
812: {1,1,4,10}
For example, 450 = prime(1)*prime(2)*prime(2)*prime(3)*prime(3) has prime indices {1,2,2,3,3} and prime factors {2,3,3,5,5}, and since 36 is divisible by 18, 450 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 1000], Divisible[Times@@primeMS[#], Total[Prime/@primeMS[#]]]&]
CROSSREFS
These are the Heinz numbers of the partitions counted by A330954.
Numbers divisible by the sum of their prime factors are A036844.
Numbers divisible by the sum of their prime indices are A324851.
Sum of prime indices divides product of prime indices: A326149.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose product divides their sum of primes are A331381.
Product of prime indices equals sum of prime factors: A331384.
Cf. A000040, A001414, A056239, A057568, A324850, A330953, A331379, A331381, A331382, A331383.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2020
STATUS
approved
A326151
Numbers whose product of prime indices is twice their sum of prime indices.
+10
11
49, 63, 65, 81, 150, 154, 190, 198, 364, 468, 580, 840, 952, 1080, 1224, 1480, 2128, 2288, 2736, 3440, 5152, 5280, 6624, 8480, 9408, 10816, 12096, 12992, 15552, 16704, 19520, 24960, 26752, 27776, 35712, 44800, 45440, 56576, 57600, 66304, 85248, 101120, 118272
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The only squarefree terms are 65, 154, and 190. See A326157 for a proof.
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.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose product of parts is twice their sum of parts. The enumeration of these partitions by sum is given by A326152.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 80 terms from Jinyuan Wang)
EXAMPLE
The sequence of terms together with their prime indices begins:
49: {4,4}
63: {2,2,4}
65: {3,6}
81: {2,2,2,2}
150: {1,2,3,3}
154: {1,4,5}
190: {1,3,8}
198: {1,2,2,5}
364: {1,1,4,6}
468: {1,1,2,2,6}
580: {1,1,3,10}
840: {1,1,1,2,3,4}
952: {1,1,1,4,7}
1080: {1,1,1,2,2,2,3}
1224: {1,1,1,2,2,7}
1480: {1,1,1,3,12}
2128: {1,1,1,1,4,8}
2288: {1,1,1,1,5,6}
2736: {1,1,1,1,2,2,8}
3440: {1,1,1,1,3,14}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Times@@primeMS[#]==2*Plus@@primeMS[#]&]
PROG
(PARI) is(k) = {my(f=factor(k)); for(i=1, #f~, f[i, 1]=primepi(f[i, 1])); factorback(f)==2*sum(i=1, #f~, f[i, 2]*f[i, 1]); } \\ Jinyuan Wang, Jun 27 2020
CROSSREFS
Satisfies A003963(a(n)) = 2 * A056239(a(n)).
Cf. A112798, A301987, A325037, A325038, A325044, A326149, A326152, A326153/A326154, A326156, A326157.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 09 2019
STATUS
approved
A326153
Numerator of the product of prime indices of n divided by the sum of prime indices of n, n > 1.
+10
11
1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 6, 1, 1, 4, 1, 3, 4, 5, 1, 2, 3, 6, 4, 2, 1, 1, 1, 1, 10, 7, 12, 2, 1, 8, 3, 1, 1, 8, 1, 5, 12, 9, 1, 1, 2, 9, 14, 3, 1, 8, 15, 4, 8, 10, 1, 6, 1, 11, 2, 1, 2, 5, 1, 7, 18, 3, 1, 4, 1, 12, 9, 4, 20, 4, 1, 3, 2, 13, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,5
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.
LINKS
Table of n, a(n) for n=2..84.
EXAMPLE
The sequence A326153/A326154 begins: 1, 1, 1/2, 1, 2/3, 1, 1/3, 1, 3/4, 1, 1/2, 1, 4/5, 6/5, 1/4, 1, 4/5, 1, 3/5, 4/3, 5/6, 1, 2/5, 3/2, 6/7, 4/3.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Numerator[Times@@primeMS[n]/Plus@@primeMS[n]], {n, 2, 100}]
CROSSREFS
Numerator of A003963(n)/A056239(n).
Positions of ones are A326155 without the first term.
Cf. A057567, A057568, A112798, A301987, A325037, A325042, A325044, A326149, A326150, A326151, A326154, A326156, A326158.
KEYWORD
nonn,frac
AUTHOR
Gus Wiseman, Jun 09 2019
STATUS
approved
A326154
Denominator of the product of prime indices of n divided by the sum of prime indices of n, n > 1.
+10
11
1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 5, 5, 4, 1, 5, 1, 5, 3, 6, 1, 5, 2, 7, 3, 3, 1, 1, 1, 5, 7, 8, 7, 3, 1, 9, 2, 2, 1, 7, 1, 7, 7, 10, 1, 3, 1, 7, 9, 4, 1, 7, 8, 7, 5, 11, 1, 7, 1, 12, 1, 6, 1, 4, 1, 9, 11, 2, 1, 7, 1, 13, 4, 5, 9, 3, 1, 7, 1, 14, 1, 1, 10
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,3
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.
LINKS
Table of n, a(n) for n=2..85.
EXAMPLE
The sequence A326153/A326154 begins: 1, 1, 1/2, 1, 2/3, 1, 1/3, 1, 3/4, 1, 1/2, 1, 4/5, 6/5, 1/4, 1, 4/5, 1, 3/5, 4/3, 5/6, 1, 2/5, 3/2, 6/7, 4/3.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Denominator[Times@@primeMS[n]/Plus@@primeMS[n]], {n, 2, 100}]
CROSSREFS
Denominator of A003963(n)/A056239(n).
Positions of ones are A326149.
Cf. A057567, A057568, A112798, A301987, A325037, A325042, A325044, A326150, A326151, A326153, A326155, A326156, A326158.
KEYWORD
nonn,frac
AUTHOR
Gus Wiseman, Jun 09 2019
STATUS
approved
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