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 1-4 of 4 results.

A342768 a(n) = A342767(n, n).

Original entry on oeis.org

1, 2, 3, 8, 5, 12, 7, 32, 27, 20, 11, 48, 13, 28, 45, 128, 17, 108, 19, 80, 63, 44, 23, 192, 125, 52, 243, 112, 29, 180, 31, 512, 99, 68, 175, 432, 37, 76, 117, 320, 41, 252, 43, 176, 405, 92, 47, 768, 343, 500, 153, 208, 53, 972, 275, 448, 171, 116, 59, 720
Offset: 1

Views

Author

Rémy Sigrist, Apr 02 2021

Keywords

Comments

This sequence has similarities with A087019.
These are the positions of first appearances of each positive integer in A346701, and also in A346703. - Gus Wiseman, Aug 09 2021

Examples

			For n = 42:
- 42 = 2 * 3 * 7, so:
          2 3 7
        x 2 3 7
        -------
          2 3 7
        2 3 3
    + 2 2 2
    -----------
      2 2 3 3 7
- hence a(42) = 2 * 2 * 3 * 3 * 7 = 252.

Links

Crossrefs

Cf. A007947, A087019, A342767.
The sum of prime indices of a(n) is 2*A056239(n) - A061395(n).
The version for even indices is A129597(n) = 2*a(n) for n > 1.
The sorted version is A346635.
These are the positions of first appearances in A346701 and in A346703.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A027193 counts partitions of odd length, ranked by A026424.
A209281 adds up the odd bisection of standard compositions (even: A346633).
A346697 adds up the odd bisection of prime indices (reverse: A346699).
Cf. A000290, A006530, A033942, A037143, A344653, A345957, A346698, A346700, A346704.

Programs

  • Mathematica
    Table[n^2/FactorInteger[n][[-1,1]],{n,100}] (* Gus Wiseman, Aug 09 2021 *)
  • PARI
    See Links section.

Formula

a(n) = n iff n = 1 or n is a prime number.
a(p^k) = p^(2*k-1) for any k > 0 and any prime number p.
A007947(a(n)) = A007947(n).
A001222(a(n)) = 2*A001222(n) - 1 for any n > 1.
From Gus Wiseman, Aug 09 2021: (Start)
A001221(a(n)) = A001221(n).
If g = A006530(n) is the greatest prime factor of n, then a(n) = n^2/g.
a(n) = A129597(n)/2.
(End)

A346635 Numbers whose division (or multiplication) by their greatest prime factor yields a perfect square. Numbers k such that k*A006530(k) is a perfect square.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 80, 83, 89, 92, 97, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 148, 149, 151, 153
Offset: 1

Views

Author

Gus Wiseman, Aug 10 2021

Keywords

Comments

This is the sorted version of A342768(n) = position of first appearance of n in A346701 (but A346703 works also).

Examples

			The terms together with their prime indices begin:
     1: {}          31: {11}            71: {20}
     2: {1}         32: {1,1,1,1,1}     73: {21}
     3: {2}         37: {12}            76: {1,1,8}
     5: {3}         41: {13}            79: {22}
     7: {4}         43: {14}            80: {1,1,1,1,3}
     8: {1,1,1}     44: {1,1,5}         83: {23}
    11: {5}         45: {2,2,3}         89: {24}
    12: {1,1,2}     47: {15}            92: {1,1,9}
    13: {6}         48: {1,1,1,1,2}     97: {25}
    17: {7}         52: {1,1,6}         99: {2,2,5}
    19: {8}         53: {16}           101: {26}
    20: {1,1,3}     59: {17}           103: {27}
    23: {9}         61: {18}           107: {28}
    27: {2,2,2}     63: {2,2,4}        108: {1,1,2,2,2}
    28: {1,1,4}     67: {19}           109: {29}
    29: {10}        68: {1,1,7}        112: {1,1,1,1,4}

Crossrefs

Removing 1 gives a subset of A026424.
The unsorted even version is A129597.
The unsorted version is A342768(n) = A342767(n,n).
Except the first term, the even version is 2*a(n).
A000290 lists squares.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A006530 gives the greatest prime factor.
A061395 gives the greatest prime index.
A027193 counts partitions of odd length.
A056239 adds up prime indices, row sums of A112798.
A209281 = odd bisection sum of standard compositions (even: A346633).
A316524 = alternating sum of prime indices (sign: A344617, rev.: A344616).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
A346697 = odd bisection sum of prime indices (weights of A346703).
A346699 = odd bisection sum of reversed prime indices (weights of A346701).
Cf. A028260, A033942, A035363, A037143, A341446, A344653, A345957, A345958, A345959, A346698, A346700, A346704.

Programs

  • Maple
    filter:= proc(n) issqr(n/max(numtheory:-factorset(n))) end proc:
filter(1):= true:
select(filter, [$1..200]); # Robert Israel, Nov 26 2022
  • Mathematica
    sqrQ[n_]:=IntegerQ[Sqrt[n]];
Select[Range[100],sqrQ[#*FactorInteger[#][[-1,1]]]&]
  • PARI
    isok(m) = (m==1) || issquare(m/vecmax(factor(m)[,1])); \\ Michel Marcus, Aug 12 2021

Formula

a(n) = A129597(n)/2 for n > 1.

A342765 Array T(n, k), n, k > 0, read by antidiagonals; T(n, k) = max(A006530(n), A006530(k)) * T(n/A006530(n), k/A006530(k)) with T(1, 1) = 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 3, 4, 5, 6, 5, 6, 6, 5, 6, 7, 6, 5, 4, 5, 6, 7, 8, 7, 6, 10, 10, 6, 7, 8, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 9, 12, 14, 10, 10, 14, 12, 9, 10, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 20, 14, 14, 20, 9, 10, 11, 12
Offset: 1

Views

Author

Rémy Sigrist, Apr 02 2021

Keywords

Comments

To compute T(n, k):
- write the prime factors of n and of k in ascending order with multiplicities on two lines, right aligned,
- take the largest prime number in each column and multiply back,
- for example, for T(12, 14):
12 -> 2 2 3
14 -> 2 7
-----
2 2 7 -> 28 = T(12, 14)
This sequence is closely related to lunar addition (A087061):
- let n and k be two p-smooth numbers,
- let f be the function that associates to a p-smooth number, say m, the unique number whose (p+1)-base digits are prime, nondecreasing and whose product is m,
- let g be the inverse of f,
- then for any p-smooth numbers n and k, T(n, k) = g(f(n) "+" f(k)) where "+" denotes lunar addition in base p+1,
- see A342767 for the corresponding multiplication.

Examples

			Array T(n, k) begins:
  n\k|   1   2   3   4   5   6   7   8   9  10  11  12  13  14
  ---+--------------------------------------------------------
    1|   1   2   3   4   5   6   7   8   9  10  11  12  13  14
    2|   2   2   3   4   5   6   7   8   9  10  11  12  13  14
    3|   3   3   3   6   5   6   7  12   9  10  11  12  13  14
    4|   4   4   6   4  10   6  14   8   9  10  22  12  26  14
    5|   5   5   5  10   5  10   7  20  15  10  11  20  13  14
    6|   6   6   6   6  10   6  14  12   9  10  22  12  26  14
    7|   7   7   7  14   7  14   7  28  21  14  11  28  13  14
    8|   8   8  12   8  20  12  28   8  18  20  44  12  52  28
    9|   9   9   9   9  15   9  21  18   9  15  33  18  39  21
   10|  10  10  10  10  10  10  14  20  15  10  22  20  26  14
   11|  11  11  11  22  11  22  11  44  33  22  11  44  13  22
   12|  12  12  12  12  20  12  28  12  18  20  44  12  52  28
   13|  13  13  13  26  13  26  13  52  39  26  13  52  13  26
   14|  14  14  14  14  14  14  14  28  21  14  22  28  26  14

Links

Crossrefs

Cf. A001222, A006530, A087061, A342766, A342767.

Programs

  • PARI
    gpf(n) = if (n==1, 1, my (p=factor(n)[,1]~); p[#p])
T(n, k) = if (n==1 || k==1, max(n, k), my (p=gpf(n), q=gpf(k)); max(p, q)*T(n/p, k/q))

Formula

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 1) = n.
T(n, n) = n.
A001222(T(n, k)) = max(A001222(n), A001222(k)).
A006530(T(n, k)) = max(A006530(n), A006530(k)).

A343033 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime exponents of numbers (see Comments section for precise definition).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 5, 2, 1, 1, 5, 3, 3, 5, 1, 1, 6, 7, 4, 7, 6, 1, 1, 7, 15, 5, 5, 15, 7, 1, 1, 2, 11, 6, 11, 6, 11, 2, 1, 1, 3, 3, 7, 35, 35, 7, 3, 3, 1, 1, 10, 5, 4, 13, 30, 13, 4, 5, 10, 1, 1, 11, 21, 9, 5, 77, 77, 5, 9, 21, 11, 1
Offset: 1

Views

Author

Rémy Sigrist and N. J. A. Sloane, Apr 03 2021

Keywords

Comments

To compute T(n, k):
- write the prime exponents of n and of k on two lines, right aligned (these lines correspond to rows of A067255 in reversed order),
- to "multiply" two prime numbers: take the smallest,
- to "add" two prime numbers: take the largest,
- for example, for T(12, 14):
(11 7 5 3 2)
12 --> 1 2
14 --> x 1 0 0 1
---------
1 1
0 0
0 0
+ 1 1
-----------
1 1 0 1 1 --> 462 = T(12, 14)
This sequence is closely related to lunar multiplication (A087062):
- for any b > 1, let S_b be the set of nonnegative integers m such that A051903(m)< b,
- there is a natural bijection f from S_b to the set of nonnegative integers:
f(Product_{k >= 0} prime(k)^d(k)) = Sum_{k >= 0} d(k) * b^k,
- let g be the inverse of f,
- then for any numbers n and k in S_b, we have:
T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base b,
- the corresponding addition table is A003990.

Examples

			Array T(n, k) begins:
  n\k|  1   2   3   4   5    6    7   8   9   10   11   12   13   14
  ----  -  --  --  --  --  ---  ---  --  --  ---  ---  ---  ---  ---
    1|  1   1   1   1   1    1    1   1   1    1    1    1    1    1
    2|  1   2   3   2   5    6    7   2   3   10   11    6   13   14  --> A007947
    3|  1   3   5   3   7   15   11   3   5   21   13   15   17   33  --> A328915
    4|  1   2   3   4   5    6    7   4   9   10   11   12   13   14  --> A007948
    5|  1   5   7   5  11   35   13   5   7   55   17   35   19   65
    6|  1   6  15   6  35   30   77   6  15  210  143   30  221  462
    7|  1   7  11   7  13   77   17   7  11   91   19   77   23  119
    8|  1   2   3   4   5    6    7   8   9   10   11   12   13   14
    9|  1   3   5   9   7   15   11   9  25   21   13   45   17   33
   10|  1  10  21  10  55  210   91  10  21  110  187  210  247  910
   11|  1  11  13  11  17  143   19  11  13  187   23  143   29  209
   12|  1   6  15  12  35   30   77  12  45  210  143   60  221  462
   13|  1  13  17  13  19  221   23  13  17  247   29  221   31  299
   14|  1  14  33  14  65  462  119  14  33  910  209  462  299  238

Links

Crossrefs

Cf. A003990, A007947, A007948, A051903, A067255, A087062, A328915, A342767, A343035.

Programs

  • PARI
    T(n,k) = { my (r=1, pp=factor(n)[,1]~, qq=factor(k)[,1]~); for (i=1, #pp, for (j=1, #qq, my (p=prime(primepi(pp[i])+primepi(qq[j])-1), v=valuation(r, p), w=min(valuation(n, pp[i]), valuation(k, qq[j]))); if (w>v, r*=p^(w-v)))); r }

Formula

T(n, k) = T(k, n).
T(n, 1) = 1.
T(n, 2) = A007947(n).
T(n, 3) = A328915(n).
T(n, 4) = A007948(n).
T(n, n) = A343035(n).
A051903(T(n, k)) = min(A051903(n), A051903(k)).
Showing 1-4 of 4 results.

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

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