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A325369 Numbers with no two prime exponents appearing the same number of times in the prime signature. 6
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 84, 85, 86 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities appear with distinct multiplicities. The enumeration of these partitions by sum is given by A325329.

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

Most small numbers are in the sequence. However the sequence of non-terms together with their prime indices begins:

  12: {1,1,2}

  18: {1,2,2}

  20: {1,1,3}

  24: {1,1,1,2}

  28: {1,1,4}

  40: {1,1,1,3}

  44: {1,1,5}

  45: {2,2,3}

  48: {1,1,1,1,2}

  50: {1,3,3}

  52: {1,1,6}

  54: {1,2,2,2}

  56: {1,1,1,4}

  63: {2,2,4}

  68: {1,1,7}

  72: {1,1,1,2,2}

  75: {2,3,3}

  76: {1,1,8}

  80: {1,1,1,1,3}

  88: {1,1,1,5}

For example, the prime indices of 1260 are {1,1,2,2,3,4}, whose multiplicities give the prime signature {1,1,2,2}, and since 1 and 2 appear the same number of times, 1260 is not in the sequence.

MATHEMATICA

Select[Range[100], UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]

CROSSREFS

Cf. A056239, A098859, A112798, A118914, A130091, A317090, A319161, A325326, A325329, A325331, A325337, A325370, A325371.

Sequence in context: A263837 A283550 A271113 * A317101 A304449 A085924

Adjacent sequences:  A325366 A325367 A325368 * A325370 A325371 A325372

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved

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Last modified January 18 13:30 EST 2020. Contains 331007 sequences. (Running on oeis4.)