A069900 -id:A069900 - 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

A081306 Numbers n with prime factors less than 2*spf(n), 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, 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

Reinhard Zumkeller, Mar 17 2003

nonn

A081303(a(n)) < 0, A006530(a(n)) < A020639(a(n))*2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006530, A020639, A081303.
Complement of A069900.
Union of {1} and A069899. [R. J. Mathar, Sep 18 2008]

filter:= proc(n) local F;
  F:= numtheory:-factorset(n);
  max(F) < 2*min(F);
end proc:
select(filter, [$1..200]); # Robert Israel, Mar 28 2018

Select[Range[200], Max[F = FactorInteger[#][[All, 1]]] < 2 Min[F]&] (* Jean-François Alcover, Mar 04 2019 *)

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

A081305 Number of numbers m <= n with at least one prime factor greater than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 10, 10, 11, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 17, 18, 19, 19, 19, 20, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 31, 31, 32, 32, 33, 33, 34, 34, 35

Offset: 1

Reinhard Zumkeller, Mar 17 2003

nonn

a(n)+A081304(n)=n; a(114)=A081304(114)=57;

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

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A069900, A081303.

f:= proc(n) local R; R:= numtheory:-factorset(n); if max(R) > 2*min(R) then 1 else 0 fi end proc:
ListTools:-PartialSums(map(f, [$1..100])); # Robert Israel, Jul 27 2020

pfg[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},If[Last[f]> 2*First[ f], 1,0]]; Accumulate[Array[pfg,80]] (* Harvey P. Dale, Apr 28 2014 *)

A307516 Numbers whose maximum prime index and minimum prime index differ by more than 1.

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, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

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 maximum and minimum parts differ by more than 1. The enumeration of these partitions by sum is given by A000094.

Differs from A069900 first at n = 43.

			The sequence of terms together with their prime indices begins:
   10: {1,3}
   14: {1,4}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   38: {1,8}
   39: {2,6}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   46: {1,9}
   50: {1,3,3}
   51: {2,7}
   52: {1,1,6}
   55: {3,5}

Positions of numbers > 1 in A243055. Complement of A000961 and A256617.

Cf. A000094, A000245, A001222, A052126, A056239, A061395, A064989, A069900, A105441, A112798, A307517, A325196.

with(numtheory):
q:= n-> (l-> pi(l[-1])-pi(l[1])>1)(sort([factorset(n)[]])):
select(q, [$2..200])[];  # Alois P. Heinz, Apr 12 2019

Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]>1&]

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[#]

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

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A081306 Numbers n with prime factors less than 2*spf(n), where spf(m) is the smallest prime factor of m (A020639).

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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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A081305 Number of numbers m <= n with at least one prime factor greater than 2*spf(m), where spf(m) is the smallest prime factor of m (A020639).

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A307516 Numbers whose maximum prime index and minimum prime index differ by more than 1.

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

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