A380751 -id:A380751 - cp's OEIS frontend

A380783 Lexicographically earliest sequence of positive integers such that for any value k, no two sets of one or more indices at which k occurs have the same product.

1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 4, 2, 2, 3, 2, 4, 4, 3, 2, 4, 2, 3, 4, 5, 2, 4, 2, 5, 4, 3, 5, 5, 2, 3, 4, 6, 2, 5, 2, 5, 6, 3, 2, 6, 2, 3, 4, 5, 2, 3, 6, 4, 4, 3, 2, 5, 2, 3, 6, 3, 6, 7, 2, 5, 4, 6, 2, 6, 2, 3, 4, 5, 7, 7, 2, 7, 2, 3, 2, 5, 6, 3, 4

Offset: 1

Neal Gersh Tolunsky, Feb 02 2025

nonn

The Fermi-Dirac primes (A050376) are the indices of 2s in this sequence.

			a(8) = 3: We cannot have 1 here because the set of indices i = 8 and i = 1,8 would have the same product. We cannot have a(8) = 2 because i = 8 would have the same product as i = 2,4. So a(8) = 3.

Dominic McCarty, Table of n, a(n) for n = 1..10000
Dominic McCarty, Java program for A380783

Cf. A050376, A380751, A380921 (indices of records).

A380968 Lexicographically earliest sequence of positive integers such that for any value k, no two sets of one or more indices at which k occurs have the same mean.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 3, 2, 4, 4, 5, 3, 1, 4, 5, 5, 6, 6, 7, 4, 6, 7, 2, 5, 8, 6, 3, 7, 1, 7, 5, 8, 8, 4, 9, 8, 9, 9, 10, 10, 6, 10, 9, 11, 11, 10, 11, 2, 8, 12, 11, 3, 7, 10, 12, 5, 12, 9, 11, 4, 13, 13, 14, 13, 12, 6, 14, 13, 14, 10, 15, 15, 16, 15, 11, 13

Offset: 1

Neal Gersh Tolunsky, Feb 09 2025

nonn

A260873 gives the indices of 1s in the sequence.
The longest run in the sequence has length 2.
No three equal terms will appear at indices in arithmetic progression.
For any value k, the distances between pairs of k will be distinct.

			a(7) = 3: a(7) cannot be 1 because i = 4; i = 1,7; and i = 1,4,7 would all have the same mean index 4. a(7) cannot be 2 because i = 6; i = 5,6,7; and i = 5,7 would have the same mean index 6. So a(7) = 3.
a(19) cannot be 1, 2, or 3. a(19) = 4 does not work either because i = 13,19 would have the same mean (namely 16) as i = 12,17,19. So a(19) = 5.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A380783, A380751.

A381856 Lexicographically earliest sequence of positive integers such that for any value k, no two sets of two or more indices at which k occurs have the same standard deviation.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 4, 3, 3, 4, 4, 1, 5, 2, 5, 3, 4, 5, 4, 6, 1, 5, 6, 6, 2, 3, 7, 5, 6, 4, 6, 1, 7, 7, 8, 5, 7, 8, 8, 9, 6, 9, 2, 8, 3, 7, 4, 5, 9, 9, 8, 10, 9, 10, 10, 11, 7, 1, 8, 10, 11, 11, 6, 11, 9, 12, 10, 2, 12, 8, 11, 13, 12, 12, 3, 10, 13, 13

Offset: 1

Neal Gersh Tolunsky, Mar 08 2025

nonn

A382381 gives the indices of 1s in this sequence.

If the definition is modified to compare all sets of indices whose terms are equal (not just those sets with the same value k), we get A337226.

			a(13) = 3: a(13) cannot be 1 as i = 4,13 would have the same standard deviation as i = 1,4,8,13 (namely 4.5). We cannot have a(13) = 2 because i = 3,6 would have the same standard deviation as i = 10,13 (namely 1.5). With a(13) = 3, we find that no two subsets of i = 7,9,12,13 have the same standard deviation, so a(13) = 3.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A382381, A337226, A380968, A380783, A380751.

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A380783 Lexicographically earliest sequence of positive integers such that for any value k, no two sets of one or more indices at which k occurs have the same product.

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A380968 Lexicographically earliest sequence of positive integers such that for any value k, no two sets of one or more indices at which k occurs have the same mean.

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A381856 Lexicographically earliest sequence of positive integers such that for any value k, no two sets of two or more indices at which k occurs have the same standard deviation.

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