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.

Previous Showing 11-17 of 17 results.

A361393 Positive integers k such that 2*omega(k) > bigomega(k).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85
Offset: 1

Views

Author

Gus Wiseman, Mar 16 2023

Keywords

Comments

First differs from A317090 in having 120 and lacking 360.
There are numbers like 1, 120, 168, 180, 252,... which are not in A179983 but in here, and others like 360, 504, 540, 600,... which are in A179983 but not in here. - R. J. Mathar, Mar 21 2023

Examples

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
The prime indices of 120 are {1,1,1,2,3}, with 3 distinct parts and 5 parts, and 2*3 > 5, so 120 is in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, with 3 distinct parts and 6 parts, and 2*3 is not greater than 6, so 360 is not in the sequence.

Crossrefs

These partitions are counted by A237365.
The complement is A361204.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
A360005 gives median of prime indices (times 2), distinct A360457.
Comparing twice the number of distinct parts to the number of parts:
less: A360254, ranks A360558
equal: A239959, ranks A067801
greater: A237365, ranks A361393
less or equal: A237363, ranks A361204
greater or equal: A361394, ranks A361395
Cf. A046660, A061395, A067340, A111907.
Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

Programs

  • Maple
    isA361393 := proc(n)
    if 2*A001221(n) > numtheory[bigomega](n) then
        true;
    else
        false ;
    end if:
end proc:
for n from 1 to 100 do
    if isA361393(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 21 2023
  • Mathematica
    Select[Range[1000],2*PrimeNu[#]>PrimeOmega[#]&]

Formula

{k: 2*A001221(k) > A001222(k)}. - R. J. Mathar, Mar 21 2023

A361909 Positive integers > 1 whose prime indices satisfy: (maximum) = 2*(length).

Original entry on oeis.org

3, 14, 21, 35, 49, 52, 78, 117, 130, 152, 182, 195, 228, 273, 286, 325, 338, 342, 380, 429, 455, 464, 507, 513, 532, 570, 637, 696, 715, 798, 836, 845, 855, 950, 988, 1001, 1044, 1160, 1183, 1184, 1197, 1254, 1292, 1330, 1425, 1444, 1482, 1566, 1573, 1624
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:
     3: {2}
    14: {1,4}
    21: {2,4}
    35: {3,4}
    49: {4,4}
    52: {1,1,6}
    78: {1,2,6}
   117: {2,2,6}
   130: {1,3,6}
   152: {1,1,1,8}
   182: {1,4,6}
   195: {2,3,6}
   228: {1,1,2,8}
   273: {2,4,6}
   286: {1,5,6}
   325: {3,3,6}
   338: {1,6,6}
   342: {1,2,2,8}

Crossrefs

The LHS is A061395 (greatest prime index), least A055396.
Without multiplying by 2 in the RHS, we have A106529.
For omega instead of bigomega we have A111907, counted by A239959.
Partitions of this type are counted by A237753.
The RHS is A255201 (twice bigomega).
For mean instead of length we have A361855, counted by A361853.
For median instead of length we have A361856, counted by A361849.
For minimum instead of length we have A361908, counted by A118096.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors with multiplicity.
A112798 lists prime indices, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A326567/A326568 gives mean of prime indices.
Cf. A067801, A324521, A237752, A237820, A360005, A361205, A361907.

Programs

  • Mathematica
    Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]==2*PrimeOmega[#]&]

A362050 Numbers whose prime indices satisfy: (length) = 2*(median).

Original entry on oeis.org

4, 54, 81, 90, 100, 126, 135, 140, 189, 198, 220, 234, 260, 297, 306, 340, 342, 351, 380, 414, 459, 460, 513, 522, 558, 580, 620, 621, 666, 738, 740, 774, 783, 820, 837, 846, 860, 940, 954, 999, 1060, 1062, 1098, 1107, 1161, 1180, 1206, 1220, 1269, 1278, 1314
Offset: 1

Views

Author

Gus Wiseman, Apr 20 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).
All terms are squarefree.

Examples

			The terms together with their prime indices begin:
    4: {1,1}
   54: {1,2,2,2}
   81: {2,2,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  126: {1,2,2,4}
  135: {2,2,2,3}
  140: {1,1,3,4}
  189: {2,2,2,4}
  198: {1,2,2,5}

Crossrefs

The LHS is A001222 (bigomega).
The RHS is A360005 (twice median).
Before multiplying the median by 2, A361800 counts partitions of this type.
For maximum instead of length we have A361856, counted by A361849.
Partitions of this type are counted by A362049.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A067801, A106529, A111907, A255201, A316413, A361868, A361908.

Programs

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

A079712 Numbers m such that bigomega(m) = 3*omega(m).

Original entry on oeis.org

1, 8, 27, 96, 125, 144, 160, 216, 224, 324, 343, 352, 400, 416, 486, 544, 608, 736, 784, 928, 992, 1000, 1184, 1215, 1312, 1331, 1376, 1504, 1696, 1701, 1888, 1920, 1936, 1952, 2025, 2144, 2197, 2272, 2336, 2500, 2528, 2656, 2673, 2688, 2704, 2744, 2848
Offset: 1

Views

Author

Benoit Cloitre, Jan 31 2003

Keywords

Comments

A cube k is a term iff k belongs to A062838; in this case, k = p_1^3 * p_2^3 *...* p_r^3 and bigomega(k) = 3*omega(k) = 3*r. - Bernard Schott, May 09 2022

Links

Crossrefs

Cf. A067801, A062838 (subsequence of cubes).

Programs

  • Mathematica
    Select[Range[3000], PrimeOmega[#] == 3*PrimeNu[#] &] (* Amiram Eldar, Jun 29 2022 *)
  • PARI
    is(n)=my(f=factor(n)[,2]); vecsum(f)==3*#f \\ Charles R Greathouse IV, Oct 16 2015

A307682 Products of four primes, two of which are distinct.

Original entry on oeis.org

24, 36, 40, 54, 56, 88, 100, 104, 135, 136, 152, 184, 189, 196, 225, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 441, 459, 472, 484, 488, 513, 536, 568, 584, 621, 632, 664, 676, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029
Offset: 1

Views

Author

Kalle Siukola, Apr 21 2019

Keywords

Comments

Numbers with exactly four prime factors (counted with multiplicity) and exactly two distinct prime factors.
Numbers n such that bigomega(n) = 4 and omega(n) = 2.
Products of a prime and the cube of a different prime (pq^3) together with squares of squarefree semiprimes (p^2*q^2).

Crossrefs

Union of A065036 and A085986.
Intersection of A007774 and A067801.
Intersection of A007774 and A195086.
Intersection of A014613 and A067801.
Intersection of A014613 and A195086.
Cf. A307342.

Programs

  • Mathematica
    Select[Range@ 1050, And[PrimeNu@ # == 2, PrimeOmega@ # == 4] &] (* Michael De Vlieger, Apr 21 2019 *)
  • PARI
    isok(n) = (bigomega(n) == 4) && (omega(n) == 2); \\ Michel Marcus, Apr 22 2019
  • Python
    import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) == 2])

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)).
Previous Showing 11-17 of 17 results.

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