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.

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A379845 Even squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

Original entry on oeis.org

2, 30, 154, 190, 390, 442, 506, 658, 714, 874, 1110, 1118, 1254, 1330, 1430, 1786, 1794, 1798, 1958, 2310, 2414, 2442, 2470, 2730, 2958, 3034, 3066, 3266, 3390, 3534, 3710, 3770, 3874, 3914, 4042, 4466, 4526, 4758, 4930, 5106, 5434, 5474, 5642, 6090, 6106
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2025

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. The sum and product of prime indices are A056239 and A003963 respectively.

Examples

			The terms together with their prime indices begin:
     2: {1}
    30: {1,2,3}
   154: {1,4,5}
   190: {1,3,8}
   390: {1,2,3,6}
   442: {1,6,7}
   506: {1,5,9}
   658: {1,4,15}
   714: {1,2,4,7}
   874: {1,8,9}
  1110: {1,2,3,12}

Crossrefs

Even squarefree case of A326149.
For nonprime instead of even we have A326158.
Squarefree case of A379319.
Even case of A379844.
Partitions of this type are counted by A380221, see A379733, A379735.
A003963 multiplies together prime indices.
A005117 lists the squarefree numbers.
A056239 adds up prime indices.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A112798, A175508, A324850, A324851, A326150, A326151, A326153/A326154, A326156, A326157.

Programs

  • Mathematica
    Select[Range[2,1000],EvenQ[#]&&SquareFreeQ[#]&&Divisible[Times@@prix[#],Plus@@prix[#]]&]

A380216 Numbers whose prime indices have (product)/(sum) equal to an integer > 1.

Original entry on oeis.org

49, 63, 65, 81, 125, 150, 154, 165, 169, 190, 198, 259, 273, 333, 351, 361, 364, 385, 390, 435, 442, 468, 481, 490, 495, 506, 525, 561, 580, 595, 609, 630, 658, 675, 700, 714, 741, 765, 781, 783, 810, 840, 841, 846, 874, 900, 918, 925, 931, 935, 952, 988
Offset: 1

Views

Author

Gus Wiseman, Jan 23 2025

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. The sum and product of prime indices are A056239 and A003963 respectively.

Examples

			The terms together with their prime indices begin:
   49: {4,4}
   63: {2,2,4}
   65: {3,6}
   81: {2,2,2,2}
  125: {3,3,3}
  150: {1,2,3,3}
  154: {1,4,5}
  165: {2,3,5}
  169: {6,6}
  190: {1,3,8}
  198: {1,2,2,5}
  259: {4,12}
  273: {2,4,6}
  333: {2,2,12}
  351: {2,2,2,6}
  361: {8,8}
  364: {1,1,4,6}
For example, 198 has prime indices {1,2,2,5}, and 20/10 is an integer > 1, so 198 is in the sequence.

Crossrefs

The fraction A003963(n)/A056239(n) reduces to A326153(n)/A326154(n).
The non-proper version is A326149, superset of A326150.
Also a superset of A326151.
The squarefree case is A326158 without first term.
Partitions of this type are counted by A380219.
A379666 counts partitions by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379735, A380217.
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A028422, A036844, A112798, A301988, A319000, A324850, A324851, A326156, A379319, A379844.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Divisible[Times@@prix[#],Total[prix[#]]]&&!SameQ[Times@@prix[#],Total[prix[#]]]&]

A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 3, 3, 0, 4, 0, 9, 6, 4, 0, 22, 5, 6, 15, 28, 0, 54, 0, 49, 30, 14, 57, 134, 0, 22, 58, 219, 0, 242, 0, 180, 349, 44, 0, 722, 113, 369, 196, 404, 0, 994, 556, 1363, 338, 111, 0, 3016, 0, 150, 2569, 3150, 1485, 2815, 0
Offset: 1

Views

Author

Gus Wiseman, Jan 22 2025

Keywords

Comments

Also the number of strict integer partitions of n - 1 not containing 1 whose product of parts is a multiple of n. These are strict integer factorizations of multiples of n summing to n - 1.

Examples

			The a(6) = 1 through a(16) = 3 partitions:
  (3,2,1) . . . (5,4,1) . (8,3,1)   . (7,6,1)   (9,5,1)     (8,4,3,1)
                          (6,3,2,1)   (7,4,2,1) (6,5,3,1)   (8,5,2,1)
                                                (5,4,3,2,1) (6,4,3,2,1)

Crossrefs

Positions of 0 after 9 appear to be the prime numbers A000040.
The non-strict version is A379320 shifted right, ranks A380217 = A379319/2.
Not requiring 1 gives A379733.
For n instead of n+1 we have A379735 shifted left, non-strict A379734.
Partitions of this type are ranked by A379845.
The case of equality for non-strict partitions is A380218 shifted left.
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A069016, A319000, A319916, A325042, A326150, A326152, A326156, A379671, A379678.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],MemberQ[#,1]&&UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]

A379318 Odd numbers whose product of prime indices is a multiple of their sum of prime indices.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239
Offset: 1

Views

Author

Gus Wiseman, Jan 16 2025

Keywords

Comments

Contains all odd primes.
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. The sum and product of prime indices are A056239 and A003963 respectively.

Examples

			The terms together with their prime indices begin:
     2: {1}         53: {16}           109: {29}
     3: {2}         59: {17}           113: {30}
     5: {3}         61: {18}           125: {3,3,3}
     7: {4}         63: {2,2,4}        127: {31}
     9: {2,2}       65: {3,6}          131: {32}
    11: {5}         67: {19}           137: {33}
    13: {6}         71: {20}           139: {34}
    17: {7}         73: {21}           149: {35}
    19: {8}         79: {22}           150: {1,2,3,3}
    23: {9}         81: {2,2,2,2}      151: {36}
    29: {10}        83: {23}           154: {1,4,5}
    30: {1,2,3}     84: {1,1,2,4}      157: {37}
    31: {11}        89: {24}           163: {38}
    37: {12}        97: {25}           165: {2,3,5}
    41: {13}       101: {26}           167: {39}
    43: {14}       103: {27}           169: {6,6}
    47: {15}       107: {28}           173: {40}
    49: {4,4}      108: {1,1,2,2,2}    179: {41}

Crossrefs

Including evens gives A326149, counted by A057568.
For nonprime instead of odd we get A326150.
For even instead of odd we get A379319, counted by A379320.
Partitions of this type are counted by A379734, strict A379735, see A379733.
For squarefree instead of odd we get A379844, even case A379845.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A301988, A318950, A319000, A319916, A324851, A325041, A326152, A379671, A379678.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],OddQ[#]&&Divisible[Times@@prix[#],Total[prix[#]]]&]
Previous Showing 11-14 of 14 results.

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