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.

A379306 Number of squarefree prime indices of n.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Dec 25 2024

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 prime indices of 39 are {2,6}, so a(39) = 2.
The prime indices of 70 are {1,3,4}, so a(70) = 2.
The prime indices of 98 are {1,4,4}, so a(98) = 1.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Crossrefs

Positions of first appearances are A000079.
Positions of zero are A379307, counted by A114374 (strict A256012).
Positions of one are A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379310 nonsquarefree, see A302478.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A057627, A068361, A070321, A071403, A072284, A112925, A112929, A120327, A377430, A378086.

Programs

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

Formula

Totally additive with a(prime(k)) = A008966(k).

A379317 Positive integers with a unique even prime index.

Original entry on oeis.org

3, 6, 7, 12, 13, 14, 15, 19, 24, 26, 28, 29, 30, 33, 35, 37, 38, 43, 48, 51, 52, 53, 56, 58, 60, 61, 65, 66, 69, 70, 71, 74, 75, 76, 77, 79, 86, 89, 93, 95, 96, 101, 102, 104, 106, 107, 112, 113, 116, 119, 120, 122, 123, 130, 131, 132, 138, 139, 140, 141, 142
Offset: 1

Views

Author

Gus Wiseman, Dec 29 2024

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}
   6: {1,2}
   7: {4}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  19: {8}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  43: {14}
  48: {1,1,1,1,2}

Crossrefs

Partitions of this type are counted by A038348 (strict A096911).
For all even parts we have A066207, counted by A035363 (strict A000700).
For no even parts we have A066208, counted by A000009 (strict A035457).
Positions of 1 in A257992.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A349158.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A038550, A087436.

Programs

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

A331914 Numbers with at most one prime prime index, counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 52, 53, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 86, 87
Offset: 1

Views

Author

Gus Wiseman, Feb 08 2020

Keywords

Comments

First differs from A324935 in having 39.
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 sequence of terms together with their prime indices begins:
   1: {}           24: {1,1,1,2}      52: {1,1,6}
   2: {1}          26: {1,6}          53: {16}
   3: {2}          28: {1,1,4}        56: {1,1,1,4}
   4: {1,1}        29: {10}           57: {2,8}
   5: {3}          31: {11}           58: {1,10}
   6: {1,2}        32: {1,1,1,1,1}    59: {17}
   7: {4}          34: {1,7}          61: {18}
   8: {1,1,1}      35: {3,4}          62: {1,11}
  10: {1,3}        37: {12}           64: {1,1,1,1,1,1}
  11: {5}          38: {1,8}          65: {3,6}
  12: {1,1,2}      39: {2,6}          67: {19}
  13: {6}          40: {1,1,1,3}      68: {1,1,7}
  14: {1,4}        41: {13}           69: {2,9}
  16: {1,1,1,1}    42: {1,2,4}        70: {1,3,4}
  17: {7}          43: {14}           71: {20}
  19: {8}          44: {1,1,5}        73: {21}
  20: {1,1,3}      46: {1,9}          74: {1,12}
  21: {2,4}        47: {15}           76: {1,1,8}
  22: {1,5}        48: {1,1,1,1,2}    77: {4,5}
  23: {9}          49: {4,4}          78: {1,2,6}

Crossrefs

These are numbers n such that A257994(n) <= 1.
Prime-indexed primes are A006450, with products A076610.
The number of distinct prime prime indices is A279952.
Numbers with at least one prime prime index are A331386.
The set S of numbers with at most one prime index in S are A331784.
The set S of numbers with at most one distinct prime index in S are A331912.
Numbers with exactly one prime prime index are A331915.
Numbers with exactly one distinct prime prime index are A331916.
Numbers with at most one distinct prime prime index are A331995.
Cf. A000040, A000720, A007097, A018252, A112798, A320628, A330945, A331785.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[primeMS[#],_?PrimeQ]<=1&]

A331916 Numbers with exactly one distinct prime prime index.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 17, 18, 20, 21, 22, 24, 25, 27, 31, 34, 35, 36, 39, 40, 41, 42, 44, 48, 50, 54, 57, 59, 62, 63, 65, 67, 68, 69, 70, 72, 77, 78, 80, 81, 82, 83, 84, 87, 88, 95, 96, 100, 108, 109, 111, 114, 115, 117, 118, 119, 121, 124, 125, 126, 127
Offset: 1

Views

Author

Gus Wiseman, Feb 08 2020

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 sequence of terms together with their prime indices begins:
    3: {2}           40: {1,1,1,3}       81: {2,2,2,2}
    5: {3}           41: {13}            82: {1,13}
    6: {1,2}         42: {1,2,4}         83: {23}
    9: {2,2}         44: {1,1,5}         84: {1,1,2,4}
   10: {1,3}         48: {1,1,1,1,2}     87: {2,10}
   11: {5}           50: {1,3,3}         88: {1,1,1,5}
   12: {1,1,2}       54: {1,2,2,2}       95: {3,8}
   17: {7}           57: {2,8}           96: {1,1,1,1,1,2}
   18: {1,2,2}       59: {17}           100: {1,1,3,3}
   20: {1,1,3}       62: {1,11}         108: {1,1,2,2,2}
   21: {2,4}         63: {2,2,4}        109: {29}
   22: {1,5}         65: {3,6}          111: {2,12}
   24: {1,1,1,2}     67: {19}           114: {1,2,8}
   25: {3,3}         68: {1,1,7}        115: {3,9}
   27: {2,2,2}       69: {2,9}          117: {2,2,6}
   31: {11}          70: {1,3,4}        118: {1,17}
   34: {1,7}         72: {1,1,1,2,2}    119: {4,7}
   35: {3,4}         77: {4,5}          121: {5,5}
   36: {1,1,2,2}     78: {1,2,6}        124: {1,1,11}
   39: {2,6}         80: {1,1,1,1,3}    125: {3,3,3}

Crossrefs

These are numbers n such that A279952(n) = 1.
Prime-indexed primes are A006450, with products A076610.
The number of prime prime indices is A257994.
Numbers with at least one prime prime index are A331386.
The set S of numbers with exactly one prime index in S are A331785.
The set S of numbers with exactly one distinct prime index in S are A331913.
Numbers with at most one prime prime index are A331914.
Numbers with at most one distinct prime prime index are A331995.
Cf. A000040, A000720, A007097, A007821, A018252, A112798, A289509, A320628, A330944, A330945, A331784.

Programs

  • Mathematica
    Select[Range[100],Count[PrimePi/@First/@FactorInteger[#],_?PrimeQ]==1&]

A295665 Fully multiplicative with a(prime(m)) = prime(k) when m = prime(k), and a(prime(m)) = 1 when m is not a prime.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Nov 26 2017

Keywords

Comments

The number of applications to reach 1 is A322027(n). Positions of first appearances are A076610. - Gus Wiseman, Jan 17 2020

Examples

			For n = 360 = 2^3 * 3^2 * 5 = prime(1)^3 * prime(2)^2 * prime(3), 1 is not a prime, but 2 and 3 are, thus a(360) = 2^2 * 3 = 12.

Links

Crossrefs

Cf. A000040, A000720, A010051, A295666.
Cf. also A003963, A257538.
Positions of 1's are A320628.
Positions of terms > 1 are A331386.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
The number of prime prime indices is A257994.
The number of nonprime prime indices is A330944.
Numbers whose prime indices are not all prime are A330945.
Cf. A001222, A007097, A018252, A056239, A112798, A320633, A322027, A330947, A330948.

Programs

  • Mathematica
    Table[Times@@Cases[FactorInteger[n],{p_?(PrimeQ[PrimePi[#]]&),k_}:>PrimePi[p]^k],{n,40}] (* Gus Wiseman, Jan 17 2020 *)
  • Scheme
    (definec (A295665 n) (if (= 1 n) 1 (let ((k (A055396 n))) (* (if (zero? (A010051 k)) 1 k) (A295665 (A032742 n))))))

Formula

Multiplicative with a(p^e) = A000720(p)^(e*A010051(A000720(p))).
a(1) = 1; for n > 1, if A055396(n) is a prime, then a(n) = A055396(n) * a(A032742(n)), otherwise a(n) = a(A032742(n)).
Other identities. For all n >= 1:
a(A006450(n)) = A000040(n).
a(A007097(n)) = A007097(n-1).
a(A294876(n)) = A295666(n).

A331995 Numbers with at most one distinct prime prime index.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 76
Offset: 1

Views

Author

Gus Wiseman, Feb 08 2020

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 sequence of terms together with their prime indices begins:
   1: {}           22: {1,5}          44: {1,1,5}
   2: {1}          23: {9}            46: {1,9}
   3: {2}          24: {1,1,1,2}      47: {15}
   4: {1,1}        25: {3,3}          48: {1,1,1,1,2}
   5: {3}          26: {1,6}          49: {4,4}
   6: {1,2}        27: {2,2,2}        50: {1,3,3}
   7: {4}          28: {1,1,4}        52: {1,1,6}
   8: {1,1,1}      29: {10}           53: {16}
   9: {2,2}        31: {11}           54: {1,2,2,2}
  10: {1,3}        32: {1,1,1,1,1}    56: {1,1,1,4}
  11: {5}          34: {1,7}          57: {2,8}
  12: {1,1,2}      35: {3,4}          58: {1,10}
  13: {6}          36: {1,1,2,2}      59: {17}
  14: {1,4}        37: {12}           61: {18}
  16: {1,1,1,1}    38: {1,8}          62: {1,11}
  17: {7}          39: {2,6}          63: {2,2,4}
  18: {1,2,2}      40: {1,1,1,3}      64: {1,1,1,1,1,1}
  19: {8}          41: {13}           65: {3,6}
  20: {1,1,3}      42: {1,2,4}        67: {19}
  21: {2,4}        43: {14}           68: {1,1,7}

Crossrefs

These are numbers n such that A279952(n) <= 1.
Prime-indexed primes are A006450, with products A076610.
Numbers whose prime indices are not all prime are A330945.
Numbers with at least one prime prime index are A331386.
The set S of numbers with at most one prime index in S are A331784.
The set S of numbers with at most one distinct prime index in S are A331912.
Numbers with at most one prime prime index are A331914.
Numbers with exactly one prime prime index are A331915.
Numbers with exactly one distinct prime prime index are A331916.
Cf. A000040, A000720, A001221, A007097, A007821, A112798, A257994, A320628, A330944, A331785, A331912, A331913.

Programs

  • Mathematica
    Select[Range[100],Count[PrimePi/@First/@FactorInteger[#],_?PrimeQ]<=1&]

A379313 Positive integers whose prime indices are not all composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82
Offset: 1

Views

Author

Gus Wiseman, Dec 28 2024

Keywords

Comments

Or, positive integers whose prime indices include at least one 1 or prime number.
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:
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    17: {7}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}

Crossrefs

Partitions of this type are counted by A000041 - A023895.
The "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
The complement is A320629, counted by A023895 (strict A204389).
For a unique prime we have A331915, counted by A379304 (strict A379305).
Positions of nonzeros in A379311.
For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
A377033 gives k-th differences of composite numbers, see A073445, A377034.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
`Cf. A038550, A087436, A173390, A379300, A379301.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@CompositeQ/@prix[#]&]
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