A329631 Irregular triangle read by rows where row n lists the prime indices of the n-th squarefree number.

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

Offset

1,2

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.

Links

Table of n, a(n) for n=1..80.

Example

Triangle begins:
   1: {}         33: {2,5}      66: {1,2,5}     97: {25}
   2: {1}        34: {1,7}      67: {19}       101: {26}
   3: {2}        35: {3,4}      69: {2,9}      102: {1,2,7}
   5: {3}        37: {12}       70: {1,3,4}    103: {27}
   6: {1,2}      38: {1,8}      71: {20}       105: {2,3,4}
   7: {4}        39: {2,6}      73: {21}       106: {1,16}
  10: {1,3}      41: {13}       74: {1,12}     107: {28}
  11: {5}        42: {1,2,4}    77: {4,5}      109: {29}
  13: {6}        43: {14}       78: {1,2,6}    110: {1,3,5}
  14: {1,4}      46: {1,9}      79: {22}       111: {2,12}
  15: {2,3}      47: {15}       82: {1,13}     113: {30}
  17: {7}        51: {2,7}      83: {23}       114: {1,2,8}
  19: {8}        53: {16}       85: {3,7}      115: {3,9}
  21: {2,4}      55: {3,5}      86: {1,14}     118: {1,17}
  22: {1,5}      57: {2,8}      87: {2,10}     119: {4,7}
  23: {9}        58: {1,10}     89: {24}       122: {1,18}
  26: {1,6}      59: {17}       91: {4,6}      123: {2,13}
  29: {10}       61: {18}       93: {2,11}     127: {31}
  30: {1,2,3}    62: {1,11}     94: {1,15}     129: {2,14}
  31: {11}       65: {3,6}      95: {3,8}      130: {1,3,6}

Mathematica

Table[PrimePi/@First/@If[k==1, {}, FactorInteger[k]], {k, Select[Range[30], SquareFreeQ]}]

Crossrefs

Row sums are A319246.

Row lengths are A072047.

Same as A319247 with rows reversed.

Composition of A000720 and A265668.

Looking at all numbers instead of just squarefree numbers gives A112798.

Cf. A000009, A005117, A048672, A056239, A299755, A302590.

Sequence in context: A303945 A026793 A344089 * A239304 A072193 A233359

Adjacent sequences: A329628 A329629 A329630 * A329632 A329633 A329634

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 18 2019

Status

approved