A305215 -id:A305215 - cp's OEIS frontend

A305325 Irregular triangle read by rows T(n, k), n >= 1 and 1 <= k <= A305215(n): T(n, k) is the k-th positive number with largest prime power factor equal to A000961(n).

1, 2, 3, 6, 4, 12, 5, 10, 15, 20, 30, 60, 7, 14, 21, 28, 35, 42, 70, 84, 105, 140, 210, 420, 8, 24, 40, 56, 120, 168, 280, 840, 9, 18, 36, 45, 63, 72, 90, 126, 180, 252, 315, 360, 504, 630, 1260, 2520, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 132, 154, 165

Offset: 1

Rémy Sigrist, May 30 2018

The largest prime power factor of a number n is given by A034699(n).

When interpreted as a flat sequence we obtain a permutation of the natural numbers.

			Triangle begins:
  1: [1]
  2: [2]
  3: [3, 6]
  4: [4, 12]
  5: [5, 10, 15, 20, 30, 60]
  6: [7, 14, 21, 28, 35, 42, 70, 84, 105, 140, 210, 420]
  7: [8, 24, 40, 56, 120, 168, 280, 840]
  8: [9, 18, 36, 45, 63, 72, 90, 126, 180, 252, 315, 360, 504, 630, 1260, 2520]

Rémy Sigrist, Table of n, a(n) for n = 1..15360
Rémy Sigrist, PARI program for A305325
Index entries for sequences that are permutations of the natural numbers

Cf. A000961, A034699, A051451, A305215, A336816 (inverse).

PARI
```
See Links section.
```

T(n, 1) = A000961(n).

T(n, A305215(n)) = A051451(n).

A336816 Inverse permutation to A305325.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 13, 25, 33, 8, 49, 6, 97, 14, 9, 193, 241, 34, 481, 10, 15, 50, 961, 26, 1921, 98, 2881, 16, 3841, 11, 7681, 15361, 51, 242, 17, 35, 18433, 482, 99, 27, 36865, 18, 73729, 52, 36, 962, 147457, 194, 294913, 1922, 243, 100, 442369, 2882, 53, 28

Offset: 1

Rémy Sigrist, Nov 21 2020

nonn

			A305325(18) = 42, so a(42) = 18.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A336816
Index entries for sequences that are permutations of the natural numbers

Cf. A000961, A305215, A305325.

PARI
```
See Links section.
```

a(A000961(n)) = 1 + Sum_{k = 1..n-1} A305215(k).

A056795 Number of divisors of k as k runs through sequence of distinct values of LCM(1,..,n).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 32, 48, 96, 192, 240, 480, 960, 1920, 2880, 3840, 7680, 15360, 18432, 36864, 73728, 147456, 294912, 442368, 884736, 1769472, 3538944, 4128768, 8257536, 16515072, 33030144, 66060288, 82575360, 165150720, 330301440

Offset: 1

Labos Elemer, Aug 28 2000

nonn

Values of LCM's in A003418 and accordingly their number of divisors jump at powers of primes (A000961). Here divisor-numbers of LCM's are displayed without repetition.

			For x = 19,20,21,22 the value of A003418(x) = A051451(13) = LCM(1,..,x) = 232792560, of which the total number of divisors is 960, so a(13) = 960.

Amiram Eldar, Table of n, a(n) for n = 1..3365

Cf. A000005, A003418, A051451, A000961.

Partial sums of A305215.

f(n) = lcm(vector(n, i, i)); \\ A003418
lista(nn) = {my(last = 0); for (n=1, nn, my(new = f(n)); if (new != last, print1(numdiv(new), ", "); last = new););} \\ Michel Marcus, Oct 08 2020

a(n) = A000005(A051451(n)).

cp's OEIS Frontend

A305325 Irregular triangle read by rows T(n, k), n >= 1 and 1 <= k <= A305215(n): T(n, k) is the k-th positive number with largest prime power factor equal to A000961(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A336816 Inverse permutation to A305325.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A056795 Number of divisors of k as k runs through sequence of distinct values of LCM(1,..,n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula