cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A361856 Positive integers whose prime indices satisfy (maximum) = 2*(median).

Original entry on oeis.org

12, 24, 42, 48, 60, 63, 72, 96, 126, 130, 140, 144, 189, 192, 195, 252, 266, 288, 308, 325, 330, 360, 378, 384, 399, 420, 432, 495, 546, 567, 572, 576, 588, 600, 630, 638, 650, 665, 756, 768, 819, 864, 882, 884, 931, 945, 957, 962, 975, 1122, 1134, 1152, 1190
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2023

Keywords

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.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
These are Heinz numbers of partitions satisfying (maximum) = 2*(median).

Examples

			The terms together with their prime indices begin:
    12: {1,1,2}
    24: {1,1,1,2}
    42: {1,2,4}
    48: {1,1,1,1,2}
    60: {1,1,2,3}
    63: {2,2,4}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}
   126: {1,2,2,4}
   130: {1,3,6}
   140: {1,1,3,4}
   144: {1,1,1,1,2,2}
The prime indices of 126 are {1,2,2,4}, with maximum 4 and median 2, so 126 is in the sequence.
The prime indices of 308 are {1,1,4,5}, with maximum 5 and median 5/2, so 308 is in the sequence.

Crossrefs

The LHS (greatest prime index) is A061395.
The RHS (twice median) is A360005, distinct A360457.
These partitions are counted by A361849.
For mean instead of median we have A361855, counted by A361853.
For minimum instead of median we have A361908, counted by A118096.
For length instead of median we have A361909, counted by A237753.
A000975 counts subsets with integer median.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A112798 lists prime indices, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
Cf. A027193, A067801, A111907, A361205, A361848, A361858.

Programs

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

Formula

A061395(a(n)) = 2*A360005(a(n)).

A361855 Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).

Original entry on oeis.org

28, 40, 78, 84, 171, 190, 198, 220, 240, 252, 280, 351, 364, 390, 406, 435, 714, 748, 756, 765, 777, 784, 814, 840, 850, 925, 988, 1118, 1197, 1254, 1330, 1352, 1419, 1425, 1440, 1505, 1564, 1600, 1638, 1716, 1755, 1794, 1802, 1820, 1950, 2067, 2204, 2254
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2023

Keywords

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.
Also positive integers whose prime indices satisfy (maximum) = 2*(mean).
Also Heinz numbers of partitions of the same size as their complement (see example).

Examples

			The terms together with their prime indices begin:
   28: {1,1,4}
   40: {1,1,1,3}
   78: {1,2,6}
   84: {1,1,2,4}
  171: {2,2,8}
  190: {1,3,8}
  198: {1,2,2,5}
  220: {1,1,3,5}
  240: {1,1,1,1,2,3}
  252: {1,1,2,2,4}
  280: {1,1,1,3,4}
The prime indices of 84 are {1,1,2,4}, with maximum 4, length 4, and sum 8, and 4*4 = 2*8, so 84 is in the sequence.
The prime indices of 120 are {1,1,1,2,3}, with maximum 3, length 5, and sum 8, and 3*5 != 2*8, so 120 is not in the sequence.
The prime indices of 252 are {1,1,2,2,4}, with maximum 4, length 5, and sum 10, and 4*5 = 2*10, so 252 is in the sequence.
The partition (5,2,2,1) with Heinz number 198 has diagram:
  o o o o o
  o o . . .
  o o . . .
  o . . . .
Since the partition and its complement (shown in dots) both have size 10, 198 is in the sequence.

Crossrefs

These partitions are counted by A361853, strict A361854.
For median instead of mean we have A361856, counted by A361849.
For minimum instead of mean we have A361908, counted by A118096.
For length instead of mean we have A361909, counted by A237753.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A061395 gives greatest prime index.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A067801, A237824, A316413, A324521, A326844, A361205, A361851, A361906.

Programs

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

Formula

A061395(a(n)) * A001222(a(n)) = 2*A056239(a(n)).

A361908 Positive integers > 1 whose prime indices satisfy (maximum) = 2*(minimum).

Original entry on oeis.org

6, 12, 18, 21, 24, 36, 48, 54, 63, 65, 72, 96, 105, 108, 133, 144, 147, 162, 189, 192, 216, 288, 315, 319, 324, 325, 384, 432, 441, 455, 481, 486, 525, 567, 576, 648, 715, 731, 735, 768, 845, 864, 931, 945, 972, 1007, 1029, 1152, 1296, 1323, 1403, 1458, 1463
Offset: 1

Views

Author

Gus Wiseman, Apr 05 2023

Keywords

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.

Examples

			The terms together with their prime indices begin:
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    21: {2,4}
    24: {1,1,1,2}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    63: {2,2,4}
    65: {3,6}
    72: {1,1,1,2,2}
    96: {1,1,1,1,1,2}

Links

Crossrefs

The RHS is 2*A055396 (twice minimum).
The LHS is A061395 (greatest prime index).
Partitions of this type are counted by A118096.
For mean instead of minimum we have A361855, counted by A361853.
For median instead of minimum we have A361856, counted by A361849.
For length instead of minimum we have A361909, counted by A237753.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors with multiplicity.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A237820, A237821, A361858.

Programs

  • Maple
    filter:= proc(n) local F,b;
  if n::even then b:= padic:-ordp(n,3);
     if b = 0 then return false else return n = 2^padic:-ordp(n,2) * 3^b fi
  fi;
  F:= ifactors(n)[2][..,1];
  nops(F) >= 2 and numtheory:-pi(max(F)) = 2*numtheory:-pi(min(F))
end proc:
select(filter, [$1..2000]); # Robert Israel, Mar 11 2025
  • Mathematica
    Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]==2*PrimePi[FactorInteger[#][[1,1]]]&]

A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

Original entry on oeis.org

20, 28, 40, 44, 52, 56, 66, 68, 76, 78, 80, 84, 88, 92, 99, 102, 104, 112, 114, 116, 117, 120, 124, 132, 136, 138, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 190, 198, 200, 204, 207, 208, 212, 220, 222, 224, 228, 230, 232, 234
Offset: 1

Views

Author

Gus Wiseman, Apr 05 2023

Keywords

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

Examples

			The prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 > 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   20: {1,1,3}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   52: {1,1,6}
   56: {1,1,1,4}
   66: {1,2,5}
   68: {1,1,7}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   92: {1,1,9}
   99: {2,2,5}

Crossrefs

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal version is A361856, counted by A361849.
These partitions are counted by A361857, reverse A361858.
Including the equal case gives A361868, counted by A361859.
For mean instead of median we have A361907.
A000975 counts subsets with integer median.
A001222 counts prime factors, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A111907, A237751, A237820, A359893, A361848, A361855, A361908, A361909.

Programs

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

A363218 Positive integers whose prime indices satisfy: (length) = 2*(maximum).

Original entry on oeis.org

4, 24, 36, 54, 81, 160, 240, 360, 400, 540, 600, 810, 896, 900, 1000, 1215, 1344, 1350, 1500, 2016, 2025, 2240, 2250, 2500, 3024, 3136, 3360, 3375, 3750, 4536, 4704, 5040, 5600, 5625, 5632, 6250, 6804, 7056, 7560, 7840, 8400, 8448, 9375, 10206, 10584, 10976
Offset: 1

Views

Author

Gus Wiseman, May 23 2023

Keywords

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.

Examples

			The terms together with their prime indices begin:
      4: {1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     54: {1,2,2,2}
     81: {2,2,2,2}
    160: {1,1,1,1,1,3}
    240: {1,1,1,1,2,3}
    360: {1,1,1,2,2,3}
    400: {1,1,1,1,3,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    810: {1,2,2,2,2,3}
    896: {1,1,1,1,1,1,1,4}
    900: {1,1,2,2,3,3}
   1000: {1,1,1,3,3,3}
   1215: {2,2,2,2,2,3}
   1344: {1,1,1,1,1,1,2,4}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   2016: {1,1,1,1,1,2,2,4}
   2025: {2,2,2,2,3,3}
   2240: {1,1,1,1,1,1,3,4}

Crossrefs

The LHS (number of prime indices) is A001222.
The RHS is twice A061395.
Before multiplying by 2 we had A106529.
Partitions of this type are counted by A237753.
For sum instead of length we have A344415, counted by A035363.
An adjoint version is A361909, also counted by A237753.
For minimum instead of maximum we have A363134, counted by A237757.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A001221, A067801, A118096, A324521, A237752, A361855, A361908, A363222.

Programs

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

Formula

Disjoint from A361909.

A363222 Numbers whose multiset of prime indices satisfies (maximum) - (minimum) = (length).

Original entry on oeis.org

10, 21, 28, 42, 55, 70, 88, 91, 98, 99, 132, 165, 187, 198, 208, 220, 231, 247, 308, 312, 325, 330, 351, 363, 391, 455, 462, 468, 484, 520, 544, 550, 551, 585, 702, 713, 715, 726, 728, 770, 780, 816, 819, 833, 845, 975, 1073, 1078, 1092, 1144, 1170, 1210, 1216
Offset: 1

Views

Author

Gus Wiseman, May 29 2023

Keywords

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.

Examples

			The terms together with their prime indices begin:
    10: {1,3}
    21: {2,4}
    28: {1,1,4}
    42: {1,2,4}
    55: {3,5}
    70: {1,3,4}
    88: {1,1,1,5}
    91: {4,6}
    98: {1,4,4}
    99: {2,2,5}
   132: {1,1,2,5}
   165: {2,3,5}
   187: {5,7}
   198: {1,2,2,5}

Crossrefs

The RHS is A001222.
Partitions of this type are counted by A237832.
The LHS (maximum minus minimum) is A243055.
A001221 (omega) counts distinct prime factors.
A112798 lists prime indices, sum A056239.
A360005 gives median of prime indices, distinct A360457.
Cf. A067801, A111907, A118096, A237821, A361205, A361908, A361909.

Programs

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

Formula

A061395(a(n)) - A055396(a(n)) = A001222(a(n)).
Showing 1-6 of 6 results.

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