A342765 -id:A342765 - cp's OEIS frontend

A342766 a(1) = 1, for any n > 1, a(n) = A342765(a(n-1), n).

1, 2, 3, 6, 10, 10, 14, 28, 42, 42, 66, 66, 78, 78, 78, 156, 204, 204, 228, 228, 228, 228, 276, 276, 460, 460, 690, 690, 870, 870, 930, 1860, 1860, 1860, 1860, 1860, 2220, 2220, 2220, 2220, 2460, 2460, 2580, 2580, 2580, 2580, 2820, 2820, 3948, 3948, 3948, 3948

Offset: 1

Rémy Sigrist, Apr 02 2021

nonn

This sequence has similarities with A087052.
This sequence is nondecreasing.
A new value is introduced at each power of prime (A000961).
The ordinal transform of the sequence is A276781.
The RUNS transform of the sequence is A057820.

			The first terms, alongside their prime factorizations, are:
  n   a(n)  n          a(n)
  --  ----  ---------  ----------
   1     1          1           1
   2     2          2           2
   3     3          3           3
   4     6      2 * 2       2 * 3
   5    10          5       2 * 5
   6    10      2 * 3       2 * 5
   7    14          7       2 * 7
   8    28  2 * 2 * 2   2 * 2 * 7
   9    42      3 * 3   2 * 3 * 7
  10    42      2 * 5   2 * 3 * 7
  11    66         11   2 * 3 * 11
  12    66  2 * 2 * 3   2 * 3 * 11

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342766
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A000961, A057820, A087052, A276781, A342765.

PARI
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See Links section.
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A342767 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime factorizations of numbers (see Comments section for precise definition).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 3, 4, 1, 1, 2, 4, 4, 2, 1, 1, 4, 3, 8, 3, 4, 1, 1, 2, 6, 4, 4, 6, 2, 1, 1, 8, 3, 8, 5, 8, 3, 8, 1, 1, 4, 8, 4, 6, 6, 4, 8, 4, 1, 1, 4, 9, 16, 5, 12, 5, 16, 9, 4, 1, 1, 2, 6, 8, 8, 6, 6, 8, 8, 6, 2, 1, 1, 8, 3, 8, 9, 16, 7, 16, 9, 8, 3, 8, 1

Offset: 1

Rémy Sigrist, Apr 02 2021

To compute T(n, k):
- write the prime factors of n and of k in ascending order with multiplicities on two lines, right aligned,
- to "multiply" two prime numbers: take the smallest,
- to "add" two prime numbers: take the largest,
- for example, for T(12, 14):
     12   ->   2 2 3
     14   -> x   2 7
             -------
               2 2 3
           + 2 2 2
           ---------
             2 2 2 3   ->  24 = T(12, 14)
This sequence is closely related to lunar multiplication (A087062):
- 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 product in base p+1,
- as T(n, p) = n for any prime number >= A006530(n), we don't have prime numbers here,
- however, if we consider only p-smooth numbers (for some prime number p), then p is the "unit" and the semiprimes p*q (with q <= p) are "prime".

			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   2   4   2   4   2   8   4   4   2   8   2   4  ->  A061142
    3|  1  2   3   4   3   6   3   8   9   6   3  12   3   6  ->  A079065
    4|  1  4   4   8   4   8   4  16   8   8   4  16   4   8
    5|  1  2   3   4   5   6   5   8   9  10   5  12   5  10
    6|  1  4   6   8   6  12   6  16  18  12   6  24   6  12
    7|  1  2   3   4   5   6   7   8   9  10   7  12   7  14
    8|  1  8   8  16   8  16   8  32  16  16   8  32   8  16
    9|  1  4   9   8   9  18   9  16  27  18   9  36   9  18
   10|  1  4   6   8  10  12  10  16  18  20  10  24  10  20
   11|  1  2   3   4   5   6   7   8   9  10  11  12  11  14
   12|  1  8  12  16  12  24  12  32  36  24  12  48  12  24
   13|  1  2   3   4   5   6   7   8   9  10  11  12  13  14
   14|  1  4   6   8  10  12  14  16  18  20  14  24  14  28

Rémy Sigrist, Table of n, a(n) for n = 1..10011
Rémy Sigrist, PARI program for A342767
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A006530, A061142, A079065, A087062, A342765, A342768.

PARI
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See Links section.
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T(n, k) = T(k, n).
T(n, n) = A342768(n).
T(n, 1) = 1.
T(n, 2) = A061142(n).
T(n, 3) = A079065(n).
T(n, p) = n for any prime number p >= A006530(n).

cp's OEIS Frontend

A342766 a(1) = 1, for any n > 1, a(n) = A342765(a(n-1), n).

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