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 1-7 of 7 results.

A327403 Number of steps to reach a fixed point starting with n and repeatedly taking the quotient by the maximum stable divisor (A327393, A327402).

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Sep 15 2019

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. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476. The maximum stable divisor of n is A327393(n).

Examples

			We have 798 -> 42 -> 6 -> 2 -> 1, so a(798) = 4.

Links

Crossrefs

See link for additional cross-references.
Positions of first appearance of each integer are A325782.
Cf. A000005, A303362, A323014, A327393, A327402.

Programs

  • Mathematica
    stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[FixedPointList[#/Max[Select[Divisors[#],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&]]&,n]]-2,{n,100}]
  • PARI
    A327403(n) = for(k=0,oo,my(nextn=A327402(n)); if(nextn==n,return(k)); n = nextn); \\ Antti Karttunen, Jan 28 2025

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

A316476 Stable numbers. Numbers whose distinct prime indices are pairwise indivisible.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 64, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 128, 131, 135, 137
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2018

Keywords

Comments

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

Examples

			The prime indices of 45 are {2,2,3}, so the distinct prime indices are {2,3}, which are pairwise indivisible, so 45 belongs to the sequence.
The prime indices of 105 are {2,3,4}, which are not pairwise indivisible (2 divides 4), so 105 does not belong to the sequence.

Links

Crossrefs

Cf. A056239, A112798, A285572, A285573, A303362, A304713, A316468, A316475, A327393, A327394, A378442 (characteristic function).

Programs

  • Mathematica
    Select[Range[100],Select[Tuples[If[#===1,{},Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]]],2],UnsameQ@@#&&Divisible@@#&]=={}&]
  • PARI
    ok(n)={my(v=apply(primepi, factor(n)[,1])); for(j=2, #v, for(i=1, j-1, if(v[j]%v[i]==0, return(0)))); 1} \\ Andrew Howroyd, Aug 26 2018

A328677 Numbers whose distinct prime indices are relatively prime and pairwise indivisible.

Original entry on oeis.org

2, 4, 8, 15, 16, 32, 33, 35, 45, 51, 55, 64, 69, 75, 77, 85, 93, 95, 99, 119, 123, 128, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 205, 207, 209, 215, 217, 219, 221, 225, 245, 249, 253, 255, 256, 265, 275, 279, 287, 291, 295, 297, 309, 323
Offset: 1

Views

Author

Gus Wiseman, Oct 30 2019

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. Stable numbers are listed in A316476.

Examples

			The sequence of terms together with their prime indices begins:
    2: {1}
    4: {1,1}
    8: {1,1,1}
   15: {2,3}
   16: {1,1,1,1}
   32: {1,1,1,1,1}
   33: {2,5}
   35: {3,4}
   45: {2,2,3}
   51: {2,7}
   55: {3,5}
   64: {1,1,1,1,1,1}
   69: {2,9}
   75: {2,3,3}
   77: {4,5}
   85: {3,7}
   93: {2,11}
   95: {3,8}
   99: {2,2,5}
  119: {4,7}

Crossrefs

These are the Heinz numbers of the partitions counted by A328676.
Numbers whose prime indices are relatively prime are A289509.
Partitions whose distinct parts are pairwise indivisible are A305148.
The version for binary indices (instead of prime indices) is A328671.
Cf. A000837, A056239, A112798, A285573, A289508, A303362, A304713, A327393, A328460.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],GCD@@primeMS[#]==1&&stableQ[primeMS[#],Divisible]&]

Formula

Intersection of A316476 and A289509.

A328676 Number of relatively prime integer partitions of n whose distinct parts are pairwise indivisible.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 3, 5, 5, 11, 7, 16, 14, 18, 22, 34, 30, 47, 45, 59, 66, 89, 90, 118, 125, 159, 169, 218, 225, 289, 304, 369, 400, 486, 520, 636, 680, 806, 873, 1051, 1105, 1333, 1424, 1664, 1803, 2122, 2253, 2659, 2841, 3283, 3560, 4118, 4388, 5096
Offset: 1

Views

Author

Gus Wiseman, Oct 29 2019

Keywords

Examples

			The a(4) = 1 through a(11) = 11 partitions:
  1111  32     111111  43       53        54         73          65
        11111          52       332       72         433         74
                       322      11111111  522        532         83
                       1111111            3222       3322        92
                                          111111111  1111111111  443
                                                                 533
                                                                 722
                                                                 3332
                                                                 5222
                                                                 32222
                                                                 11111111111

Crossrefs

The Heinz numbers of these partitions are given by A328677.
The strict case is A328678.
The binary index version is A328671.
Relatively prime partitions are A000837.
Partitions whose distinct parts are pairwise indivisible are A305148.
Cf. A285573, A289509, A303362, A316476, A326912, A327393, A328171.

Programs

  • Mathematica
    stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&stableQ[#,Divisible]&]],{n,30}]

A327394 Number of stable divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 4, 5, 2, 4, 2, 4, 3, 3, 2, 5, 3, 3, 4, 4, 2, 5, 2, 6, 4, 3, 4, 5, 2, 3, 3, 5, 2, 4, 2, 4, 6, 3, 2, 6, 3, 4, 4, 4, 2, 5, 4, 5, 3, 3, 2, 6, 2, 3, 4, 7, 3, 5, 2, 4, 4, 5, 2, 6, 2, 3, 6, 4, 4, 4, 2, 6, 5, 3, 2, 5, 4, 3, 3, 5, 2, 7, 4, 4, 4, 3, 4, 7, 2, 4, 6, 5, 2, 5, 2, 5, 6, 3, 2, 6, 2, 5, 3, 6, 2, 4, 3, 4, 4, 3, 4, 7, 3
Offset: 1

Views

Author

Gus Wiseman, Sep 15 2019

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. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476. Maximum stable divisor is A327393.

Examples

			The stable divisors of 60 are {1, 2, 3, 4, 5, 15}, so a(60) = 6.

Links

Crossrefs

See link for additional cross-references.
Cf. A000005, A033273, A303362, A316476, A327393.
Inverse Möbius transform of A378442.

Programs

  • Mathematica
    stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Divisors[n],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&]],{n,100}]
  • PARI
    A378442(n)={my(v=apply(primepi, factor(n)[, 1])); for(j=2, #v, for(i=1, j-1, if(v[j]%v[i]==0, return(0)))); 1}; \\ From the function "ok" in A316476 by Andrew Howroyd, Aug 26 2018
A327394(n) = sumdiv(n,d,A378442(d)); \\ Antti Karttunen, Nov 27 2024

Formula

a(n) = Sum_{d|n} A378442(d). - Antti Karttunen, Nov 27 2024

Extensions

More terms from Antti Karttunen, Nov 27 2024

A327402 Quotient of n over the maximum stable divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 3, 5, 1, 6, 1, 4, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 7, 3, 2, 1, 4, 1, 2, 7, 1, 5, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 6, 1, 5, 1, 2, 1, 12, 1, 2, 3, 8, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 8, 3
Offset: 1

Views

Author

Gus Wiseman, Sep 15 2019

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. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476.

Examples

			The stable divisors of 60 are {1, 2, 3, 4, 5, 15}, so a(60) = 60/15 = 4.

Links

Crossrefs

See link for additional cross-references.
Cf. A000005, A033273, A303362, A316476, A327403, A378442.

Programs

  • Mathematica
    stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[n/Max[Select[Divisors[n],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&]],{n,100}]
  • PARI
    A378442(n)={my(v=apply(primepi, factor(n)[, 1])); for(j=2, #v, for(i=1, j-1, if(v[j]%v[i]==0, return(0)))); 1}; \\ From the function "ok" in A316476 by Andrew Howroyd, Aug 26 2018
A327402(n) = fordiv(n,d,if(A378442(n/d),return(d))); \\ Antti Karttunen, Jan 28 2025

Formula

a(n) = n/A327393(n).

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

A329366 Numbers whose distinct prime indices are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197
Offset: 1

Views

Author

Gus Wiseman, Nov 12 2019

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 Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are Heinz numbers of stable intersecting partitions.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Crossrefs

Intersection of A316476 and A328867.
Heinz numbers of the partitions counted by A328871.
Replacing "intersecting" with "relatively prime" gives A328677.
Cf. A056239, A112798, A285573, A289509, A303362, A304713, A327393, A328671.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],stableQ[Union[primeMS[#]],GCD[#1,#2]==1&]&&stableQ[Union[primeMS[#]],Divisible]&]
Showing 1-7 of 7 results.

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