A368728 -id:A368728

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A368729

Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.

+10
3

1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 51, 55, 59, 63, 65, 67, 69, 73, 75, 77, 79, 81, 83, 85, 87, 91, 93, 97, 99, 101, 105, 109, 115, 117, 119, 121, 123, 125, 127, 129, 135, 137, 139, 141, 143, 145, 147, 149

(list; graph; refs; listen; history; text; internal format)

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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

LINKS

Table of n, a(n) for n=1..61.

EXAMPLE

The terms together with the corresponding multigraphs begin:

1: {}

3: {{1}}

5: {{2}}

7: {{1,1}}

9: {{1},{1}}

11: {{3}}

13: {{1,2}}

15: {{1},{2}}

17: {{4}}

21: {{1},{1,1}}

23: {{2,2}}

25: {{2},{2}}

27: {{1},{1},{1}}

29: {{1,3}}

31: {{5}}

33: {{1},{3}}

35: {{2},{1,1}}

39: {{1},{1,2}}

41: {{6}}

43: {{1,4}}

45: {{1},{1},{2}}

47: {{2,3}}

49: {{1,1},{1,1}}

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], OddQ[#]&&Max@@Length/@prix/@prix[#]<=2&]

CROSSREFS

In the unlabeled case these multigraphs are counted by A320663.

These are products of primes indexed by elements of A037143 greater than 1.

For just primes we have A076610, squarefree A302590.

For just semiprimes we have A339112, squarefree A340020.

For just half-loops we have A340019.

This is the odd case of A368728, complement A368833.

A000607 counts partitions into primes, with ones allowed A034891.

A001358 lists semiprimes, squarefree A006881.

A006450, A106349, A322551, A368732 list selected primes.

A056239 adds up prime indices, row sums of A112798.

A101048 counts partitions into semiprimes.

Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320912, A339113.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 07 2024

STATUS

approved

A368732

Primes whose index is one, another prime number, or a semiprime.

+10
3

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 47, 59, 67, 73, 79, 83, 97, 101, 109, 127, 137, 139, 149, 157, 163, 167, 179, 191, 199, 211, 227, 233, 241, 257, 269, 271, 277, 283, 293, 313, 331, 347, 353, 367, 373, 389, 401, 421, 431, 439, 443, 449, 461, 467, 487

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..57.

MATHEMATICA

Prime/@Select[Range[100], PrimeOmega[#]<=2&]

CROSSREFS

For just primes we have A006450, products A076610, strict A302590.

These indices are A037143.

For just semiprimes we have A106349, products A339112, strict A340020.

Products of these primes are A368728, odd A368729, odd strict A340019.

Products of the complementary primes are A368833.

A000607 counts partitions into primes, with ones allowed A034891.

A001358 lists semiprimes, squarefree A006881.

A056239 adds up prime indices, row sums of A112798.

A101048 counts partitions into semiprimes.

A322551 lists primes of squarefree semiprime index.

Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320911, A320912, A339113.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 08 2024

STATUS

approved

A368833

Numbers whose prime indices are not 1, prime, or semiprime.

+10
3

19, 37, 38, 53, 57, 61, 71, 74, 76, 89, 95, 103, 106, 107, 111, 113, 114, 122, 131, 133, 142, 148, 151, 152, 159, 171, 173, 178, 181, 183, 185, 190, 193, 197, 206, 209, 212, 213, 214, 222, 223, 226, 228, 229, 239, 244, 247, 251, 259, 262, 263, 265, 266, 267

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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..54.

EXAMPLE

The terms together with their prime indices begin:

19: {8}

37: {12}

38: {1,8}

53: {16}

57: {2,8}

61: {18}

71: {20}

74: {1,12}

76: {1,1,8}

89: {24}

95: {3,8}

103: {27}

106: {1,16}

107: {28}

111: {2,12}

113: {30}

114: {1,2,8}

122: {1,18}

131: {32}

133: {4,8}

142: {1,20}

148: {1,1,12}

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], Max@@PrimeOmega/@prix[#]>2&]

CROSSREFS

These are non-products of primes indexed by elements of A037143.

The complement for just primes is A076610, strict A302590.

The complement for just semiprimes is A339112, strict A340020.

The complement for just squarefree semiprimes is A339113, strict A309356.

The complement is A368728.

The complement for just primes and semiprimes is A368729, strict A340019.

A000607 counts partitions into primes, with ones allowed A034891.

A001358 lists semiprimes, squarefree A006881.

A006450, A106349, A322551, A368732 list selected primes.

A056239 adds up prime indices, row sums of A112798.

A101048 counts partitions into semiprimes.

Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 08 2024

STATUS

approved

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