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.

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A302521 Odd numbers whose prime indices are squarefree and have disjoint prime indices. Numbers n such that the n-th multiset multisystem is a set partition.

Original entry on oeis.org

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 41, 43, 47, 51, 55, 59, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 215, 219, 221, 233, 241, 249, 255, 257, 269, 271
Offset: 1

Views

Author

Gus Wiseman, Apr 09 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set partitions.
01: {}
03: {{1}}
05: {{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
55: {{2},{3}}
59: {{7}}
67: {{8}}
73: {{2,4}}
79: {{1,5}}
83: {{9}}
85: {{2},{4}}
93: {{1},{5}}

Crossrefs

Cf. A000110, A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279790, A281113, A294788, A301750, A302242, A302243, A302505.

Programs

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

A302601 Numbers that are powers of a prime number whose prime index is also a prime power (not including 1).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 17, 19, 23, 25, 27, 31, 41, 49, 53, 59, 67, 81, 83, 97, 103, 109, 121, 125, 127, 131, 157, 179, 191, 211, 227, 241, 243, 277, 283, 289, 311, 331, 343, 353, 361, 367, 401, 419, 431, 461, 509, 529, 547, 563, 587, 599, 617, 625, 661, 691, 709
Offset: 1

Views

Author

Gus Wiseman, Apr 10 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			49 is in the sequence because 49 = prime(4)^2 = prime(prime(1)^2)^2.
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset multisystems.
001: {}
003: {{1}}
005: {{2}}
007: {{1,1}}
009: {{1},{1}}
011: {{3}}
017: {{4}}
019: {{1,1,1}}
023: {{2,2}}
025: {{2},{2}}
027: {{1},{1},{1}}
031: {{5}}
041: {{6}}
049: {{1,1},{1,1}}
053: {{1,1,1,1}}
059: {{7}}
067: {{8}}
081: {{1},{1},{1},{1}}
083: {{9}}
097: {{3,3}}
103: {{2,2,2}}
109: {{10}}
121: {{3},{3}}
125: {{2},{2},{2}}
127: {{11}}
131: {{1,1,1,1,1}}

Crossrefs

Cf. A000961, A001222, A003963, A005117, A007425, A007716, A056239, A275024, A281113, A295931, A301764, A302242, A302243, A302498.

Programs

  • Mathematica
    Select[Range[1000],#===1||MatchQ[FactorInteger[#],{{?(PrimePowerQ[PrimePi[#]]&),}}]&]
  • PARI
    isok(n) = (n==1) || ((isprimepower(n, &p)) && isprimepower(primepi(p))); \\ Michel Marcus, Apr 10 2018

A318994 Totally additive with a(prime(n)) = n + 1.

Original entry on oeis.org

0, 2, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 8, 8, 8, 10, 9, 8, 9, 9, 9, 11, 9, 12, 10, 9, 10, 9, 10, 13, 11, 10, 10, 14, 10, 15, 10, 10, 12, 16, 11, 10, 10, 11, 11, 17, 11, 10, 11, 12, 13, 18, 11, 19, 14, 11, 12, 11, 11, 20, 12, 13, 11, 21, 12, 22
Offset: 1

Views

Author

Gus Wiseman, Sep 07 2018

Keywords

Links

Crossrefs

Cf. A000040, A001222, A003963, A056239, A275024, A299757, A302242, A302243, A318995.

Programs

  • Maple
    a:= n-> add((1+numtheory[pi](i[1]))*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 07 2018
  • Mathematica
    Table[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>(PrimePi[p]+1)*k]//Total,{n,100}]
  • PARI
    a(n)={my(f=factor(n)); sum(i=1, #f~, my([p,e]=f[i,]); (primepi(p)+1)*e)} \\ Andrew Howroyd, Sep 07 2018

A324324 MM-numbers of crossing set partitions.

Original entry on oeis.org

2117, 3973, 4843, 5891, 6757, 7181, 7801, 10019, 10063, 11051, 11567, 13021, 13193, 13459, 14123, 14921, 17603, 18407, 18761, 18877, 19307, 19633, 20941, 21083, 21251, 21457, 22849, 23519, 23533, 24727, 26101, 27133, 27169, 27173, 27413, 29111, 30479, 31261
Offset: 1

Views

Author

Gus Wiseman, Feb 22 2019

Keywords

Comments

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 in 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}}.
A multiset multisystem is crossing if it contains two parts of the form {{...x...y...},{...z...t...}} with x < z < y < t or z < x < t < y.

Crossrefs

Cf. A000108 (non-crossing set partitions), A001055, A001222, A003963, A005117, A016098 (crossing set partitions), A054726, A056239, A112798, A302242, A302243, A302505, A302521 (MM-numbers of set partitions).
Cf. A324166, A324167, A324169, A324170, A324171, A324326.

Programs

  • Mathematica
    croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xTable[PrimePi[p],{k}]]]];
setptnQ[bks_]:=UnsameQ@@Join@@bks&&!MemberQ[bks,{}];
Select[Range[10000],And[croXQ[primeMS/@primeMS[#]],setptnQ[primeMS/@primeMS[#]]]&]

A305830 Combined weight of the n-th FDH set-system. Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then add up their FD-weights (A064547).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 3, 2, 2, 2, 2, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 1, 3, 3, 2, 3, 2, 2, 2, 2, 2, 3, 1, 2, 3, 2, 3, 2, 3, 3, 3, 2, 1, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 1, 1, 3, 3, 2, 3
Offset: 1

Views

Author

Gus Wiseman, Jun 10 2018

Keywords

Comments

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. Then a(n) = w(s_1) + ... + w(s_k) where w = A064547.

Examples

			Sequence of FDH set-systems (a list containing all finite sets of finite sets of positive integers) begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{2}}
   5: {{3}}
   6: {{},{1}}
   7: {{4}}
   8: {{},{2}}
   9: {{1,2}}
  10: {{},{3}}
  11: {{5}}
  12: {{1},{2}}
  13: {{1,3}}
  14: {{},{4}}
  15: {{1},{3}}
  16: {{6}}
  17: {{1,4}}
  18: {{},{1,2}}
  19: {{7}}
  20: {{2},{3}}
  21: {{1},{4}}
  22: {{},{5}}
  23: {{2,3}}
  24: {{},{1},{2}}
  25: {{8}}
  26: {{},{1,3}}
  27: {{1},{1,2}}

Crossrefs

Cf. A003963, A050376, A056239, A061775, A064547, A213925, A279065, A279614, A299755, A299756, A299757, A302242, A302243, A305829, A305832.

Programs

  • Mathematica
    nn=100;
FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Total[Length/@(FDfactor/@(FDfactor[n]/.FDrules))],{n,nn}]

A302496 Products of distinct primes of prime-power index.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 38, 41, 42, 46, 51, 53, 55, 57, 59, 62, 66, 67, 69, 70, 77, 82, 83, 85, 93, 95, 97, 102, 103, 105, 106, 109, 110, 114, 115, 118, 119, 123, 127, 131, 133, 134, 138, 154, 155, 157, 159
Offset: 1

Views

Author

Gus Wiseman, Apr 09 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of constant-multiset systems.
01: {}
02: {{}}
03: {{1}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
10: {{},{2}}
11: {{3}}
14: {{},{1,1}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
30: {{},{1},{2}}
31: {{5}}
33: {{1},{3}}
34: {{},{4}}
35: {{2},{1,1}}
38: {{},{1,1,1}}

Crossrefs

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279786, A281113, A296131, A301767, A302242, A302243, A302493, A302494.

Programs

  • Mathematica
    Select[Range[nn],Or[#===1,SquareFreeQ[#]&&And@@PrimePowerQ/@PrimePi/@DeleteCases[FactorInteger[#][[All,1]],2]]&]
  • PARI
    is(n) = if(bigomega(n)!=omega(n), return(0), my(f=factor(n)[, 1]~); for(k=1, #f, if(!isprimepower(primepi(f[k])) && primepi(f[k])!=1, return(0)))); 1 \\ Felix Fröhlich, Apr 10 2018

A302596 Powers of prime numbers of prime index.

Original entry on oeis.org

1, 3, 5, 9, 11, 17, 25, 27, 31, 41, 59, 67, 81, 83, 109, 121, 125, 127, 157, 179, 191, 211, 241, 243, 277, 283, 289, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 625, 709, 729, 739, 773, 797, 859, 877, 919, 961, 967, 991, 1031, 1063, 1087, 1153
Offset: 1

Views

Author

Gus Wiseman, Apr 10 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
001: {}
003: {{1}}
005: {{2}}
009: {{1},{1}}
011: {{3}}
017: {{4}}
025: {{2},{2}}
027: {{1},{1},{1}}
031: {{5}}
041: {{6}}
059: {{7}}
067: {{8}}
081: {{1},{1},{1},{1}}
083: {{9}}
109: {{10}}
121: {{3},{3}}
125: {{2},{2},{2}}

Links

Crossrefs

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A047968, A281113, A295924, A301760, A302242, A302243.

Programs

Formula

Sum_{n>=1} 1/a(n) = 1 + Sum_{p in A006450} 1/(p-1). - Amiram Eldar, Sep 19 2022

A357139 Take the weakly increasing prime indices of each prime index of n, then concatenate.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 5, 1, 3, 4, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 6, 1, 1, 1, 1, 4, 3, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 1, 4, 1, 2
Offset: 1

Views

Author

Gus Wiseman, Sep 29 2022

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

			Triangle begins:
   1:
   2:
   3:  1
   4:
   5:  2
   6:  1
   7:  1 1
   8:
   9:  1 1
  10:  2
  11:  3
  12:  1
  13:  1 2
For example, the weakly increasing prime indices of 105 are (2,3,4), with prime indices ((1),(2),(1,1)), so row 105 is (1,2,1,1).

Crossrefs

Row lengths are A302242.
Positions of strict rows are A302505.
Positions of constant rows are A302593.
Row sums are A325033, products A325032.
The version for standard compositions is A357135, rank A357134.
A000961 lists prime powers.
A003963 multiples prime indices.
A056239 adds up prime indices.
Cf. A000720, A001221, A001222, A007716, A058891, A109082, A275024, A302243, A324926, A325034.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Join@@primeMS/@primeMS[n],{n,100}]

A302498 Numbers that are a power of a prime number whose prime index is itself a power of a prime number.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 41, 49, 53, 59, 64, 67, 81, 83, 97, 103, 109, 121, 125, 127, 128, 131, 157, 179, 191, 211, 227, 241, 243, 256, 277, 283, 289, 311, 331, 343, 353, 361, 367, 401, 419, 431, 461, 509, 512, 529, 547, 563
Offset: 1

Views

Author

Gus Wiseman, Apr 09 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			49 is in the sequence because 49 = prime(prime(1)^2)^2.
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of constant constant-multiset multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
07: {{1,1}}
08: {{},{},{}}
09: {{1},{1}}
11: {{3}}
16: {{},{},{},{}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
25: {{2},{2}}
27: {{1},{1},{1}}
31: {{5}}
32: {{},{},{},{},{}}
41: {{6}}
49: {{1,1},{1,1}}
53: {{1,1,1,1}}
59: {{7}}
64: {{},{},{},{},{},{}}

Links

Crossrefs

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A076610, A275024, A279789, A295920, A301763, A302242, A302243, A302492, A302493, A302496, A302497.

Programs

  • Mathematica
    Select[Range[100],Or[#===1,PrimePowerQ[#]&&And@@(Or[#===1,PrimePowerQ[#]]&/@PrimePi/@FactorInteger[#][[All,1]])]&]
  • PARI
    ok(n)={my(p); n == 1 || (isprimepower(n, &p) && (p == 2 || isprimepower(primepi(p))))} \\ Andrew Howroyd, Aug 26 2018

A291686 Numbers whose prime indices other than 1 are distinct prime numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 17, 20, 22, 24, 30, 31, 32, 33, 34, 40, 41, 44, 48, 51, 55, 59, 60, 62, 64, 66, 67, 68, 80, 82, 83, 85, 88, 93, 96, 102, 109, 110, 118, 120, 123, 124, 127, 128, 132, 134, 136, 155, 157, 160, 164, 165, 166, 170, 176, 177
Offset: 1

Views

Author

Gus Wiseman, Apr 09 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.

Examples

			9 is not in the sequence because the prime indices of 9 = prime(2)*prime(2) are {2,2} which are prime numbers but not distinct.
15 is in the sequence because the prime indices of 15 = prime(2)*prime(3) are {2,3} which are distinct prime numbers.
21 is not in the sequence because the prime indices of 21 = prime(2)*prime(4) are {2,4} which are distinct but not all prime numbers.
24 is in the sequence because the prime indices of 24 = prime(1)*prime(1)*prime(1)*prime(2) are {1,1,1,2} which without the 1s are distinct prime numbers.

Links

Crossrefs

Cf. A000009, A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A279790, A281113, A294788, A294788, A301750, A302242, A302243.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#===1,UnsameQ@@DeleteCases[primeMS[#],1]&&And@@(PrimeQ/@DeleteCases[primeMS[#],1])]&]
  • PARI
    ok(n)={my(t=n>>valuation(n,2)); issquarefree(t) && !#select(p->!isprime(primepi(p)), factor(t)[,1])} \\ Andrew Howroyd, Aug 26 2018

Formula

Sum_{n>=1} 1/a(n) = 2 * Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021
Previous Showing 11-20 of 32 results. Next

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