A320629 -id:A320629 - cp's OEIS frontend

A330947 Nonprime numbers whose prime indices are all prime numbers.

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]]&]

A329554 Smallest MM-number of a set of n nonempty sets with no singletons.

Original entry on oeis.org

1, 13, 377, 16211, 761917, 55619941, 4393975339, 443791509239, 50148440544007, 6870336354528959, 954976753279525301, 142291536238649269849, 23193520406899830985387, 3873317907952271774559629, 701070541339361191195292849, 139513037726532877047863276951

Offset: 0

Gus Wiseman, Nov 17 2019

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding systems begins:
       1: {}
      13: {{1,2}}
     377: {{1,2},{1,3}}
   16211: {{1,2},{1,3},{1,4}}
  761917: {{1,2},{1,3},{1,4},{2,3}}

The smallest BII-number of a set of n sets is A000225(n).
BII-numbers of set-systems with no singletons are A326781.
MM-numbers of sets of nonempty sets are the odd terms of A302494.
MM-numbers of multisets of nonempty non-singleton sets are A320629.
The version with empty edges is A329556.
The version with singletons is A329557.
The version with empty edges and singletons is A329558.
Cf. A056239, A072639, A112798, A279952, A302242, A326031, A329552, A329556.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

sqvs=Select[Range[2,30],SquareFreeQ[#]&&!PrimeQ[#]&];
Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]

a(n) = Product_{i = 1..n} prime(A120944(i)).

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[#]]&]

A371444 Numbers whose binary indices are composite numbers.

Original entry on oeis.org

8, 32, 40, 128, 136, 160, 168, 256, 264, 288, 296, 384, 392, 416, 424, 512, 520, 544, 552, 640, 648, 672, 680, 768, 776, 800, 808, 896, 904, 928, 936, 2048, 2056, 2080, 2088, 2176, 2184, 2208, 2216, 2304, 2312, 2336, 2344, 2432, 2440, 2464, 2472, 2560, 2568

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
     8:           1000 ~ {4}
    32:         100000 ~ {6}
    40:         101000 ~ {4,6}
   128:       10000000 ~ {8}
   136:       10001000 ~ {4,8}
   160:       10100000 ~ {6,8}
   168:       10101000 ~ {4,6,8}
   256:      100000000 ~ {9}
   264:      100001000 ~ {4,9}
   288:      100100000 ~ {6,9}
   296:      100101000 ~ {4,6,9}
   384:      110000000 ~ {8,9}
   392:      110001000 ~ {4,8,9}
   416:      110100000 ~ {6,8,9}
   424:      110101000 ~ {4,6,8,9}
   512:     1000000000 ~ {10}
   520:     1000001000 ~ {4,10}
   544:     1000100000 ~ {6,10}
   552:     1000101000 ~ {4,6,10}
   640:     1010000000 ~ {8,10}
   648:     1010001000 ~ {4,8,10}
   672:     1010100000 ~ {6,8,10}

For powers of 2 instead of composite numbers we have A253317.
For prime indices we have the even case of A320628.
For prime instead of composite we have A326782.
This is the even case of A371444.
An opposite version is A371449.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A001222, A005117, A055887, A320629, A326781, A368109, A368533, A371289, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],EvenQ[#]&&And@@Not/@PrimeQ/@bpe[#]&]

A340104 Products of distinct primes of nonprime index (A007821).

Original entry on oeis.org

1, 2, 7, 13, 14, 19, 23, 26, 29, 37, 38, 43, 46, 47, 53, 58, 61, 71, 73, 74, 79, 86, 89, 91, 94, 97, 101, 103, 106, 107, 113, 122, 131, 133, 137, 139, 142, 146, 149, 151, 158, 161, 163, 167, 173, 178, 181, 182, 193, 194, 197, 199, 202, 203, 206, 214, 223, 226

Offset: 1

Gus Wiseman, Mar 12 2021

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 the corresponding prime indices of prime indices begins:
     1: {}              58: {{},{1,3}}        113: {{1,2,3}}
     2: {{}}            61: {{1,2,2}}         122: {{},{1,2,2}}
     7: {{1,1}}         71: {{1,1,3}}         131: {{1,1,1,1,1}}
    13: {{1,2}}         73: {{2,4}}           133: {{1,1},{1,1,1}}
    14: {{},{1,1}}      74: {{},{1,1,2}}      137: {{2,5}}
    19: {{1,1,1}}       79: {{1,5}}           139: {{1,7}}
    23: {{2,2}}         86: {{},{1,4}}        142: {{},{1,1,3}}
    26: {{},{1,2}}      89: {{1,1,1,2}}       146: {{},{2,4}}
    29: {{1,3}}         91: {{1,1},{1,2}}     149: {{3,4}}
    37: {{1,1,2}}       94: {{},{2,3}}        151: {{1,1,2,2}}
    38: {{},{1,1,1}}    97: {{3,3}}           158: {{},{1,5}}
    43: {{1,4}}        101: {{1,6}}           161: {{1,1},{2,2}}
    46: {{},{2,2}}     103: {{2,2,2}}         163: {{1,8}}
    47: {{2,3}}        106: {{},{1,1,1,1}}    167: {{2,6}}
    53: {{1,1,1,1}}    107: {{1,1,4}}         173: {{1,1,1,3}}

These primes (of nonprime index) are listed by A007821.
The non-strict version is A320628, with odd case A320629.
The odd case is A340105.
The prime instead of nonprime version:
  primes: A006450
  products: A076610
  strict: A302590
The semiprime instead of nonprime version:
  primes: A106349
  products: A339112
  strict: A340020
The squarefree semiprime instead of nonprime version:
  strict: A309356
  primes: A322551
  products: A339113
A056239 gives the sum of prime indices, which are listed by A112798.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A330944 counts nonprime prime indices.
A330945 lists numbers with a nonprime prime index (nonprime case: A330948).
A339561 lists products of distinct squarefree semiprimes (A339560).
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A018252, A289509, A320461, A320631.

Select[Range[100],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeQ[PrimePi[p]]]&]

Equals A005117 /\ A320628.

A340105 Odd products of distinct primes of nonprime index (A007821).

Original entry on oeis.org

1, 7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 79, 89, 91, 97, 101, 103, 107, 113, 131, 133, 137, 139, 149, 151, 161, 163, 167, 173, 181, 193, 197, 199, 203, 223, 227, 229, 233, 239, 247, 251, 257, 259, 263, 269, 271, 281, 293, 299, 301, 307, 311, 313, 317

Offset: 1

Gus Wiseman, Mar 12 2021

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 the corresponding sets of multisets begins:
     1: {}              91: {{1,1},{1,2}}      173: {{1,1,1,3}}
     7: {{1,1}}         97: {{3,3}}            181: {{1,2,4}}
    13: {{1,2}}        101: {{1,6}}            193: {{1,1,5}}
    19: {{1,1,1}}      103: {{2,2,2}}          197: {{2,2,3}}
    23: {{2,2}}        107: {{1,1,4}}          199: {{1,9}}
    29: {{1,3}}        113: {{1,2,3}}          203: {{1,1},{1,3}}
    37: {{1,1,2}}      131: {{1,1,1,1,1}}      223: {{1,1,1,1,2}}
    43: {{1,4}}        133: {{1,1},{1,1,1}}    227: {{4,4}}
    47: {{2,3}}        137: {{2,5}}            229: {{1,3,3}}
    53: {{1,1,1,1}}    139: {{1,7}}            233: {{2,7}}
    61: {{1,2,2}}      149: {{3,4}}            239: {{1,1,6}}
    71: {{1,1,3}}      151: {{1,1,2,2}}        247: {{1,2},{1,1,1}}
    73: {{2,4}}        161: {{1,1},{2,2}}      251: {{1,2,2,2}}
    79: {{1,5}}        163: {{1,8}}            257: {{3,5}}
    89: {{1,1,1,2}}    167: {{2,6}}            259: {{1,1},{1,1,2}}

These primes (of nonprime index) are listed by A007821.
The non-strict version is A320629, with not necessarily odd version A320628.
The not necessarily odd version is A340104.
The prime instead of odd nonprime version:
  primes: A006450
  products: A076610
  strict: A302590
The squarefree semiprime instead of odd nonprime version:
  strict: A309356
  primes: A322551
  products: A339113
The semiprime instead of odd nonprime version:
  primes: A106349
  products: A339112
  strict: A340020
A001358 lists semiprimes.
A056239 gives the sum of prime indices, which are listed by A112798.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A330944 counts nonprime prime indices.
A330945 lists numbers with a nonprime prime index (nonprime case: A330948).
A339561 lists products of distinct squarefree semiprimes.
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A018252, A289509, A320461, A320631, A320911, A320912.

Select[Range[1,100,2],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeQ[PrimePi[p]]]&]

Equals A005117 /\ A005408 /\ A320628 = A005117 /\ A320629.

A320634 Odd numbers whose multiset multisystem is a multiset partition spanning an initial interval of positive integers (odd = no empty sets).

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 21, 27, 35, 37, 39, 45, 49, 53, 57, 61, 63, 65, 69, 75, 81, 89, 91, 95, 105, 111, 113, 117, 131, 133, 135, 141, 143, 145, 147, 151, 159, 161, 165, 169, 171, 175, 183, 185, 189, 195, 207, 223, 225, 243, 245, 247, 251, 259, 265, 267, 273

Offset: 1

Gus Wiseman, Oct 18 2018

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 n-th multiset multisystem is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the 78th multiset multisystem is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
    1: {}
    3: {{1}}
    7: {{1,1}}
    9: {{1},{1}}
   13: {{1,2}}
   15: {{1},{2}}
   19: {{1,1,1}}
   21: {{1},{1,1}}
   27: {{1},{1},{1}}
   35: {{2},{1,1}}
   37: {{1,1,2}}
   39: {{1},{1,2}}
   45: {{1},{1},{2}}
   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}}
   75: {{1},{2},{2}}
   81: {{1},{1},{1},{1}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
   95: {{2},{1,1,1}}
  105: {{1},{2},{1,1}}
  111: {{1},{1,1,2}}
  113: {{1,2,3}}
  117: {{1},{1},{1,2}}
  131: {{1,1,1,1,1}}
  133: {{1,1},{1,1,1}}
  135: {{1},{1},{1},{2}}

Odd terms of A320456.

Cf. A003963, A055932, A056239, A112798, A255906, A302242, A305052, A320532, A320629.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1,100,2],normQ[primeMS/@primeMS[#]]&]

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

Gus Wiseman, Dec 28 2024

nonn

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.

			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}

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.

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

cp's OEIS Frontend

A330947 Nonprime numbers whose prime indices are all prime numbers.

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A320631 Products of odd primes of nonprime squarefree index.

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A329554 Smallest MM-number of a set of n nonempty sets with no singletons.

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

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A371444 Numbers whose binary indices are composite numbers.

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A340104 Products of distinct primes of nonprime index (A007821).

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A340105 Odd products of distinct primes of nonprime index (A007821).

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A320634 Odd numbers whose multiset multisystem is a multiset partition spanning an initial interval of positive integers (odd = no empty sets).

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A379313 Positive integers whose prime indices are not all composite.

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