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 1-6 of 6 results.

A328451 Sorted positions of first appearances in A328219, where if n = A000040(i_1) * ... * A000040(i_k), then A328219(n) = LCM(1+i_1,...,1+i_k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 13, 14, 15, 17, 19, 21, 26, 29, 35, 37, 38, 39, 42, 47, 51, 53, 58, 61, 65, 74, 78, 79, 87, 89, 91, 95, 101, 105, 106, 107, 111, 113, 119, 122, 127, 133, 141, 145, 151, 158, 159, 173, 174, 178, 181, 182, 183, 185, 195, 199, 202, 203, 214, 221
Offset: 1

Views

Author

Gus Wiseman, Oct 17 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.
Indices of 1's in the ordinal transform of A328219. - Antti Karttunen, Oct 18 2019

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  26: {1,6}
  29: {10}
  35: {3,4}
  37: {12}
  38: {1,8}
  39: {2,6}
  42: {1,2,4}
  47: {15}

Crossrefs

A subsequence of A005117.
Sorted positions of first appearances in A328219.
The GCD of the prime indices of n, all plus 1, is A328169(n).
The LCM of the prime indices of n, all minus 1, is A328456(n).
Partitions whose parts plus 1 are relatively prime are A318980.
Numbers whose prime indices plus 1 are relatively prime are A318981.
Cf. A001222, A056239, A112798, A289508, A289509, A290103, A328163, A328167, A328168, A328170.

Programs

  • Mathematica
    dav=Table[If[n==1,1,LCM@@(PrimePi/@First/@FactorInteger[n]+1)],{n,100}];
Table[Position[dav,i][[1,1]],{i,dav//.{A___,x_,B___,x_,C___}:>{A,x,B,C}}]
  • PARI
    up_to = 1024;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A290103(n) = lcm(apply(p->primepi(p),factor(n)[,1]));
A328219(n) = A290103(A003961(n));
vord_trans = ordinal_transform(vector(up_to,n,A328219(n)));
for(n=1,up_to,if(1==vord_trans[n], print1(n,", "))); \\ Antti Karttunen, Oct 18 2019

A328169 GCD of the prime indices of n, all plus 1.

Original entry on oeis.org

0, 2, 3, 2, 4, 1, 5, 2, 3, 2, 6, 1, 7, 1, 1, 2, 8, 1, 9, 2, 1, 2, 10, 1, 4, 1, 3, 1, 11, 1, 12, 2, 3, 2, 1, 1, 13, 1, 1, 2, 14, 1, 15, 2, 1, 2, 16, 1, 5, 2, 1, 1, 17, 1, 2, 1, 3, 1, 18, 1, 19, 2, 1, 2, 1, 1, 20, 2, 1, 1, 21, 1, 22, 1, 1, 1, 1, 1, 23, 2, 3, 2
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2019

Keywords

Comments

Zeros are ignored when computing GCD, and the empty set has GCD 0.
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

			85 has prime indices {3,7}, so a(85) = GCD(4,8) = 4.

Crossrefs

Positions of 0's and 1's are A318981.
Positions of records (first appearances) appear to be A116974.
The GCD of the prime indices of n, all minus 1, is A328167(n).
The LCM of the prime indices of n, all plus 1, is A328219(n).
Partitions whose parts plus 1 are relatively prime are A318980.
Cf. A000837, A056239, A112798, A258409, A289508, A289509, A290103, A328168.

Programs

  • Mathematica
    Table[GCD@@(PrimePi/@First/@If[n==1,{},FactorInteger[n]]+1),{n,100}]

Formula

a(n) = A289508(A003961(n)).

A060766 Least common multiple of differences between consecutive divisors of n (ordered by size).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 6, 15, 10, 6, 12, 35, 10, 8, 16, 9, 18, 10, 28, 99, 22, 12, 20, 143, 18, 42, 28, 60, 30, 16, 88, 255, 28, 18, 36, 323, 130, 60, 40, 21, 42, 154, 60, 483, 46, 24, 42, 75, 238, 234, 52, 27, 132, 84, 304, 783, 58, 60, 60, 899, 84, 32, 104, 165, 66, 442
Offset: 2

Views

Author

Labos Elemer, Apr 24 2001

Keywords

Examples

			For n=98, divisors={1,2,7,14,49,98}; differences={1,5,7,35,49}; a(98) = LCM of differences = 245.

Links

Crossrefs

The GCD version appears to be A258409.
The LCM of the prime indices of n is A290103(n).
The differences between consecutive divisors of n are row n of A193829.
Cf. A000005, A027750, A060680, A060681, A060683, A328023, A328026, A328219.

Programs

  • Mathematica
    a[n_ ] := LCM@@(Drop[d=Divisors[n], 1]-Drop[d, -1])
Table[LCM@@Differences[Divisors[n]],{n,2,70}] (* Harvey P. Dale, Oct 08 2012 *)

Formula

a(n) = A290103(A328023(n)). - Gus Wiseman, Oct 16 2019

Extensions

Edited by Dean Hickerson, Jan 22 2002

A333492 Position of first appearance of n in A271410 (LCM of binary indices).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 64, 128, 256, 18, 1024, 12, 4096, 66, 20, 32768, 65536, 258, 262144, 24, 68, 1026, 4194304, 132, 16777216, 4098, 67108864, 72, 268435456, 22, 1073741824, 2147483648, 1028, 65538, 80, 264, 68719476736, 262146, 4100, 144, 1099511627776, 70, 4398046511104
Offset: 1

Views

Author

Gus Wiseman, Mar 28 2020

Keywords

Comments

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.

Examples

			The sequence together with the corresponding binary expansions and binary indices begins:
      1:                 1 ~ {1}
      2:                10 ~ {2}
      4:               100 ~ {3}
      8:              1000 ~ {4}
     16:             10000 ~ {5}
      6:               110 ~ {2,3}
     64:           1000000 ~ {7}
    128:          10000000 ~ {8}
    256:         100000000 ~ {9}
     18:             10010 ~ {2,5}
   1024:       10000000000 ~ {11}
     12:              1100 ~ {3,4}
   4096:     1000000000000 ~ {13}
     66:           1000010 ~ {2,7}
     20:             10100 ~ {3,5}
  32768:  1000000000000000 ~ {16}
  65536: 10000000000000000 ~ {17}
    258:         100000010 ~ {2,9}

Links

Crossrefs

The version for prime indices is A330225.
The version for standard compositions is A333225.
Let q(k) be the binary indices of k:
- The sum of q(k) is A029931(k).
- The elements of q(k) are row k of A048793.
- The product of q(k) is A096111(k).
- The LCM of q(k) is A271410(k).
- The GCD of q(k) is A326674(k).
GCD of prime indices is A289508.
LCM of prime indices is A290103.
LCM of standard compositions is A333226.
Cf. A000120, A066099, A070939, A074761, A076078, A124767, A285572, A324837, A328219, A328451, A331579, A333227.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
q=Table[LCM@@bpe[n],{n,10000}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

Extensions

Terms a(23) and beyond from Giovanni Resta, Mar 29 2020

A328456 LCM of the prime indices of 2n + 1, all minus 1; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 1, 4, 5, 2, 6, 7, 3, 8, 2, 1, 9, 10, 4, 6, 11, 5, 12, 13, 2, 14, 3, 6, 15, 4, 7, 16, 17, 3, 10, 18, 8, 19, 20, 2, 12, 21, 1, 22, 6, 9, 23, 15, 10, 14, 24, 4, 25, 26, 6, 27, 28, 11, 29, 8, 5, 6, 4, 12, 2, 30, 13, 31, 21, 2, 32, 33, 14, 20, 18, 3, 34
Offset: 0

Views

Author

Gus Wiseman, Oct 17 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.

Examples

			The prime indices of 2 * 17 + 1 = 35, all minus 1, are {2,3}, with LCM 6, so a(17) = 6.

Crossrefs

Positions of records (first appearances) are A006005.
The GCD of the prime indices of n, all minus 1, is A328167(n).
The LCM of the prime indices of n, all plus 1, is A328219(n).
Partitions whose parts minus 1 are relatively prime are A328170.
Numbers whose prime indices minus 1 are relatively prime are A328168.
Cf. A056239, A112798, A258409, A289508, A289509, A290103, A318981, A328169, A328451.

Programs

  • Mathematica
    Table[If[n==1,0,LCM@@(PrimePi/@First/@FactorInteger[n]-1)],{n,1,100,2}]

A330225 Position of first appearance of n in A290103 = LCM of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271
Offset: 1

Views

Author

Gus Wiseman, Mar 26 2020

Keywords

Comments

Appears to be the prime numbers (A000040) with 2 replaced by 1 and 37 replaced by 35.
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.

Crossrefs

The version for product instead of lcm is A318871
The version for standard compositions is A333225.
The version for binary indices is A333492.
Let q(k) be the prime indices of k:
- The product of q(k) is A003963(k).
- The sum of q(k) is A056239(k).
- The terms of q(k) are row k of A112798.
- The GCD of q(k) is A289508(k).
- The LCM of q(k) is A290103(k).
- The LCM of q(k) + 1 is A328219(k).
Cf. A000837, A074761, A074971, A076078, A285572, A289509, A290104, A319333, A324837, A328451, A331579, A333226.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[If[n==1,1,LCM@@primeMS[n]],{n,100}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]
Showing 1-6 of 6 results.

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