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.

A347457 Heinz numbers of integer partitions with integer alternating product.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 71, 72, 73, 74, 75, 76, 78
Offset: 1

Views

Author

Gus Wiseman, Sep 26 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also numbers whose multiset of prime indices has integer reverse-alternating product.

Examples

			The prime indices of 525 are {2,3,3,4}, with reverse-alternating product 2, so 525 is in the sequence
The prime indices of 135 are {2,2,2,3}, with reverse-alternating product 3/2, so 135 is not in the sequence.

Crossrefs

The reciprocal version is A028982.
Allowing any alternating product > 1 gives A028983, reverse A347465.
Factorizations of this type are counted by A347437.
These partitions are counted by A347446.
The reverse reciprocal version A347451.
The odd-length case is A347453.
The reverse version is A347454.
The complement is A347455.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A347461 counts possible alternating products of partitions, reverse A347462.
Cf. A001105, A001222, A028260, A119620, A119899, A316523, A344606, A344617, A346703, A346704, A347448, A347450.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],IntegerQ[altprod[Reverse[primeMS[#]]]]&]

A347454 Numbers whose multiset of prime indices has integer alternating product.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113
Offset: 1

Views

Author

Gus Wiseman, Sep 26 2021

Keywords

Comments

First differs from A265640 in having 42.
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.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also Heinz numbers of partitions with integer reverse-alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The terms and their prime indices begin:
      1: {}            20: {1,1,3}         47: {15}
      2: {1}           23: {9}             48: {1,1,1,1,2}
      3: {2}           25: {3,3}           49: {4,4}
      4: {1,1}         27: {2,2,2}         50: {1,3,3}
      5: {3}           28: {1,1,4}         52: {1,1,6}
      7: {4}           29: {10}            53: {16}
      8: {1,1,1}       31: {11}            59: {17}
      9: {2,2}         32: {1,1,1,1,1}     61: {18}
     11: {5}           36: {1,1,2,2}       63: {2,2,4}
     12: {1,1,2}       37: {12}            64: {1,1,1,1,1,1}
     13: {6}           41: {13}            67: {19}
     16: {1,1,1,1}     42: {1,2,4}         68: {1,1,7}
     17: {7}           43: {14}            71: {20}
     18: {1,2,2}       44: {1,1,5}         72: {1,1,1,2,2}
     19: {8}           45: {2,2,3}         73: {21}

Crossrefs

The even-length case is A000290.
The additive version is A026424.
Allowing any alternating product < 1 gives A119899, strict A028260.
Allowing any alternating product >= 1 gives A344609, multiplicative A347456.
Factorizations of this type are counted by A347437.
These partitions are counted by A347445, reverse A347446.
Allowing any alternating product <= 1 gives A347450.
The reciprocal version is A347451.
The odd-length case is A347453.
The version for reversed prime indices is A347457, complement A347455.
Allowing any alternating product > 1 gives A347465, reverse A028983.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A001105, A001222, A028982, A119620, A236913, A316523, A344653, A346703, A346704, A347443, A347439.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],IntegerQ[altprod[primeMS[#]]]&]

A347453 Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 42, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 114, 116, 117, 124, 125, 127, 128, 130
Offset: 1

Views

Author

Gus Wiseman, Sep 24 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also numbers whose multiset of prime indices has odd length and integer alternating product, where a prime index of n is a number m such that prime(m) divides n.

Examples

			The terms and their prime indices begin:
      2: {1}         29: {10}            61: {18}
      3: {2}         31: {11}            63: {2,2,4}
      5: {3}         32: {1,1,1,1,1}     67: {19}
      7: {4}         37: {12}            68: {1,1,7}
      8: {1,1,1}     41: {13}            71: {20}
     11: {5}         42: {1,2,4}         72: {1,1,1,2,2}
     12: {1,1,2}     43: {14}            73: {21}
     13: {6}         44: {1,1,5}         75: {2,3,3}
     17: {7}         45: {2,2,3}         76: {1,1,8}
     18: {1,2,2}     47: {15}            78: {1,2,6}
     19: {8}         48: {1,1,1,1,2}     79: {22}
     20: {1,1,3}     50: {1,3,3}         80: {1,1,1,1,3}
     23: {9}         52: {1,1,6}         83: {23}
     27: {2,2,2}     53: {16}            89: {24}
     28: {1,1,4}     59: {17}            92: {1,1,9}

Crossrefs

The reciprocal version is A000290.
Allowing any alternating product <= 1 gives A001105.
Allowing any alternating product gives A026424.
Factorizations of this type are counted by A347441.
These partitions are counted by A347444.
Allowing any length gives A347454.
Allowing any alternating product > 1 gives A347465.
A027193 counts odd-length partitions.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347446 counts partitions with integer alternating product.
A347457 ranks partitions with integer alt product, complement A347455.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A001222, A028260, A028982, A028983, A339890, A344617, A344653, A345958, A346703, A346704, A347437, A347443, A347450, A347451.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],OddQ[PrimeOmega[#]]&&IntegerQ[altprod[primeMS[#]]]&]

A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 14, 16, 18, 21, 22, 24, 25, 26, 32, 34, 36, 38, 39, 40, 46, 49, 50, 54, 56, 57, 58, 62, 64, 65, 72, 74, 81, 82, 84, 86, 87, 88, 90, 94, 96, 98, 100, 104, 106, 111, 115, 118, 121, 122, 126, 128, 129, 133, 134, 136, 142, 144, 146, 150, 152
Offset: 1

Views

Author

Gus Wiseman, Sep 24 2021

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.
We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
Also Heinz numbers integer partitions with integer reverse-reciprocal alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The terms and their prime indices begin:
      1: {}            32: {1,1,1,1,1}       65: {3,6}
      2: {1}           34: {1,7}             72: {1,1,1,2,2}
      4: {1,1}         36: {1,1,2,2}         74: {1,12}
      6: {1,2}         38: {1,8}             81: {2,2,2,2}
      8: {1,1,1}       39: {2,6}             82: {1,13}
      9: {2,2}         40: {1,1,1,3}         84: {1,1,2,4}
     10: {1,3}         46: {1,9}             86: {1,14}
     14: {1,4}         49: {4,4}             87: {2,10}
     16: {1,1,1,1}     50: {1,3,3}           88: {1,1,1,5}
     18: {1,2,2}       54: {1,2,2,2}         90: {1,2,2,3}
     21: {2,4}         56: {1,1,1,4}         94: {1,15}
     22: {1,5}         57: {2,8}             96: {1,1,1,1,1,2}
     24: {1,1,1,2}     58: {1,10}            98: {1,4,4}
     25: {3,3}         62: {1,11}           100: {1,1,3,3}
     26: {1,6}         64: {1,1,1,1,1,1}    104: {1,1,1,6}

Crossrefs

The version for reversed prime indices is A028982, counted by A119620.
The additive version is A119899, strict A028260.
Allowing any alternating product >= 1 gives A344609.
Factorizations of this type are counted by A347439.
Allowing any alternating product <= 1 gives A347450.
The non-reciprocal version is A347454.
Allowing any alternating product > 1 gives A347465, reverse A028983.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347457 ranks partitions with integer alternating product.
Cf. A001222, A236913, A316523, A344617, A345958, A345959, A346703, A346704, A347437, A347446, A347455.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],IntegerQ[1/altprod[primeMS[#]]]&]
Showing 1-4 of 4 results.

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