table of polite number representations for 1<n<101

table of polite number representations for

1

<

n

<

101

There clearly are patterns to the number of ways to represent a

positive integer

as a sum of consecutive

nonnegative integers

. There is only one way to represent

odd primes

in this manner, whereas

composite

odd numbers

tend to have more

representations

.

To try to make the relationship between

integer factorization

and number of representations as a sum of consecutive

integers

, the following table, in

addition

to listing the different sums and tallying them, also gives the value of the number of (nondistinct)

prime factors

function

(

http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction

)

Ω

⁢

(

n

)

and the

difference

between the two. But to avoid needless repetition, the sums given are only of positive numbers; the only cases this makes a difference is for the

triangular numbers

T

n

, which in addition to being representable as

∑

i

=

1

n

i

are also representable as

∑

i

=

0

n

i

.

For sums with more than three addends, the middle addends have been replaced by three dots.

n

Rep 1

Rep 2

Rep 3

Rep 4

Rep 5

N

p

⁢

(

n

)

Ω

⁢

(

n

)

Ω

⁢

(

n

)

-

N

p

⁢

(

n

)

2

0

1

1

3

1 + 2

1

1

0

4

0

2

2

5

2 + 3

1

1

0

6

1 + 2 + 3

1

2

1

7

3 + 4

1

1

0

8

0

3

3

9

4 + 5

2 + 3 + 4

2

2

0

10

1 … 4

1

2

1

11

5 + 6

1

1

0

12

3 + 4 + 5

1

3

2

13

6 + 7

1

1

0

14

2 … 5

1

2

1

15

7 + 8

4 + 5 + 6

1 … 5

3

2

1

16

0

4

4

17

8 + 9

1

1

0

18

5 + 6 + 7

3 … 6

2

3

1

19

9 + 10

1

1

0

20

2 … 6

1

3

2

21

10 + 11

6 + 7 + 8

1 … 6

3

2

-

1

22

4 … 7

1

2

1

23

11 + 12

1

1

0

24

7 + 8 + 9

1

4

3

25

12 + 13

3 … 7

2

2

0

26

5 … 8

1

2

1

27

13 + 14

8 + 9 + 10

2 … 7

3

3

0

28

1 … 7

1

3

2

29

14 + 15

1

1

0

30

9 + 10 + 11

6 … 9

4 … 8

3

3

0

31

15 + 16

1

1

0

32

0

5

5

33

16 + 17

10 + 11 + 12

3 … 8

3

2

-

1

34

7 … 10

1

2

1

35

17 + 18

5 … 9

2 … 8

3

2

-

1

36

11 + 12 + 13

1 … 8

2

4

-

2

37

18 + 19

1

1

0

38

8 … 11

1

2

1

39

19 + 20

12 + 13 + 14

4 … 9

3

2

-

1

40

6 … 10

1

4

3

41

20 + 21

1

1

0

42

13 + 14 + 15

9 … 12

3 … 9

3

3

0

43

21 + 22

1

1

0

44

7 … 11

2 … 9

2

3

1

45

22 + 23

14 + 15 + 16

5 … 10

1 … 9

4

3

-

1

46

10 … 13

1

2

1

47

23 + 24

1

1

0

48

15 + 16 + 17

1

5

4

49

24 + 25

4 … 10

2

2

0

50

11 … 14

8 … 12

2

3

1

51

25 + 26

16 + 17 + 18

6 … 11

3

2

-

1

52

3 … 10

1

3

2

53

26 + 27

1

1

0

54

17 + 18 + 19

12 … 15

2 … 10

3

4

1

55

27 + 28

9 … 13

1 … 10

3

2

-

1

56

5 … 11

1

4

3

57

28 + 29

18 + 19 + 20

7 … 12

3

2

-

1

58

13 … 16

1

2

1

59

29 + 30

1

1

0

60

19 + 20 + 21

10 … 14

4 … 11

3

4

1

61

30 + 31

1

1

0

62

14 … 17

1

2

1

63

31 + 32

20 + 21 + 22

8 … 13

6 … 12

3 … 11

5

3

-

2

64

0

6

6

65

32 + 33

11 … 15

2 … 11

3

2

-

1

66

21 + 22 + 23

15 … 18

1 … 11

3

3

0

67

33 + 34

1

1

0

68

5 … 12

1

3

2

69

34 + 35

22 + 23 + 24

9 … 14

3

2

-

1

70

16 … 19

12 … 16

7 … 13

3

3

0

71

35 + 36

1

1

0

72

23 + 24 + 25

4 … 12

2

5

3

73

36 + 37

1

1

0

74

17 … 20

1

2

1

75

37 + 38

24 + 25 + 26

13 … 17

10 … 15

3 … 12

5

3

-

2

76

6 … 13

1

3

2

77

38 + 39

8 … 14

2 … 12

3

2

-

1

78

25 + 26 + 27

18 … 21

1 … 12

3

3

0

79

39 + 40

1

1

0

80

14 … 18

1

6

5

81

40 + 41

26 + 27 + 28

11 … 16

5 … 13

4

4

0

82

19 … 22

1

2

1

83

41 + 42

1

1

0

84

27 + 28 + 29

9 … 15

7 … 14

3

4

1

85

42 + 43

15 … 19

4 … 13

3

2

-

1

86

20 … 23

1

2

1

87

43 + 44

28 + 29 + 30

12 … 17

3

2

-

1

88

3 … 13

1

4

3

89

44 + 45

1

1

0

90

29 + 30 + 31

21 … 24

16 … 20

6 … 14

2 … 13

5

4

-

1

91

45 + 46

10 … 16

1 … 13

3

2

-

1

92

8 … 15

1

3

2

93

46 + 47

30 + 31 + 32

13 … 18

3

2

-

1

94

22 … 25

1

2

1

95

47 + 48

17 … 21

5 … 14

3

2

-

1

96

31 + 32 + 33

1

6

5

97

48 + 49

1

1

0

98

23 … 26

11 … 17

2

3

1

99

49 + 50

32 + 33 + 34

14 … 19

7 … 15

4 … 14

5

3

-

2

100

18 … 22

9 … 16

2

3

1

Title

table of

polite number

representations for

1

<

n

<

101

Canonical name

TableOfPoliteNumberRepresentationsFor1N101

Date of creation

2013-03-22 18:45:12

Last modified on

2013-03-22 18:45:12

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A25

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