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 1-3 of 3 results.

A368729 Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 51, 55, 59, 63, 65, 67, 69, 73, 75, 77, 79, 81, 83, 85, 87, 91, 93, 97, 99, 101, 105, 109, 115, 117, 119, 121, 123, 125, 127, 129, 135, 137, 139, 141, 143, 145, 147, 149
Offset: 1

Views

Author

Gus Wiseman, Jan 07 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. The multiset of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			The terms together with the corresponding multigraphs begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
  47: {{2,3}}
  49: {{1,1},{1,1}}

Crossrefs

In the unlabeled case these multigraphs are counted by A320663.
These are products of primes indexed by elements of A037143 greater than 1.
For just primes we have A076610, squarefree A302590.
For just semiprimes we have A339112, squarefree A340020.
For just half-loops we have A340019.
This is the odd case of A368728, complement A368833.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320912, A339113.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&Max@@Length/@prix/@prix[#]<=2&]

A368732 Primes whose index is one, another prime number, or a semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 47, 59, 67, 73, 79, 83, 97, 101, 109, 127, 137, 139, 149, 157, 163, 167, 179, 191, 199, 211, 227, 233, 241, 257, 269, 271, 277, 283, 293, 313, 331, 347, 353, 367, 373, 389, 401, 421, 431, 439, 443, 449, 461, 467, 487
Offset: 1

Views

Author

Gus Wiseman, Jan 08 2024

Keywords

Crossrefs

For just primes we have A006450, products A076610, strict A302590.
These indices are A037143.
For just semiprimes we have A106349, products A339112, strict A340020.
Products of these primes are A368728, odd A368729, odd strict A340019.
Products of the complementary primes are A368833.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
A322551 lists primes of squarefree semiprime index.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320911, A320912, A339113.

Programs

  • Mathematica
    Prime/@Select[Range[100],PrimeOmega[#]<=2&]

A368833 Numbers whose prime indices are not 1, prime, or semiprime.

Original entry on oeis.org

19, 37, 38, 53, 57, 61, 71, 74, 76, 89, 95, 103, 106, 107, 111, 113, 114, 122, 131, 133, 142, 148, 151, 152, 159, 171, 173, 178, 181, 183, 185, 190, 193, 197, 206, 209, 212, 213, 214, 222, 223, 226, 228, 229, 239, 244, 247, 251, 259, 262, 263, 265, 266, 267
Offset: 1

Views

Author

Gus Wiseman, Jan 08 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:
   19: {8}
   37: {12}
   38: {1,8}
   53: {16}
   57: {2,8}
   61: {18}
   71: {20}
   74: {1,12}
   76: {1,1,8}
   89: {24}
   95: {3,8}
  103: {27}
  106: {1,16}
  107: {28}
  111: {2,12}
  113: {30}
  114: {1,2,8}
  122: {1,18}
  131: {32}
  133: {4,8}
  142: {1,20}
  148: {1,1,12}

Crossrefs

These are non-products of primes indexed by elements of A037143.
The complement for just primes is A076610, strict A302590.
The complement for just semiprimes is A339112, strict A340020.
The complement for just squarefree semiprimes is A339113, strict A309356.
The complement is A368728.
The complement for just primes and semiprimes is A368729, strict A340019.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Max@@PrimeOmega/@prix[#]>2&]
Showing 1-3 of 3 results.

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