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 21-24 of 24 results.

A339740 Non-products of distinct primes or squarefree semiprimes.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336
Offset: 1

Views

Author

Gus Wiseman, Dec 20 2020

Keywords

Comments

Differs from A293243 and A212164 in having 1080, with prime indices {1,1,1,2,2,2,3} and factorization into distinct squarefree numbers 2*3*6*30.

Examples

			The sequence of terms together with their prime indices begins:
      4: {1,1}             80: {1,1,1,1,3}
      8: {1,1,1}           81: {2,2,2,2}
      9: {2,2}             88: {1,1,1,5}
     16: {1,1,1,1}         96: {1,1,1,1,1,2}
     24: {1,1,1,2}        104: {1,1,1,6}
     25: {3,3}            108: {1,1,2,2,2}
     27: {2,2,2}          112: {1,1,1,1,4}
     32: {1,1,1,1,1}      121: {5,5}
     40: {1,1,1,3}        125: {3,3,3}
     48: {1,1,1,1,2}      128: {1,1,1,1,1,1,1}
     49: {4,4}            135: {2,2,2,3}
     54: {1,2,2,2}        136: {1,1,1,7}
     56: {1,1,1,4}        144: {1,1,1,1,2,2}
     64: {1,1,1,1,1,1}    152: {1,1,1,8}
     72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
For example, a complete list of strict factorizations of 72 is: (2*3*12), (2*4*9), (2*36), (3*4*6), (3*24), (4*18), (6*12), (8*9), (72); but since none of these consists of only primes or squarefree semiprimes, 72 is in the sequence.

Crossrefs

A013929 allows only primes.
A320894 does not allow primes (but omega is assumed even).
A339741 is the complement.
A339742 has zeros at these positions.
A339840 allows squares of primes.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050326 into distinct squarefree numbers.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339661 into distinct squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339617 counts non-graphical partitions of 2n (A339618).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
The following count partitions/factorizations of even length and give their Heinz numbers:
- A027187/A339846 counts all of even length (A028260).
- A096373/A339737 cannot be partitioned into strict pairs (A320891).
- A338915/A339662 cannot be partitioned into distinct pairs (A320892).
- A339559/A339564 cannot be partitioned into distinct strict pairs (A320894).
Cf. A001221, A005117, A050320, A320893, A320911, A320912, A320922, A320924, A339113, A339561.

Programs

  • Mathematica
    sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqps[#]=={}&]

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

Views

Author

Gus Wiseman, Mar 12 2021

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 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}}

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Gus Wiseman, Mar 12 2021

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 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}}

Crossrefs

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.

Programs

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

Formula

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

A340017 Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.

Original entry on oeis.org

36, 100, 196, 216, 225, 360, 441, 484, 504, 540, 600, 676, 756, 792, 936, 1000, 1089, 1156, 1176, 1188, 1224, 1225, 1296, 1350, 1368, 1400, 1404, 1444, 1500, 1521, 1656, 1836, 1960, 2052, 2088, 2116, 2160, 2200, 2232, 2250, 2484, 2600, 2601, 2646, 2664, 2744
Offset: 1

Views

Author

Gus Wiseman, Dec 30 2020

Keywords

Comments

Of course, every number is a product of squarefree numbers (A050320).
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
All terms have even Omega (A001222, A028260).

Examples

			The sequence of terms together with their prime indices begins:
      36: {1,1,2,2}        1000: {1,1,1,3,3,3}
     100: {1,1,3,3}        1089: {2,2,5,5}
     196: {1,1,4,4}        1156: {1,1,7,7}
     216: {1,1,1,2,2,2}    1176: {1,1,1,2,4,4}
     225: {2,2,3,3}        1188: {1,1,2,2,2,5}
     360: {1,1,1,2,2,3}    1224: {1,1,1,2,2,7}
     441: {2,2,4,4}        1225: {3,3,4,4}
     484: {1,1,5,5}        1296: {1,1,1,1,2,2,2,2}
     504: {1,1,1,2,2,4}    1350: {1,2,2,2,3,3}
     540: {1,1,2,2,2,3}    1368: {1,1,1,2,2,8}
     600: {1,1,1,2,3,3}    1400: {1,1,1,3,3,4}
     676: {1,1,6,6}        1404: {1,1,2,2,2,6}
     756: {1,1,2,2,2,4}    1444: {1,1,8,8}
     792: {1,1,1,2,2,5}    1500: {1,1,2,3,3,3}
     936: {1,1,1,2,2,6}    1521: {2,2,6,6}
For example, a complete list of all factorizations of 7560 into squarefree semiprimes is:
  7560 = (6*6*6*35) = (6*6*10*21) = (6*6*14*15),
but since none of these is strict, 7560 is in the sequence.

Links

Crossrefs

See link for additional cross references.
The distinct prime shadows (under A181819) of these terms are A339842.
Factorizations into squarefree semiprimes are counted by A320656.
Products of squarefree semiprimes that are not products of distinct semiprimes are A320893.
Factorizations into distinct squarefree semiprimes are A339661.
For the next four lines, we list numbers with even Omega (A028260).
- A320891 cannot be factored into squarefree semiprimes.
- A320894 cannot be factored into distinct squarefree semiprimes.
- A320911 can be factored into squarefree semiprimes.
- A339561 can be factored into distinct squarefree semiprimes.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A030229 lists squarefree numbers with even Omega.
A050320 counts factorizations into squarefree numbers.
A050326 counts factorizations into distinct squarefree numbers.
A181819 is the Heinz number of the prime signature of n (prime shadow).
A320656 counts factorizations into squarefree semiprimes.
A339560 can be partitioned into distinct strict pairs.
Cf. A001055, A001222, A005117, A007717, A096373, A112798, A300061, A322353, A339559, A338899, A339740.

Programs

  • Mathematica
    strr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strr[n/d],Min@@#>=d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[1000],Select[strr[#],UnsameQ@@#&]=={}&&strr[#]!={}&]

Formula

Equals A320894 /\ A320911.
Numbers n such that A320656(n) > 0 but A339661(n) = 0.
Previous Showing 21-24 of 24 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.