A320633 -id:A320633 - cp's OEIS frontend

A320628 Products of primes of nonprime index.

1, 2, 4, 7, 8, 13, 14, 16, 19, 23, 26, 28, 29, 32, 37, 38, 43, 46, 47, 49, 52, 53, 56, 58, 61, 64, 71, 73, 74, 76, 79, 86, 89, 91, 92, 94, 97, 98, 101, 103, 104, 106, 107, 112, 113, 116, 122, 128, 131, 133, 137, 139, 142, 146, 148, 149, 151, 152, 158, 161, 163

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

The index of a prime number n is the number m such that n is the m-th prime.

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

			The sequence of terms begins:
   1 = 1
   2 = prime(1)
   4 = prime(1)^2
   7 = prime(4)
   8 = prime(1)^3
  13 = prime(6)
  14 = prime(1)*prime(4)
  16 = prime(1)^4
  19 = prime(8)
  23 = prime(9)
  26 = prime(1)*prime(6)
  28 = prime(1)^2*prime(4)
  29 = prime(10)
  32 = prime(1)^5
  37 = prime(12)
  38 = prime(1)*prime(8)
  43 = prime(14)
  46 = prime(1)*prime(9)
  47 = prime(15)
  49 = prime(4)^2
  52 = prime(1)^2*prime(6)
  53 = prime(16)
  56 = prime(1)^3*prime(4)
  58 = prime(1)*prime(10)
  61 = prime(18)
  64 = prime(1)^6
  71 = prime(20)
  73 = prime(21)
  74 = prime(1)*prime(12)
  76 = prime(1)^2*prime(8)
  79 = prime(22)
  86 = prime(1)*prime(14)
  89 = prime(24)
  91 = prime(4)*prime(6)
  92 = prime(1)^2*prime(9)
  94 = prime(1)*prime(15)
  97 = prime(25)
  98 = prime(1)*prime(4)^2

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A018252, A056239, A290103, A302242, A302478, A320533, A320629, A320630, A320631, A320633.
Complement of A331386.
Positions of zeros in 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 this sequence.
The number of prime prime indices is given by A257994.
The number of nonprime prime indices is given by A330944.
Cf. A000040, A000720, A001222, A112798, A295665.

Select[Range[100],And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]&]

A330944 Number of nonprime prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 13 2020

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.

			24 has prime indices {1,1,1,2}, of which {1,1,1} are nonprime, so a(24) = 3.

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. A000040, A000720, A001222, A007097, A018252, A056239, A112798, A302242, A320629, A320633, A330946, A330947.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}/;!PrimeQ[PrimePi[p]]:>k]],{n,30}]

a(n) = my(f=factor(n)); sum(k=1, #f~, if(!isprime(primepi(f[k,1])), f[k,2], 0)); \\ Daniel Suteu, Jan 14 2020

a(n) + A257994(n) = A001222(n).

Additive with a(p^e) = e if primepi(p) is nonprime, and 0 otherwise. - Amiram Eldar, Nov 03 2023

A320629 Products of odd primes of nonprime index.

Original entry on oeis.org

1, 7, 13, 19, 23, 29, 37, 43, 47, 49, 53, 61, 71, 73, 79, 89, 91, 97, 101, 103, 107, 113, 131, 133, 137, 139, 149, 151, 161, 163, 167, 169, 173, 181, 193, 197, 199, 203, 223, 227, 229, 233, 239, 247, 251, 257, 259, 263, 269, 271, 281, 293, 299, 301, 307, 311

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

The index of a prime number n is the number m such that n is the m-th prime.

The asymptotic density of this sequence is (1/2) * Product_{p in A006450} (1 - 1/p) = 1/(2*Sum_{n>=1} 1/A076610(n)) < 1/6. - Amiram Eldar, Feb 02 2021

			The sequence of terms begins:
    1 = 1
    7 = prime(4)
   13 = prime(6)
   19 = prime(8)
   23 = prime(9)
   29 = prime(10)
   37 = prime(12)
   43 = prime(14)
   47 = prime(15)
   49 = prime(4)^2
   53 = prime(16)
   61 = prime(18)
   71 = prime(20)
   73 = prime(21)
   79 = prime(22)
   89 = prime(24)
   91 = prime(4)*prime(6)
   97 = prime(25)
  101 = prime(26)
  103 = prime(27)
  107 = prime(28)
  113 = prime(30)
  131 = prime(32)
  133 = prime(4)*prime(8)
  137 = prime(33)
  139 = prime(34)
  149 = prime(35)
  151 = prime(36)
  161 = prime(4)*prime(9)

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A006450, A007821, A018252, A056239, A076610, A112798, A302242, A320533, A320628, A320630, A320631, A320633.

Select[Range[1,100,2],And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]&]

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

Gus Wiseman, Jan 13 2020

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 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}}

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.

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

Gus Wiseman, Jan 17 2020

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 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

			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}

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

Select[Range[100],MemberQ[FactorInteger[#],{?(PrimeQ@*PrimePi),}]&]

A257994(a(n)) > 0.

A320630 Products of primes of nonprime squarefree index.

Original entry on oeis.org

2, 4, 8, 13, 16, 26, 29, 32, 43, 47, 52, 58, 64, 73, 79, 86, 94, 101, 104, 113, 116, 128, 137, 139, 146, 149, 158, 163, 167, 169, 172, 181, 188, 199, 202, 208, 226, 232, 233, 256, 257, 269, 271, 274, 278, 292, 293, 298, 313, 316, 317, 326, 334, 338, 344, 347

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

The index of a prime number n is the number m such that n is the m-th prime.

			The sequence of terms begins:
    2 = prime(1)
    4 = prime(1)^2
    8 = prime(1)^3
   13 = prime(6)
   16 = prime(1)^4
   26 = prime(1)*prime(6)
   29 = prime(10)
   32 = prime(1)^5
   43 = prime(14)
   47 = prime(15)
   52 = prime(1)^2*prime(6)
   58 = prime(1)*prime(10)
   64 = prime(1)^6
   73 = prime(21)
   79 = prime(22)
   86 = prime(1)*prime(14)
   94 = prime(1)*prime(15)
  101 = prime(26)
  104 = prime(1)^3*prime(6)
  113 = prime(30)
  116 = prime(1)^2*prime(10)
  128 = prime(1)^7

Cf. A000040, A006450, A007821, A018252, A056239, A076610, A112798, A302242, A302478, A320628, A320629, A320631, A320633.

Select[Range[100],With[{f=PrimePi/@First/@FactorInteger[#]},And[And@@SquareFreeQ/@f,And@@Not/@PrimeQ/@f]]&]

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

Gus Wiseman, Jan 13 2020

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 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}}

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.

Select[Range[100],!PrimeQ[#]&&!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

Gus Wiseman, Jan 13 2020

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 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}}

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.

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

A320631 Products of odd primes of nonprime squarefree index.

Original entry on oeis.org

13, 29, 43, 47, 73, 79, 101, 113, 137, 139, 149, 163, 167, 169, 181, 199, 233, 257, 269, 271, 293, 313, 317, 347, 349, 373, 377, 389, 397, 421, 439, 443, 449, 467, 487, 491, 499, 557, 559, 571, 577, 601, 607, 611, 619, 631, 647, 653, 673, 677, 727, 733, 751

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

			The sequence of terms begins:
   13 = prime(6)
   29 = prime(10)
   43 = prime(14)
   47 = prime(15)
   73 = prime(21)
   79 = prime(22)
  101 = prime(26)
  113 = prime(30)
  137 = prime(33)
  139 = prime(34)
  149 = prime(35)
  163 = prime(38)
  167 = prime(39)
  169 = prime(6)^2
  181 = prime(42)
  199 = prime(46)
  233 = prime(51)
  257 = prime(55)
  269 = prime(57)
  271 = prime(58)
  293 = prime(62)
  313 = prime(65)
  317 = prime(66)
  347 = prime(69)
  349 = prime(70)
  373 = prime(74)
  377 = prime(6)*prime(10)

Cf. A000040, A006450, A007821, A018252, A056239, A076610, A112798, A302242, A302478, A320628, A320629, A320630, A320633.

Select[Range[1,100,2],With[{f=PrimePi/@First/@FactorInteger[#]},And[And@@SquareFreeQ/@f,And@@Not/@PrimeQ/@f]]&]

A330949 Odd nonprime numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

21, 35, 39, 49, 57, 63, 65, 69, 77, 87, 91, 95, 105, 111, 115, 117, 119, 129, 133, 141, 143, 145, 147, 159, 161, 169, 171, 175, 183, 185, 189, 195, 203, 207, 209, 213, 215, 217, 219, 221, 231, 235, 237, 245, 247, 253, 259, 261, 265, 267, 273, 285, 287, 291

Offset: 1

Gus Wiseman, Jan 14 2020

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.

Also MM-numbers of multiset partitions with at least two parts, not all of which are singletons (see example).

			The sequence of terms together with their prime indices of prime indices begins:
   21: {{1},{1,1}}
   35: {{2},{1,1}}
   39: {{1},{1,2}}
   49: {{1,1},{1,1}}
   57: {{1},{1,1,1}}
   63: {{1},{1},{1,1}}
   65: {{2},{1,2}}
   69: {{1},{2,2}}
   77: {{1,1},{3}}
   87: {{1},{1,3}}
   91: {{1,1},{1,2}}
   95: {{2},{1,1,1}}
  105: {{1},{2},{1,1}}
  111: {{1},{1,1,2}}
  115: {{2},{2,2}}
  117: {{1},{1},{1,2}}
  119: {{1,1},{4}}
  129: {{1},{1,4}}
  133: {{1,1},{1,1,1}}
  141: {{1},{2,3}}

Complement of A106092 in A330945.
Including even numbers gives A330948.
Including primes gives A330946.
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 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, A320633, A330943, A330947.

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

cp's OEIS Frontend

A320628 Products of primes of nonprime index.

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A330944 Number of nonprime prime indices of n.

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A320629 Products of odd primes of nonprime index.

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A330945 Numbers whose prime indices are not all prime numbers.

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A331386 Numbers with at least one prime prime index.

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A320630 Products of primes of nonprime squarefree index.

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

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A330947 Nonprime numbers whose prime indices are all prime numbers.

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A320631 Products of odd primes of nonprime squarefree index.