A067538 -id:A067538 - cp's OEIS frontend

A316428 Heinz numbers of integer partitions such that every part is divisible by the number of parts.

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 125, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 169, 173, 179, 181, 183, 191, 193, 197

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			93499 is the Heinz number of (12,8,8,4) and belongs to the sequence because each part is divisible by 4.
Sequence of partitions such that every part is divisible by the number of parts begins (1), (2), (3), (4), (2,2), (5), (6), (7), (8), (4,2), (9).

Cf. A056239, A067538, A074761, A143773, A237984, A289509, A296150, A298423, A316413.

Select[Range[200],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>Divisible[PrimePi[p],PrimeOmega[#]]]&]

A326028 Number of factorizations of n into factors > 1 with integer geometric mean.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Jul 15 2019

nonn

First differs from A294336 and A316782 at a(36) = 5.

			The a(4) = 2 through a(36) = 5 factorizations (showing only the cases where n is a perfect power).
  (4)    (8)      (9)    (16)       (25)   (27)     (32)         (36)
  (2*2)  (2*2*2)  (3*3)  (2*8)      (5*5)  (3*3*3)  (2*2*2*2*2)  (4*9)
                         (4*4)                                   (6*6)
                         (2*2*2*2)                               (2*18)
                                                                 (3*12)

Antti Karttunen, Table of n, a(n) for n = 1..100000
Wikipedia, Geometric mean
Index entries for sequences computed from exponents in factorization of n

Positions of terms > 1 are the perfect powers A001597.
Partitions with integer geometric mean are A067539.
Subsets with integer geometric mean are A326027.
Factorizations with integer average and geometric mean are A326647.
Cf. A001055, A082553, A322794, A326514, A326515, A326516, A326622, A326623, A326624, A326625.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],IntegerQ[GeometricMean[#]]&]],{n,2,100}]

A326028(n, m=n, facmul=1, facnum=0) = if(1==n,facnum>0 && ispower(facmul,facnum), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A326028(n/d, d, facmul*d, facnum+1))); (s)); \\ Antti Karttunen, Nov 10 2024

a(2^n) = A067538(n).

a(89) onwards from Antti Karttunen, Nov 10 2024

A326621 Numbers n such that the average of the set of distinct prime indices of n is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 60, 61, 62, 63, 64, 67, 68, 71, 73, 78, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 94, 97, 100, 101, 103, 105

Offset: 1

Gus Wiseman, Jul 14 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 distinct parts have an integer average.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}

Positions of 1's in A326620.

Cf. A051293, A056239, A067538, A078174, A078175, A102627, A112798, A326567/A326568, A326619/A326620, A326622.

Select[Range[2,100],IntegerQ[Mean[PrimePi/@First/@FactorInteger[#]]]&]

A326841 Heinz numbers of integer partitions of m >= 0 using divisors of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163

Offset: 1

Gus Wiseman, Jul 26 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A018818.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}

R. J. Mathar, Table of n, a(n) for n = 1..543

The case where the length also divides m is A326847.

Cf. A001222, A018818, A056239, A067538, A112798, A316413, A326836, A326842.

isA326841 := proc(n)
    local ifs,psigsu,p,psig ;
    psigsu := A056239(n) ;
    for ifs in ifactors(n)[2] do
        p := op(1,ifs) ;
        psig := numtheory[pi](p) ;
        if modp(psigsu,psig) <> 0 then
            return false;
        end if;
    end do:
    true;
end proc:
for i from 1 to 3000 do
    if isA326841(i) then
        printf("%d %d\n",n,i);
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019

Select[Range[100],With[{y=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},And@@IntegerQ/@(Total[y]/y)]&]

A124944 Table, number of partitions of n with k as high median.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 6, 4, 1, 1, 1, 1, 1, 8, 6, 3, 1, 1, 1, 1, 1, 11, 8, 5, 1, 1, 1, 1, 1, 1, 15, 11, 7, 3, 1, 1, 1, 1, 1, 1, 20, 15, 9, 5, 1, 1, 1, 1, 1, 1, 1, 26, 21, 12, 8, 3, 1, 1, 1, 1, 1, 1, 1, 35, 27, 16, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1, 45, 37, 21, 13, 8, 3

Offset: 1

Franklin T. Adams-Watters, Nov 13 2006

For a multiset with an odd number of elements, the high median is the same as the median. For a multiset with an even number of elements, the high median is the larger of the two central elements.
This table may be read as an upper right triangle with n >= 1 as column index and k >= 1 as row index. - Peter Munn, Jul 16 2017
Arrange the parts of a partition nonincreasing order.  Remove the last part, then the first, then the last remaining part, then the first remaining part, and continue until only a single number, the high median, remains. - Clark Kimberling, May 14 2019

			For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.
From _Gus Wiseman_, Jul 12 2023: (Start)
Triangle begins:
   1
   1  1
   1  1  1
   2  1  1  1
   3  1  1  1  1
   4  3  1  1  1  1
   6  4  1  1  1  1  1
   8  6  3  1  1  1  1  1
  11  8  5  1  1  1  1  1  1
  15 11  7  3  1  1  1  1  1  1
  20 15  9  5  1  1  1  1  1  1  1
  26 21 12  8  3  1  1  1  1  1  1  1
  35 27 16 10  5  1  1  1  1  1  1  1  1
  45 37 21 13  8  3  1  1  1  1  1  1  1  1
  58 48 29 16 11  5  1  1  1  1  1  1  1  1  1
Row n = 8 counts the following partitions:
  (611)       (521)    (431)   (44)  (53)  (62)  (71)  (8)
  (5111)      (422)    (332)
  (41111)     (4211)   (3311)
  (32111)     (3221)
  (311111)    (2222)
  (221111)    (22211)
  (2111111)
  (11111111)
(End)

Row sums are A000041.
Column k = 1 is A027336(n-1), ranks A364056.
Column k = 1 in the low version is A027336, ranks A363488.
The low version of this triangle is A124943.
The rank statistic for this triangle is A363942, low version A363941.
A version for mean instead of median is A363946, low A363945.
A version for mode instead of median is A363953, low A363952.
A008284 counts partitions by length, maximum, or decreasing mean.
A026794 counts partitions by minimum, strict A026821.
A325347 counts partitions with integer median, ranks A359908.
A359893 and A359901 count partitions by median.
A360005(n)/2 returns median of prime indices.
Cf. A008289, A025065, A027193, A067538, A237984, A240219, A362608, A363740, A363943, A363944.

Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 1)/2]]] &, IntegerPartitions[#]]] &, Range[13]]  (* Peter J. C. Moses, May 14 2019 *)

A363943 Mean of the multiset of prime indices of n, rounded down.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 2, 5, 1, 6, 2, 2, 1, 7, 1, 8, 1, 3, 3, 9, 1, 3, 3, 2, 2, 10, 2, 11, 1, 3, 4, 3, 1, 12, 4, 4, 1, 13, 2, 14, 2, 2, 5, 15, 1, 4, 2, 4, 2, 16, 1, 4, 1, 5, 5, 17, 1, 18, 6, 2, 1, 4, 2, 19, 3, 5, 2, 20, 1, 21, 6, 2, 3, 4, 3, 22, 1, 2, 7

Offset: 1

Gus Wiseman, Jun 29 2023

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.

Extending the terminology introduced at A124943, this is the "low mean" of prime indices.

			The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 1.

Positions of first appearances are 1 and A000040.
Before rounding down we had A326567/A326568.
For mode instead of mean we have A363486, high A363487.
For low median instead of mean we have A363941, triangle A124943.
For high median instead of mean we have A363942, triangle A124944.
The high version is A363944, triangle A363946.
The triangle for this statistic (low mean) is A363945.
Positions of 1's are A363949(n) = 2*A344296(n), counted by A025065.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363947 ranks partitions with rounded mean 1, counted by A363948.
Cf. A102627, A327473, A327476, A327482, A348551, A359889, A363727, A363951.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
meandown[y_]:=If[Length[y]==0,0,Floor[Mean[y]]];
Table[meandown[prix[n]],{n,100}]

A340609 Numbers whose number of prime factors (A001222) is divisible by their greatest prime index (A061395).

Original entry on oeis.org

2, 4, 6, 8, 9, 16, 20, 24, 30, 32, 36, 45, 50, 54, 56, 64, 75, 81, 84, 96, 125, 126, 128, 140, 144, 160, 176, 189, 196, 210, 216, 240, 256, 264, 294, 315, 324, 350, 360, 384, 396, 400, 416, 440, 441, 486, 490, 512, 525, 540, 576, 594, 600, 616, 624, 660, 686

Offset: 1

Gus Wiseman, Jan 27 2021

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.

If n is a term, then so is n^k for k > 1. - Robert Israel, Feb 08 2021

			The sequence of terms together with their prime indices begins:
      2: {1}             64: {1,1,1,1,1,1}      216: {1,1,1,2,2,2}
      4: {1,1}           75: {2,3,3}            240: {1,1,1,1,2,3}
      6: {1,2}           81: {2,2,2,2}          256: {1,1,1,1,1,1,1,1}
      8: {1,1,1}         84: {1,1,2,4}          264: {1,1,1,2,5}
      9: {2,2}           96: {1,1,1,1,1,2}      294: {1,2,4,4}
     16: {1,1,1,1}      125: {3,3,3}            315: {2,2,3,4}
     20: {1,1,3}        126: {1,2,2,4}          324: {1,1,2,2,2,2}
     24: {1,1,1,2}      128: {1,1,1,1,1,1,1}    350: {1,3,3,4}
     30: {1,2,3}        140: {1,1,3,4}          360: {1,1,1,2,2,3}
     32: {1,1,1,1,1}    144: {1,1,1,1,2,2}      384: {1,1,1,1,1,1,1,2}
     36: {1,1,2,2}      160: {1,1,1,1,1,3}      396: {1,1,2,2,5}
     45: {2,2,3}        176: {1,1,1,1,5}        400: {1,1,1,1,3,3}
     50: {1,3,3}        189: {2,2,2,4}          416: {1,1,1,1,1,6}
     54: {1,2,2,2}      196: {1,1,4,4}          440: {1,1,1,3,5}
     56: {1,1,1,4}      210: {1,2,3,4}          441: {2,2,4,4}

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

Note: Heinz numbers are given in parentheses below.
The case of equality is A047993 (A106529).
These are the Heinz numbers of certain partitions counted by A168659.
The reciprocal version is A340610, with strict case A340828 (A340856).
If all parts (not just the greatest) are divisors we get A340693 (A340606).
A001222 counts prime factors.
A006141 counts partitions whose length equals their minimum (A324522).
A056239 adds up prime indices.
A061395 selects the maximum prime index.
A067538 counts partitions whose length divides their sum (A316413).
A067538 counts partitions whose maximum divides their sum (A326836).
A112798 lists the prime indices of each positive integer.
A200750 counts partitions with length coprime to maximum (A340608).
A325134 = A001222 + A061395.
A326845 = A056239 * A061395.
Cf. A039900, A064174, A143773, A244990/A244991, A326837, A326849 (A326848), A340653, A340787/A340788.

filter:= proc(n) local F,m,g,t;
  F:= ifactors(n)[2];
  m:= add(t[2],t=F);
  g:= numtheory:-pi(max(seq(t[1],t=F)));
  m mod g = 0;
end proc:
seelect(filter, [$2..1000]); # Robert Israel, Feb 08 2021

Select[Range[2,100],Divisible[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]&]

A061395(a(n)) divides A001222(a(n)).

A326843 Number of integer partitions of n whose length and maximum both divide n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 22, 2, 5, 11, 16, 2, 36, 2, 46, 22, 5, 2, 209, 3, 5, 42, 130, 2, 434, 2, 217, 77, 5, 52, 1400, 2, 5, 135, 1749, 2, 1782, 2, 957, 2151, 5, 2, 8355, 3, 1859, 385, 2388, 2, 6726, 2765, 10641, 627, 5, 2, 68049, 2, 5, 13424, 17142

Offset: 0

Gus Wiseman, Jul 26 2019

nonn

The Heinz numbers of these partitions are given by A326837.

			The a(1) = 1 through a(8) = 5 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (1111)           (222)                (2222)
                                     (321)                (4211)
                                     (111111)             (11111111)
The a(12) = 22 partitions:
  (12)
  (6,6)
  (4,4,4)
  (6,3,3)
  (6,4,2)
  (6,5,1)
  (3,3,3,3)
  (4,3,3,2)
  (4,4,2,2)
  (4,4,3,1)
  (6,2,2,2)
  (6,3,2,1)
  (6,4,1,1)
  (2,2,2,2,2,2)
  (3,2,2,2,2,1)
  (3,3,2,2,1,1)
  (3,3,3,1,1,1)
  (4,2,2,2,1,1)
  (4,3,2,1,1,1)
  (4,4,1,1,1,1)
  (6,2,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..180

The strict case is A326851.
The non-constant case is A326852.
The case where all parts (not just the maximum) divide n is A326842.
Cf. A018818, A047993, A067538, A326837, A326849.

Table[If[n==0,1,Length[Select[IntegerPartitions[n],Divisible[n,Length[#]]&&Divisible[n,Max[#]]&]]],{n,0,30}]

A363944 Mean of the multiset of prime indices of n, rounded up.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 2, 6, 3, 3, 1, 7, 2, 8, 2, 3, 3, 9, 2, 3, 4, 2, 2, 10, 2, 11, 1, 4, 4, 4, 2, 12, 5, 4, 2, 13, 3, 14, 3, 3, 5, 15, 2, 4, 3, 5, 3, 16, 2, 4, 2, 5, 6, 17, 2, 18, 6, 3, 1, 5, 3, 19, 3, 6, 3, 20, 2, 21, 7, 3, 4, 5, 3, 22, 2, 2, 7

Offset: 1

Gus Wiseman, Jun 30 2023

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.

Extending the terminology introduced at A124944, this is the "high mean" of prime indices.

			The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 2.

Positions of first appearances are 1 and A000040.
Positions of 1's are A000079(n>0).
Before rounding up we had A326567/A326568.
For mode instead of mean we have A363487, low A363486.
For median instead of mean we have A363942, triangle A124944.
Rounding down instead of up gives A363943, triangle A363945.
The triangle for this statistic (high mean) is A363946.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363947 ranks partitions with rounded mean 1, counted by A363948.
A363949 ranks partitions with low mean 1, counted by A025065.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A051293, A124943, A215366, A327473, A327476, A327482, A359889, A362611, A363723, A363724, A363727, A363951.

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
meanup[y_]:=If[Length[y]==0,0,Ceiling[Mean[y]]];
Table[meanup[prix[n]],{n,100}]

A360009 Numbers whose prime indices have integer mean and integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 78, 79, 81, 82, 83, 85, 87, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 105, 107, 109, 110, 111

Offset: 1

Gus Wiseman, Jan 24 2023

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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}

For just integer mean we have A316413 (counted by A067538).
The mean of prime indices is given by A326567/A326568.
The complement is A348551 \/ A359912 (counted by A349156 and A307683).
These partitions are counted by A359906.
For just integer median we have A359908 (counted by A325347).
The median of prime indices is given by A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 lists prime indices, length A001222, sum A056239.
A326622 counts factorizations with integer mean, strict A328966.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A026424, A327473, A359889, A359890, A359903, A359905, A360006.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],IntegerQ[Mean[prix[#]]]&&IntegerQ[Median[prix[#]]]&]

Intersection of A316413 and A359908.

cp's OEIS Frontend

A316428 Heinz numbers of integer partitions such that every part is divisible by the number of parts.

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A326028 Number of factorizations of n into factors > 1 with integer geometric mean.

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A326621 Numbers n such that the average of the set of distinct prime indices of n is an integer.

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A326841 Heinz numbers of integer partitions of m >= 0 using divisors of m.

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A124944 Table, number of partitions of n with k as high median.

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A363943 Mean of the multiset of prime indices of n, rounded down.

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A340609 Numbers whose number of prime factors (A001222) is divisible by their greatest prime index (A061395).

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A326843 Number of integer partitions of n whose length and maximum both divide n.

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