A307730 -id:A307730 - cp's OEIS frontend

A348253 Indices of records in A348246.

1, 2, 3, 4, 5, 7, 10, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

Cf. A307720, A307730, A348246, A348409, A348249, A348252-A348256.

A348459 a(n) = max(A307720(n), A307720(n+1)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 30 2021

nonn

This is the maximum of A348241 and A348242. It is the upper line (red or blue) in Cheswick's pictures in A348248.

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, PARI program for A348459

Cf. A307720, A307730, A348241, A348242, A348248, A348446.

PARI
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See Links section.
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A348774 A348773(2*n+1).

Original entry on oeis.org

2, 6, 12, 18, 24, 32, 42, 48, 60, 68, 74, 84, 98, 104, 110, 128, 138, 150, 158, 168, 180, 192, 198, 212, 228, 234, 242, 258, 270, 278, 284, 308, 314, 332, 348, 354, 368, 380, 390, 402, 420, 432, 440, 450, 462, 468, 488, 500, 510, 524, 548, 564, 572, 588, 600, 608, 618, 632, 644, 654

Offset: 0

N. J. A. Sloane, Nov 07 2021

nonn

The first differences are 4, 6, 6, 6, 8, ... and apart from the initial term4, appears to coincide with A155067, the differences between successive odd-indexed primes. If confirmed, this will be one of the few formulas known for A307720.

The other bisection of A348773, A348775, seems much more mysterious.

N. J. A. Sloane, Table of n, a(n) for n = 0..613

Cf. A155067, A307720, A307730, A307632, A348773, A348775.

A348775 A348773(2*n).

Original entry on oeis.org

42, 1321, 2352, 2924, 77922, 4822, 2310, 81212, 19730, 331637, 340640, 11158, 13838, 13690, 14476, 709992, 17990, 19518, 20830, 2277394, 62350, 82484, 76962, 84852, 15407670, 87388, 90636, 408240, 14526794, 7023466, 272792, 117864, 293946, 40034157, 386674, 168172, 136472, 40847194, 729008, 768646

Offset: 1

N. J. A. Sloane, Nov 07 2021

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..614

Cf. A307720, A307730, A307632, A348773, A348774.

A348250 a(n) = A348249(n) - n.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 8, 0, 10, 37, 0, 0, 0, 0, 0, 32, 0, 0, 16, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 80, 0, 34, 133, 0, 0, 0, 0, 0, 326, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 88, 2605, 46, 0, 0, 0, 0, 98, 0, 90, 0, 0, 0, 0, 0, 0, 64, 0, 62, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 228, 82, 3139, 0, 0, 260, 1135

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

A348249(n) is at least n (that is at the heart of the definition of A307720). Here we examine the difference.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A307720, A307730, A348246, A348249, A348409, A348251.

A348255 Records in A348249.

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 11, 20, 24, 52, 53, 116, 172, 371, 2662, 3226, 4800, 9546, 9804, 14171, 69148, 71863, 325494, 333873, 698509, 2262261, 15387377, 40004428, 40813283, 64314104, 550623089, 847235953, 897871194, 1259250335

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

Cf. A307720, A307730, A348246, A348409, A348249, A348252-A348256.

A348444 If A348243(n) = i, then a(n) = 1 + number of copies of i that have already appeared in A348243.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

N. J. A. Sloane, Oct 21 2021

nonn

In other words, when we see i in A348243, which i is it? The first, second, third, ...? This gives a measure of how many attempts we have to make in A307730 and A349243 before all the n instances of n have been obtained.
The first 3's appear at n = 259, 490, 585, 627, ..., the first 4's at 3161, 4230, 5989, 8207, ...
Note that A348409 gives the index of the last occurrence of each k in A307730. So until we reach A348409(k), we will not know how many attempts are needed to obtain all copies of k.

N. J. A. Sloane, Table of n, a(n) for n = 1..750

Cf. A307720, A307730, A348245, A348409.

A348458 Partial sums of A307720.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 12, 14, 16, 18, 20, 22, 25, 27, 30, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 66, 68, 72, 74, 78, 80, 84, 86, 90, 93, 97, 100, 104, 107, 111, 114, 118, 121, 125, 128, 133, 134, 139, 140, 145, 146, 153, 154, 161, 162, 169, 170, 177, 179, 184, 186, 191, 193, 198, 200, 205, 207

Offset: 1

N. J. A. Sloane, Oct 29 2021

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A307720, A307730.

A348486 a(n) = A348485(n) * A348485(n+1).

Original entry on oeis.org

1, 2, 4, 2, 3, 6, 4, 6, 3, 4, 8, 4, 6, 3, 5, 10, 6, 9, 6, 8, 12, 6, 8, 12, 9, 12, 8, 10, 5, 7, 14, 8, 12, 9, 12, 8, 10, 5, 7, 14, 8, 12, 9, 12, 8, 10, 5, 7, 14, 10, 5, 7, 14, 10, 15, 9, 12, 16, 12, 9, 12, 16, 12, 9, 12, 16, 20, 10, 14, 7, 9, 18, 10, 15, 9, 15

Offset: 1

Rémy Sigrist, Oct 21 2021

nonn

There are no consecutive equal terms.

			a(11) = A348485(11) * A348485(12) = 4 * 2 = 8.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A348486

Cf. A307730, A348485.

PARI
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See Links section.
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A348581 a(n) is the least factor among all the products A307720(k) * A307720(k+1) equal to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 4, 3, 2, 1, 3, 5, 2, 3, 4, 1, 5, 1, 4, 3, 2, 5, 4, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 7, 5, 3, 4, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 8, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 5, 4, 7, 6, 1, 8, 9, 2, 1, 7, 5, 2, 3

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Oct 24 2021

nonn

We know there are n ways to get n as a product of terms A307720(k)*A307720(k+1) for various k's. Look at these 2*n numbers from A307720. Then a(n) is the smallest of them.

			For n = 6:
- we have the following products equal to 6:
    A307720(7)  * A307720(8)  = 3 * 2 = 6
    A307720(12) * A307720(13) = 2 * 3 = 6
    A307720(13) * A307720(14) = 3 * 2 = 6
    A307720(14) * A307720(15) = 2 * 3 = 6
    A307720(15) * A307720(16) = 3 * 2 = 6
    A307720(16) * A307720(17) = 2 * 3 = 6
- the corresponding distinct factors are 2 and 3,
- hence a(6) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A348581

Cf. A307720, A307730, A348582.

C
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See Links section.
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a(p) = 1 for any prime number p.

a(n) * A348582(n) = n.

cp's OEIS Frontend

A348253 Indices of records in A348246.

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A348459 a(n) = max(A307720(n), A307720(n+1)).

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PARI

A348774 A348773(2*n+1).

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A348775 A348773(2*n).

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A348250 a(n) = A348249(n) - n.

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A348255 Records in A348249.

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A348444 If A348243(n) = i, then a(n) = 1 + number of copies of i that have already appeared in A348243.

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A348458 Partial sums of A307720.

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A348486 a(n) = A348485(n) * A348485(n+1).

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PARI

A348581 a(n) is the least factor among all the products A307720(k) * A307720(k+1) equal to n.

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Programs

C

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