A081306 -id:A081306 - cp's OEIS frontend

A237821 Number of partitions of n such that 2*(least part) <= greatest part.

0, 0, 1, 2, 4, 7, 11, 16, 25, 35, 48, 68, 92, 123, 164, 216, 282, 367, 471, 604, 769, 975, 1225, 1542, 1924, 2395, 2968, 3669, 4514, 5547, 6781, 8280, 10071, 12229, 14796, 17881, 21537, 25902, 31066, 37206, 44443, 53021, 63098, 74995, 88946, 105350, 124533

Offset: 1

Clark Kimberling, Feb 16 2014

By conjugation, also the number of integer partitions of n with different median from maximum, ranks A362980. - Gus Wiseman, May 15 2023

			a(6) = 7 counts these partitions:  51, 42, 411, 321, 3111, 2211, 21111.
From _Gus Wiseman_, May 15 2023: (Start)
The a(3) = 1 through a(8) = 16 partitions wirth 2*(least part) <= greatest part:
  (21)  (31)   (41)    (42)     (52)
        (211)  (221)   (51)     (61)
               (311)   (321)    (331)
               (2111)  (411)    (421)
                       (2211)   (511)
                       (3111)   (2221)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
The a(3) = 1 through a(8) = 16 partitions with different median from maximum:
  (21)  (31)   (32)    (42)     (43)
        (211)  (41)    (51)     (52)
               (311)   (321)    (61)
               (2111)  (411)    (322)
                       (2211)   (421)
                       (3111)   (511)
                       (21111)  (3211)
                                (4111)
                                (22111)
                                (31111)
                                (211111)
(End)

The complement is counted by A053263, ranks A081306.
These partitions have ranks A069900.
The case of equality is A118096.
For < instead of <= we have A237820, ranks A362982.
For >= instead of <= we have A237824, ranks A362981.
The conjugate partitions have ranks A362980.
A000041 counts integer partitions, strict A000009.
A325347 counts partitions with integer median, complement A307683.
Cf. A002865, A008284, A171979, A237984, A238478, A238479, A327472, A359893, A362612, A362622.

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)

G.f.: Sum_{i>=1} Sum_{j>=0} x^(3*i+j) /Product_{k=i..2*i+j} (1-x^k). - Seiichi Manyama, May 27 2023

A069900 Numbers k such that the integer quotient of largest and smallest prime factors of k is greater than one.

Original entry on oeis.org

10, 14, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Labos Elemer, Apr 10 2002

nonn

Numbers k such that A069897(k) = floor(P(k)/p(k)) > 1, where P(k) and p(k) are largest and least prime factor of k, respectively.

Also numbers having at least one prime factor greater than twice the smallest prime factor: complement of A081306. - Reinhard Zumkeller, Mar 17 2003

			Composites with at least two and sufficiently deviating prime factors are here, like 2q, where q = prime >= 5: {10, ..., 254}.
Numbers with such divisors like 30 are also included.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006530, A020639, A069897, A069899, A081303, A081306.

Select[Range@ 120, #[[-1]] > 2 #[[1]] &@ FactorInteger[#][[All, 1]] &] (* Michael De Vlieger, Dec 08 2018 *)

is(k) = if(k == 1, 0, my(p = factor(k)[,1]); p[#p] > 2*p[1]); \\ Amiram Eldar, Feb 10 2025

A081303(a(n)) > 0. - Reinhard Zumkeller, Mar 17 2003

More terms from Reinhard Zumkeller, Mar 17 2003

A081303 gpf(m) - 2*spf(m), where gpf(m) is the greatest and spf(m) is the smallest prime factor of m (A006530, A020639).

Original entry on oeis.org

-1, -2, -3, -2, -5, -1, -7, -2, -3, 1, -11, -1, -13, 3, -1, -2, -17, -1, -19, 1, 1, 7, -23, -1, -5, 9, -3, 3, -29, 1, -31, -2, 5, 13, -3, -1, -37, 15, 7, 1, -41, 3, -43, 7, -1, 19, -47, -1, -7, 1, 11, 9, -53, -1, 1, 3, 13, 25, -59, 1, -61, 27, 1, -2, 3, 7, -67, 13, 17, 3, -71, -1, -73, 33, -1, 15

Offset: 1

Reinhard Zumkeller, Mar 17 2003

sign

Cf. A046665, A081304, A081305, A081306, A069900.

A069899 Numbers k such that the integer quotient of largest and smallest prime factors of k is 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 59, 61, 64, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 91, 96, 97, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 135, 137, 139, 143, 144

Offset: 1

Labos Elemer, Apr 10 2002

nonn

Numbers k such that A069897(k) = floor(P(k)/p(k)) = 1, where P(k) and p(k) are largest and least prime factor of k, respectively.

Numbers k > 1 such that A006530(k) < 2*A020639(k). - Amiram Eldar, Feb 10 2025

			Beside primes and prime powers, composite numbers like 96 are terms because floor(3/2) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006530, A020639, A069897, A069900, A081306.

q[k_] := Module[{p = FactorInteger[k][[;;, 1]]}, p[[-1]] < 2*p[[1]]]; Select[Range[2, 150], q] (* Amiram Eldar, Feb 10 2025 *)

isok(k) = if(k == 1, 0, my(p = factor(k)[,1]); p[#p] < 2*p[1]); \\ Amiram Eldar, Feb 10 2025

a(n) = A081306(n+1). - Amiram Eldar, Feb 10 2025

A362981 Heinz numbers of integer partitions such that 2*(least part) >= greatest part.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 91, 96, 97, 101, 103, 105, 107, 108, 109, 113, 119, 121, 125

Offset: 1

Gus Wiseman, May 14 2023

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.

By conjugation, also Heinz numbers of partitions whose greatest part appears at a middle position, namely k/2, (k+1)/2, or (k+2)/2, where k is the number of parts. These partitions have ranks A362622.

			The terms together with their prime indices begin:
     1: {}         16: {1,1,1,1}      36: {1,1,2,2}
     2: {1}        17: {7}            37: {12}
     3: {2}        18: {1,2,2}        41: {13}
     4: {1,1}      19: {8}            43: {14}
     5: {3}        21: {2,4}          45: {2,2,3}
     6: {1,2}      23: {9}            47: {15}
     7: {4}        24: {1,1,1,2}      48: {1,1,1,1,2}
     8: {1,1,1}    25: {3,3}          49: {4,4}
     9: {2,2}      27: {2,2,2}        53: {16}
    11: {5}        29: {10}           54: {1,2,2,2}
    12: {1,1,2}    31: {11}           55: {3,5}
    13: {6}        32: {1,1,1,1,1}    59: {17}
    15: {2,3}      35: {3,4}          61: {18}

For prime factors instead of indices we have A081306.
Prime indices are listed by A112798, length A001222, sum A056239.
The complement is A362982, counted by A237820.
Partitions of this type are counted by A237824.
Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],2*Min@@prix[#]>=Max@@prix[#]&]

A362982 Heinz numbers of partitions such that 2*(least part) < greatest part.

Original entry on oeis.org

10, 14, 20, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 56, 57, 58, 60, 62, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 115, 116, 117, 118, 120, 122, 123, 124, 126

Offset: 1

Gus Wiseman, May 14 2023

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:
    10: {1,3}        44: {1,1,5}      70: {1,3,4}
    14: {1,4}        46: {1,9}        74: {1,12}
    20: {1,1,3}      50: {1,3,3}      76: {1,1,8}
    22: {1,5}        51: {2,7}        78: {1,2,6}
    26: {1,6}        52: {1,1,6}      80: {1,1,1,1,3}
    28: {1,1,4}      56: {1,1,1,4}    82: {1,13}
    30: {1,2,3}      57: {2,8}        84: {1,1,2,4}
    33: {2,5}        58: {1,10}       85: {3,7}
    34: {1,7}        60: {1,1,2,3}    86: {1,14}
    38: {1,8}        62: {1,11}       87: {2,10}
    39: {2,6}        66: {1,2,5}      88: {1,1,1,5}
    40: {1,1,1,3}    68: {1,1,7}      90: {1,2,2,3}
    42: {1,2,4}      69: {2,9}        92: {1,1,9}

For prime factors instead of indices we have A069900, complement A081306.
Prime indices are listed by A112798, length A001222, sum A056239.
Partitions of this type are counted by A237820.
The complement is A362981, counted by A237824.
Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],2*Min@@prix[#]

A081304 Number of numbers m <= n with prime factors less than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 17, 17, 17, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 24, 24, 25, 26, 27, 27, 27, 27, 28, 28, 29, 29, 30, 30, 31, 32, 33, 33, 33, 33, 34, 35, 35, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 39, 40, 41, 42

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

a(n)+A081305(n)=n; a(114)=A081305(114)=57;

a(n)>=n/2 for n<=114, a(n)114.

Cf. A081306, A081303.

A089341 Numbers k with lpf(k) < gpf(k) < 2*lpf(k), where lpf = A020639, gpf = A006530.

Original entry on oeis.org

6, 12, 15, 18, 24, 35, 36, 45, 48, 54, 72, 75, 77, 91, 96, 108, 135, 143, 144, 162, 175, 187, 192, 209, 216, 221, 225, 245, 247, 288, 299, 323, 324, 375, 384, 391, 405, 432, 437, 486, 493, 527, 539, 551, 576, 589, 637, 648, 667, 675, 703, 713, 768, 847, 851

Offset: 1

Reinhard Zumkeller, Jan 03 2004

nonn

A081306 without prime powers.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A005279, A006530, A020639, A046665, A081306.

Subsequence of A024619.

a089341 n = a089341_list !! (n-1)
a089341_list = filter (\x -> a006530 x < 2 * a020639 x) a024619_list
-- Reinhard Zumkeller, Sep 29 2014

q[k_] := Module[{p = FactorInteger[k][[;;, 1]]}, Length[p] > 1 && p[[-1]] < 2*p[[1]]]; Select[Range[1000], q] (* Amiram Eldar, May 16 2025 *)

isok(k) = if(k == 1, 0, my(p = factor(k)[,1], np = #p);  np > 1 && p[np] < 2*p[1]); \\ Amiram Eldar, May 16 2025

A046665(a(n)) < A020639(a(n)).

cp's OEIS Frontend

A237821 Number of partitions of n such that 2*(least part) <= greatest part.

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A069900 Numbers k such that the integer quotient of largest and smallest prime factors of k is greater than one.

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A081303 gpf(m) - 2*spf(m), where gpf(m) is the greatest and spf(m) is the smallest prime factor of m (A006530, A020639).

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A069899 Numbers k such that the integer quotient of largest and smallest prime factors of k is 1.

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A362981 Heinz numbers of integer partitions such that 2*(least part) >= greatest part.

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A362982 Heinz numbers of partitions such that 2*(least part) < greatest part.

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A081304 Number of numbers m <= n with prime factors less than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).

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A089341 Numbers k with lpf(k) < gpf(k) < 2*lpf(k), where lpf = A020639, gpf = A006530.

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