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 1-5 of 5 results.

A330945 Numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 84, 86, 87
Offset: 1

Views

Author

Gus Wiseman, Jan 13 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 of prime indices begins:
   2: {{}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
  10: {{},{2}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  20: {{},{},{2}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  28: {{},{},{1,1}}
  29: {{1,3}}

Crossrefs

Complement of A076610 (products of primes of prime index).
Numbers n such that A330944(n) > 0.
The restriction to odd terms is A330946.
The restriction to nonprimes is A330948.
The number of prime prime indices is given by A257994.
The number of nonprime prime indices is given by A330944.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000040, A000720, A001222, A018252, A056239, A112798, A302242, A320633, A330943, A330947, A330949.

Programs

  • Mathematica
    Select[Range[100],!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A331386 Numbers with at least one prime prime index.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 15, 17, 18, 20, 21, 22, 24, 25, 27, 30, 31, 33, 34, 35, 36, 39, 40, 41, 42, 44, 45, 48, 50, 51, 54, 55, 57, 59, 60, 62, 63, 65, 66, 67, 68, 69, 70, 72, 75, 77, 78, 80, 81, 82, 83, 84, 85, 87, 88, 90, 93, 95, 96, 99, 100, 102, 105, 108
Offset: 1

Views

Author

Gus Wiseman, Jan 17 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.
The asymptotic density of this sequence is 1 - Product_{p in A006450} (1 - 1/p) = 1 - 1/(Sum_{n>=1} 1/A076610(n)) > 2/3. - Amiram Eldar, Feb 02 2021

Examples

			The sequence of terms together with their prime indices begins:
    3: {2}
    5: {3}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   30: {1,2,3}
   31: {11}
   33: {2,5}
   34: {1,7}

Links

Crossrefs

Complement of A320628.
Positions of terms > 0 in A257994.
Positions of terms > 1 in A295665.
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 nonprime prime indices is given by A330944.
Cf. A000040, A000720, A001222, A018252, A056239, A076610, A112798, A302242, A320633, A330943, A330944, A330947, A330949.

Programs

Formula

A257994(a(n)) > 0.

A330948 Nonprime numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106
Offset: 1

Views

Author

Gus Wiseman, Jan 13 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 of prime indices begins:
   4: {{},{}}
   6: {{},{1}}
   8: {{},{},{}}
  10: {{},{2}}
  12: {{},{},{1}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  20: {{},{},{2}}
  21: {{1},{1,1}}
  22: {{},{3}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  28: {{},{},{1,1}}
  30: {{},{1},{2}}
  32: {{},{},{},{},{}}
  34: {{},{4}}
  35: {{2},{1,1}}
  36: {{},{},{1},{1}}
  38: {{},{1,1,1}}

Crossrefs

Complement in A330945 of A000040.
Complement in A018252 of A076610.
The restriction to odd terms is A330949.
Nonprime numbers n such that A330944(n) > 0.
Taking odds instead of nonprimes gives A330946.
The number of prime prime indices is given by A257994.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000720, A001222, A056239, A112798, A302242, A320629, A320633, A330943, A330947.

Programs

  • Mathematica
    Select[Range[100],!PrimeQ[#]&&!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A330946 Odd numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

7, 13, 19, 21, 23, 29, 35, 37, 39, 43, 47, 49, 53, 57, 61, 63, 65, 69, 71, 73, 77, 79, 87, 89, 91, 95, 97, 101, 103, 105, 107, 111, 113, 115, 117, 119, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 159, 161, 163, 167, 169, 171, 173, 175, 181, 183, 185
Offset: 1

Views

Author

Gus Wiseman, Jan 13 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.
Also MM-numbers of multiset partitions whose parts not all singletons (see example).

Examples

			The sequence of terms together with their prime indices of prime indices begins:
   7: {{1,1}}
  13: {{1,2}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  29: {{1,3}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  43: {{1,4}}
  47: {{2,3}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  57: {{1},{1,1,1}}
  61: {{1,2,2}}
  63: {{1},{1},{1,1}}
  65: {{2},{1,2}}
  69: {{1},{2,2}}
  71: {{1,1,3}}
  73: {{2,4}}

Crossrefs

Odd numbers n such that A330944(n) > 0.
Including even numbers gives A330945.
The restriction to nonprimes is A330949.
Taking nonprimes instead of odds gives A330947.
The number of prime prime indices is given by A257994.
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 set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000040, A000720, A001222, A018252, A056239, A112798, A302242, A320629, A330943, A330947, A330948.

Programs

  • Mathematica
    Select[Range[1,100,2],!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A330947 Nonprime numbers whose prime indices are all prime numbers.

Original entry on oeis.org

1, 9, 15, 25, 27, 33, 45, 51, 55, 75, 81, 85, 93, 99, 121, 123, 125, 135, 153, 155, 165, 177, 187, 201, 205, 225, 243, 249, 255, 275, 279, 289, 295, 297, 327, 335, 341, 363, 369, 375, 381, 405, 415, 425, 451, 459, 465, 471, 495, 527, 531, 537, 545, 561, 573
Offset: 1

Views

Author

Gus Wiseman, Jan 13 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 of prime indices begins:
    1: {}
    9: {{1},{1}}
   15: {{1},{2}}
   25: {{2},{2}}
   27: {{1},{1},{1}}
   33: {{1},{3}}
   45: {{1},{1},{2}}
   51: {{1},{4}}
   55: {{2},{3}}
   75: {{1},{2},{2}}
   81: {{1},{1},{1},{1}}
   85: {{2},{4}}
   93: {{1},{5}}
   99: {{1},{1},{3}}
  121: {{3},{3}}
  123: {{1},{6}}
  125: {{2},{2},{2}}
  135: {{1},{1},{1},{2}}
  153: {{1},{1},{4}}
  155: {{2},{5}}

Crossrefs

Complement in A076610 of A000040.
Complement in A018252 of A330948.
Nonprime numbers n such that A330944(n) = 0.
Taking odds instead of nonprimes gives A330946.
The number of prime prime indices is given by A257994.
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.
Numbers whose prime indices are not all prime are A330945.
Cf. A000720, A001222, A056239, A112798, A302242, A320629, A320633, A330949.

Programs

  • Mathematica
    Select[Range[100],!PrimeQ[#]&&And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]
Showing 1-5 of 5 results.

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