A360008 -id:A360008 - cp's OEIS frontend

A359893 Triangle read by rows where T(n,k) is the number of integer partitions of n with median k, where k ranges from 1 to n in steps of 1/2.

1, 1, 0, 1, 1, 1, 0, 0, 1, 2, 0, 2, 0, 0, 0, 1, 3, 0, 1, 2, 0, 0, 0, 0, 1, 4, 1, 2, 0, 3, 0, 0, 0, 0, 0, 1, 6, 1, 3, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 8, 1, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 11, 2, 7, 1, 3, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 21 2023

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

			Triangle begins:
  1
  1  0  1
  1  1  0  0  1
  2  0  2  0  0  0  1
  3  0  1  2  0  0  0  0  1
  4  1  2  0  3  0  0  0  0  0  1
  6  1  3  0  1  3  0  0  0  0  0  0  1
  8  1  6  0  2  0  4  0  0  0  0  0  0  0  1
 11  2  7  1  3  0  1  4  0  0  0  0  0  0  0  0  1
 15  2 10  3  4  0  2  0  5  0  0  0  0  0  0  0  0  0  1
 20  3 13  3  7  0  3  0  1  5  0  0  0  0  0  0  0  0  0  0  1
 26  4 19  3 11  1  4  0  2  0  6  0  0  0  0  0  0  0  0  0  0  0  1
For example, row n = 8 counts the following partitions:
  611       4211  422    .  332  .  44  .  .  .  .  .  .  .  8
  5111            521       431     53
  32111           2222              62
  41111           3221              71
  221111          3311
  311111          22211
  2111111
  11111111

Row sums are A000041.
Row lengths are 2n-1 = A005408(n-1).
Column k=1 is A027336(n+1).
For mean instead of median we have A058398, see also A008284, A327482.
The mean statistic is ranked by A326567/A326568.
Omitting half-steps gives A359901.
The odd-length case is A359902.
The median statistic is ranked by A360005(n)/2.
First appearances of medians are ranked by A360006, A360007.
A027193 counts odd-length partitions, strict A067659, ranked by A026424.
A067538 counts partitions w/ integer mean, strict A102627, ranked by A316413.
A240219 counts partitions w/ the same mean as median, complement A359894.
Cf. A325347, A349156, A359889, A359895, A359906, A359907, A360008.

Table[Length[Select[IntegerPartitions[n], Median[#]==k&]],{n,1,10},{k,1,n,1/2}]

A359889 Numbers that are 1 or whose prime indices have the same mean as median.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94

Offset: 1

Gus Wiseman, Jan 22 2023

nonn

First differs from A236510 in having 252 (prime indices {1,1,2,2,4}).
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 prime indices of 900 are {1,1,2,2,3,3}, with mean 2 and median 2, so 900 is in the sequence.

These partitions are counted by A240219, strict A359897.
The LHS (mean of prime indices) is A326567/A326568.
The complement is A359890, counted by A359894.
The odd-length case is A359891, complement A359892, counted by A359895.
The RHS (median of prime indices) is A360005/2.
A058398 counts partitions by mean, see also A008284, A327482.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A359893 and A359901 count partitions by median, odd-length A359902.
A359908 lists numbers whose prime indices have integer median.
Cf. A026424, A327473, A359899, A359903, A359905, A359909, A360006, A360007, A360008, A360009.

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

Numbers n such that A326567(n)/A326568(n) = A360005(n)/2.

A360006 Least positive integer whose prime indices have median n/2. a(1) = 1.

Original entry on oeis.org

1, 2, 6, 3, 14, 5, 26, 7, 38, 11, 58, 13, 74, 17, 86, 19, 106, 23, 122, 29, 142, 31, 158, 37, 178, 41, 202, 43, 214, 47, 226, 53, 262, 59, 278, 61, 302, 67, 326, 71, 346, 73, 362, 79, 386, 83, 398, 89, 446, 97, 458, 101, 478, 103, 502, 107, 526, 109, 542, 113

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).

Position of first appearance of n in A360005.
The sorted version is A360007, for mean A360008.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 counts partitions by median, cf. A359901, A359902.
A359908 = numbers w/ integer median of prime indices, complement A359912.
Cf. A026424, A359889, A359890, A360009.

nn=100;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[If[n==1,1,2*Median[prix[n]]],{n,nn}];
Table[Position[seq,k][[1,1]],{k,Count[Differences[Union[seq]],1]}]

Consists of 1 followed by A000040 interleaved with 2*A031215.

A359903 Numbers whose prime indices and prime signature have the same mean.

Original entry on oeis.org

1, 2, 9, 88, 100, 125, 624, 756, 792, 810, 880, 900, 1312, 2401, 4617, 4624, 6240, 7392, 7560, 7920, 8400, 9261, 9604, 9801, 10648, 12416, 23424, 33984, 37760, 45792, 47488, 60912, 66176, 71552, 73920, 75200, 78720, 83592, 89216, 89984, 91264, 91648, 99456

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.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      9: {2,2}
     88: {1,1,1,5}
    100: {1,1,3,3}
    125: {3,3,3}
    624: {1,1,1,1,2,6}
    756: {1,1,2,2,2,4}
    792: {1,1,1,2,2,5}
    810: {1,2,2,2,2,3}
    880: {1,1,1,1,3,5}
    900: {1,1,2,2,3,3}
   1312: {1,1,1,1,1,13}
   2401: {4,4,4,4}
   4617: {2,2,2,2,2,8}
   4624: {1,1,1,1,7,7}
   6240: {1,1,1,1,1,2,3,6}
   7392: {1,1,1,1,1,2,4,5}
   7560: {1,1,1,2,2,2,3,4}
   7920: {1,1,1,1,2,2,3,5}
Example: 810 has prime indices {1,2,2,2,2,3} and prime exponents (1,4,1), both of which have mean 2, so 810 is in the sequence.
Example: 78720 has prime indices {1,1,1,1,1,1,1,2,3,13} and prime exponents (7,1,1,1), both of which have mean 5/2, so 78720 is in the sequence.

Prime indices are A112798, sum A056239, mean A326567/A326568.
Prime signature is A124010, sum A001222, mean A088529/A088530.
For prime factors instead of indices we have A359904.
Partitions with these Heinz numbers are counted by A360068.
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A316413 lists numbers whose prime indices have integer mean.
A360005 gives median of prime indices (times two).
Cf. A240219, A316313, A326622, A327473, A327476, A348551, A359905, A359908, A360008, A360069.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Select[Range[1000],Mean[prix[#]]==Mean[prisig[#]]&]

A360007 Positions of first appearances in the sequence giving the median of the prime indices of n (A360005(n)/2).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 23, 26, 29, 31, 37, 38, 41, 43, 47, 53, 58, 59, 61, 67, 71, 73, 74, 79, 83, 86, 89, 97, 101, 103, 106, 107, 109, 113, 122, 127, 131, 137, 139, 142, 149, 151, 157, 158, 163, 167, 173, 178, 179, 181, 191, 193, 197, 199, 202

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).

Positions of first appearances in A360005.
The unsorted version is A360006.
For mean instead of median we have A360008.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 counts partitions by median, cf. A359901, A359902.
A359908 = numbers w/ integer median of prime indices, complement A359912.
Cf. A026424, A359889, A359890, A360009.

nn=1000;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[If[n==1,1,2*Median[prix[n]]],{n,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

Consists of 1, the primes, and all odd-indexed primes times 2.

A360614 Numerator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 1, 3, 5, 2, 6, 2, 3, 1, 7, 2, 8, 1, 2, 5, 9, 1, 3, 3, 2, 4, 10, 1, 11, 1, 5, 7, 2, 1, 12, 4, 3, 3, 13, 4, 14, 5, 1, 9, 15, 2, 2, 1, 7, 2, 16, 1, 5, 1, 4, 5, 17, 3, 18, 11, 4, 1, 3, 5, 19, 7, 9, 4, 20, 2, 21, 6, 1, 8, 5, 2, 22, 3, 1, 13, 23, 1, 7, 7, 5, 5, 24, 3, 3, 3, 11, 15, 4, 1, 25, 4, 5, 3

Offset: 1

Gus Wiseman, Feb 19 2023

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 0-prepended prime indices of 100 are {0,1,1,3,3}, with differences (1,0,2,0), with mean 3/4, so a(100) = 3.

Positions of 1's are A340609, a superset of A106529.
For twice median instead of mean we have A360555.
The denominator is A360615.
A112798 lists prime indices, length A001222, sum A056239, max A061395.
A124010 gives prime signature, mean A088529/A088530.
A316413 lists numbers with integer mean prime index, complement A348551.
A326567/A326568 gives mean of prime indices.
Cf. A334201, A340610, A360008, A360556, A360557, A360688, A366785.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Numerator[Mean[Differences[Prepend[prix[n],0]]]]],{n,100}]

A360614(n) = if(1==n,0, my(u=primepi(vecmax(factor(n)[, 1]))); (u/gcd(u, bigomega(n)))); \\ Antti Karttunen, Oct 23 2023

Numerator of A061395(n)/A001222(n).

a(1) = 0; and for n >= 1, a(n) = A061395(n) / A366785(n) = A061395(n) / gcd(A001222(n), A061395(n)). - Antti Karttunen, Oct 23 2023

Data section extended up to a(100) by Antti Karttunen, Oct 23 2023

A360615 Denominator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 3, 1, 1, 2, 4, 1, 3, 1, 1, 1, 2, 1, 2, 2, 1, 3, 3, 1, 1, 1, 5, 2, 2, 1, 2, 1, 1, 1, 4, 1, 3, 1, 3, 1, 2, 1, 5, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 4, 1, 2, 3, 6, 1, 3, 1, 3, 2, 3, 1, 5, 1, 1, 1, 3, 2, 1, 1, 5, 2, 2, 1, 1, 2, 1, 1, 4

Offset: 1

Gus Wiseman, Feb 19 2023

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 0-prepended prime indices of 100 are {0,1,1,3,3}, with differences (1,0,2,0), with mean 3/4, so a(100) = 4.

Winston de Greef, Table of n, a(n) for n = 1..10000

Positions of 1's are A340610
The numerator is A360614.
A112798 lists prime indices, length A001222, sum A056239, max A061395.
A124010 gives prime signature, mean A088529/A088530.
A316413 lists numbers with integer mean prime index, complement A348551.
A326567/A326568 gives mean of prime indices.
Cf. A106529, A334201, A340609, A360008, A360555, A360556, A360558.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Denominator[Mean[Differences[Prepend[prix[n],0]]]]],{n,100}]

a(n) = if (n==1, 0, my(f=factor(n)); denominator(primepi(vecmax(f[, 1]))/ bigomega(f))); \\ Michel Marcus, Feb 20 2023

Denominator of A061395(n)/A001222(n), for n>1.

A359904 Numbers whose prime factors and prime signature have the same mean.

Original entry on oeis.org

1, 4, 27, 400, 3125, 9072, 10800, 14580, 24057, 35721, 50625, 73984, 117760, 134400, 158976, 181440, 191488, 389376, 452709, 544000, 583680, 664848, 731136, 774400, 823543, 878592, 965888

Offset: 1

Gus Wiseman, Jan 25 2023

nonn

The multiset of prime factors of n is row n of A027746.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime factors begin:
      1: {}
      4: {2,2}
     27: {3,3,3}
    400: {2,2,2,2,5,5}
   3125: {5,5,5,5,5}
   9072: {2,2,2,2,3,3,3,3,7}
  10800: {2,2,2,2,3,3,3,5,5}
  14580: {2,2,3,3,3,3,3,3,5}
  24057: {3,3,3,3,3,3,3,11}
  35721: {3,3,3,3,3,3,7,7}
  50625: {3,3,3,3,5,5,5,5}
  73984: {2,2,2,2,2,2,2,2,17,17}

The prime factors are A027746, mean A123528/A123529.
The prime signature is A124010, mean A088529/A088530.
For prime indices instead of factors we have A359903.
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A078175 = numbers whose prime factors have integer mean, indices A316413.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A360005 gives median of prime indices (times two).
Cf. A240219, A326622, A327473, A359905, A359908, A359913, A360008, A360068.

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Select[Range[1000],Mean[prifac[#]]==Mean[prisig[#]]&]

A360669 Nonprime numbers > 1 for which the prime indices have the same mean as their first differences.

Original entry on oeis.org

10, 39, 68, 115, 138, 259, 310, 328, 387, 517, 574, 636, 793, 795, 1034, 1168, 1206, 1241, 1281, 1340, 1534, 1691, 1825, 2212, 2278, 2328, 2343, 2369, 2370, 2727, 2774, 2905, 3081, 3277, 3818, 3924, 4064, 4074, 4247, 4268, 4360, 4539, 4850, 4905, 5243, 5335

Offset: 1

Gus Wiseman, Feb 18 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 terms together with their prime indices begin:
     1: {}
    10: {1,3}
    39: {2,6}
    68: {1,1,7}
   115: {3,9}
   138: {1,2,9}
   259: {4,12}
   310: {1,3,11}
   328: {1,1,1,13}
   387: {2,2,14}
   517: {5,15}
   574: {1,4,13}
   636: {1,1,2,16}
For example, the prime indices of 138 are {1,2,9}, with mean 4, and with first differences (1,7), with mean also 4, so 138 is in the sequence.

These partitions are counted by A360670.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, mean A088529/A088530.
A301987 lists numbers whose sum of prime indices equals their product.
A316413 lists numbers whose prime indices have integer mean.
A334201 adds up all prime indices except the greatest.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A340610, A344415, A348551, A359904, A360008, A360068, A360681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Mean[prix[#]]==Mean[Differences[prix[#]]]&]

A360680 Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 6, 30, 49, 152, 210, 513, 1444, 1776, 1952, 2310, 2375, 2664, 2760, 2960, 3249, 3864, 3996, 4140, 4144, 5796, 5994, 6072, 6210, 6440, 6512, 6517, 6900, 7176, 7400, 7696, 8694, 9025, 9108, 9384, 10064, 10120, 10350, 10488, 10764, 11248, 11960, 12167

Offset: 1

Gus Wiseman, Feb 19 2023

nonn

A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     49: {4,4}
    152: {1,1,1,8}
    210: {1,2,3,4}
    513: {2,2,2,8}
   1444: {1,1,8,8}
   1776: {1,1,1,1,2,12}
   1952: {1,1,1,1,1,18}
   2310: {1,2,3,4,5}
   2375: {3,3,3,8}
   2664: {1,1,1,2,2,12}
   2760: {1,1,1,2,3,9}
   2960: {1,1,1,1,3,12}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with mean 3/2. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with mean also 3/2. So 2760 is in the sequence.

For indices instead of 0-prepended differences: A359903, counted by A360068.
For median instead of mean we have A360681.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, mean A088529/A088530.
A316413 = numbers whose prime indices have integer mean, complement A348551.
A326619/A326620 gives mean of distinct prime indices.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A340610, A359904, A359905, A360008, A360460, A360555, A360556.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Mean[Length/@Split[prix[#]]] == Mean[Differences[Prepend[prix[#],0]]]&]

cp's OEIS Frontend

A359893 Triangle read by rows where T(n,k) is the number of integer partitions of n with median k, where k ranges from 1 to n in steps of 1/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A359889 Numbers that are 1 or whose prime indices have the same mean as median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A360006 Least positive integer whose prime indices have median n/2. a(1) = 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A359903 Numbers whose prime indices and prime signature have the same mean.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360007 Positions of first appearances in the sequence giving the median of the prime indices of n (A360005(n)/2).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A360614 Numerator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A360615 Denominator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A359904 Numbers whose prime factors and prime signature have the same mean.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs