[go: up one dir, main page]
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a324851 -id:a324851
Displaying 1-10 of 18 results found. page 1 2
     Sort: relevance | references | number | modified | created      Format: long | short | data
A330950 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by n. +10
27
1, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 22, 15, 30, 42, 77, 42, 101, 56, 176, 176, 231, 135, 490, 490, 490, 792, 1002, 490, 1575, 627, 3010, 2436, 2436, 3718, 5604, 1958, 4565, 6842, 12310, 3718, 14883, 4565, 21637, 26015, 17977, 8349, 53174, 44583, 63261 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
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..50.
EXAMPLE
The a(1) = 1 through a(10) = 11 partitions:
1 11 21 211 32 321 43 5111 522 631
1111 311 2211 421 32111 3222 3331
21111 4111 41111 4221 4321
221111 22221 5311
311111 32211 32221
2111111 222111 33211
11111111 2211111 43111
322111
331111
3211111
31111111
For example, the Heinz number of (3,2) is 15, which is divisible by 5, so (3,2) is counted under a(5).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Divisible[Times@@Prime/@#, n]&]], {n, 20}]
CROSSREFS
The Heinz numbers of these partitions are given by 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.
Cf. A056239, A112798, A196050, A324850, A324924, A330953, A330954, A331379, A331381, A331383, A331384.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2020
STATUS
approved
A036844 Numbers k such that k / sopfr(k) is an integer, where sopfr = sum-of-prime-factors, A001414. +10
26
2, 3, 4, 5, 7, 11, 13, 16, 17, 19, 23, 27, 29, 30, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 70, 71, 72, 73, 79, 83, 84, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 149, 150, 151, 157, 163, 167, 173, 179, 180, 181, 191, 193, 197, 199, 211, 220, 223 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Union of A046346 and the primes. - T. D. Noe, Feb 20 2007
These are the Heinz numbers of the partitions counted by A330953. - Gus Wiseman, Jan 17 2020
Alladi and Erdős (1977) noted that sopfr(k) = k if k is a prime or k = 4. They called the terms for which k/sopfr(k) > 1 "special numbers", and proved that there are infinitely many such terms that are squarefree. - Amiram Eldar, Nov 02 2020
REFERENCES
Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring-2000.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Mohan Lal, Iterates of a number-theoretic function, Math. Comp., Vol. 23, No. 105 (1969), pp. 181-183.
FORMULA
A238525(a(n)) = 0. - Reinhard Zumkeller, Jul 21 2014
EXAMPLE
a(12) = 27 because sopfr(27) = 3 + 3 + 3 = 9 and 27 is divisible by 9.
MATHEMATICA
Select[Range[2, 224], Divisible[#, Plus @@ Times @@@ FactorInteger[#]] &] (* Jayanta Basu, Aug 13 2013 *)
PROG
(PARI) is_A036844(n)=n>1 && !(n%A001414(n)) \\ M. F. Hasler, Mar 01 2014
(Haskell)
a036844 n = a036844_list !! (n-1)
a036844_list = filter ((== 0). a238525) [2..]
-- Reinhard Zumkeller, Jul 21 2014
CROSSREFS
sopfr(n) is defined in A001414.
The version for prime indices instead of prime factors is A324851.
Partitions whose Heinz number is divisible by their sum: A330950.
Partitions whose Heinz number is divisible by their sum of primes: A330953.
Partitions whose product is divisible by their sum of primes: A330954.
Partitions whose product divides their sum of primes: A331381.
Product of prime indices is divisible by sum of prime factors: A331378.
Sum of prime factors is divisible by sum of prime indices: A331380.
Product of prime indices equals sum of prime factors: A331384.
Cf. A056239, A112798, A120383, A238525, A331379, A331382, A331383.
KEYWORD
nonn
AUTHOR
Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 09 2002
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
A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length. +10
12
1, 1, 2, 1, 2, 3, 3, 2, 4, 5, 6, 6, 7, 8, 11, 10, 13, 17, 18, 21, 24, 27, 30, 35, 39, 46, 53, 61, 68, 79, 87, 97, 110, 123, 139, 157, 175, 196, 222, 247, 278, 312, 347, 385, 433, 476, 531, 586, 651, 720, 800, 883, 979, 1085, 1200, 1325, 1464, 1614, 1777 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..59.
FindStat, St000010: The length of the partition.
FindStat, St000147: The largest part of an integer partition.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.
EXAMPLE
The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):
1 2 3 4 5 6 7 8 9 A B C D E F G
21 41 42 43 62 63 64 65 84 85 86 87 A6
321 61 81 82 83 A2 A3 A4 A5 C4
621 631 A1 642 C1 C2 C3 E2
4321 632 651 643 653 E1 943
641 921 652 932 654 952
931 941 942 961
8321 951 C31
C21 8431
8421 8521
54321
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Divisible[Max@@#, Length[#]]&]], {n, 30}]
CROSSREFS
Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict version is A168659 (A340609/A340610).
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A096401 counts strict partition with length equal to minimum.
A102627 counts strict partitions with length dividing sum.
A326842 counts partitions whose length and parts all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340829 counts strict partitions with Heinz number divisible by sum.
A340830 counts strict partitions with all parts divisible by length.
Cf. A003114, A006141, A064174, A117409, A143773 (A316428), A200750, A326843 (A326837), A330950 (A324851).
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 01 2021
STATUS
approved
A340830 Number of strict integer partitions of n such that every part is a multiple of the number of parts. +10
11
1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 4, 1, 6, 1, 5, 2, 6, 1, 8, 1, 7, 4, 7, 1, 12, 1, 8, 6, 9, 1, 16, 1, 10, 9, 11, 1, 21, 1, 12, 13, 12, 1, 28, 1, 13, 17, 16, 1, 33, 1, 19, 22, 15, 1, 45, 1, 16, 28, 25, 1, 47, 1, 28, 34, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
Table of n, a(n) for n=1..70.
FORMULA
a(n) = Sum_{d|n} A008289(n/d, d).
EXAMPLE
The a(n) partitions for n = 1, 6, 10, 14, 18, 20, 24, 26, 30:
1 6 10 14 18 20 24 26 30
4,2 6,4 8,6 10,8 12,8 16,8 18,8 22,8
8,2 10,4 12,6 14,6 18,6 20,6 24,6
12,2 14,4 16,4 20,4 22,4 26,4
16,2 18,2 22,2 24,2 28,2
9,6,3 14,10 14,12 16,14
12,9,3 16,10 18,12
15,6,3 20,10
15,9,6
18,9,3
21,6,3
15,12,3
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&And@@IntegerQ/@(#/Length[#])&]], {n, 30}]
CROSSREFS
Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A143773 (A316428).
The case where length divides sum also is A340827.
The version for factorizations is A340851.
Factorization of this type are counted by A340853.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions whose length/max divide sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340828 counts strict partitions with length divisible by maximum.
A340829 counts strict partitions with Heinz number divisible by sum.
Cf. A114638, A168659, A326641, A326843 (A326837), A326849, A326852 (A326838), A330950 (A324851), A340852.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 02 2021
STATUS
approved
page 1 2

Search completed in 0.014 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 29 06:09 EDT 2024. Contains 375510 sequences. (Running on oeis4.)