cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A361667 a(n) = A361665(A361666(n))/n.

Original entry on oeis.org

1, 1, 1, 1, 4, 8, 16, 1, 28, 2, 4, 4, 1, 8, 5, 1, 32, 14, 4, 1, 12, 2, 16, 2, 3, 1, 1104, 4, 2944, 454, 804, 1, 4, 16, 164, 7, 544, 2, 64, 32, 140, 6, 32, 1, 6812, 8, 768, 1, 752, 286, 204, 4, 64, 552, 260, 2, 3904, 1472, 32, 227, 16, 402, 4, 1, 220, 2, 372, 8
Offset: 1

Views

Author

Pontus von Brömssen, Mar 20 2023

Keywords

Crossrefs

Cf. A361664, A361665, A361666, A361668.

Formula

a(n) = A361664(n) if n is not in A361668. The equality holds also for some n in A361668; for example, a(256) = A361664(256) = 1.

A361668 Numbers k such that A361662(k) != A181821(A361666(k)).

Original entry on oeis.org

30, 51, 60, 89, 102, 105, 113, 119, 120, 128, 135, 145, 149, 150, 153, 168, 178, 179, 181, 191, 200, 204, 210, 215, 219, 221, 224, 226, 238, 240, 245, 248, 256, 257, 267, 270, 277, 281, 290, 298, 299, 300, 305, 306, 313, 317, 323, 336, 343, 345, 349, 356, 357
Offset: 1

Views

Author

Pontus von Brömssen, Mar 20 2023

Keywords

Comments

Equivalently, numbers k such that A361666(k) != A181820(A361663(k)).

Crossrefs

Cf. A181820, A181821, A361662, A361663, A361666.

A361662 Least number k >= 1 such that A074206(k) is divisible by n.

Original entry on oeis.org

1, 4, 6, 8, 24, 48, 96, 12, 216, 24, 60, 48, 30, 96, 210, 32, 288, 216, 72, 24, 216, 60, 240, 48, 210, 36, 6480, 96, 15552, 4320, 7560, 64, 120, 288, 2520, 216, 5040, 72, 960, 768, 2520, 216, 576, 60, 83160, 240, 7680, 48, 18480, 13860, 7776, 144, 1152, 6480
Offset: 1

Views

Author

Pontus von Brömssen, Mar 20 2023

Keywords

Comments

a(n) exists for all n. (This is problem 5 of the first round of the British Mathematical Olympiad 2022/2023.)
All terms are in A025487.

Links

Crossrefs

Cf. A025487, A074206, A361663, A361664, A361666.

Programs

  • PARI
    f(n)={if(!n, 0, my(sig=factor(n)[, 2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r, k)*(-1)^(r-k))))}; \\ A074206
a(n) = my(k=1); while (f(k) % n, k++); k; \\ Michel Marcus, Mar 23 2023

Formula

a(n) = A025487(A361663(n)).

A361665 Number of ordered factorizations of p_1^x_1 * ... * p_k^x_k, where (x_1, ..., x_k) is the partition with Heinz number n and p_1, ..., p_k are distinct primes.

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 8, 13, 26, 20, 16, 44, 32, 48, 76, 75, 64, 176, 128, 132, 208, 112, 256, 308, 252, 256, 818, 368, 512, 604, 1024, 541, 544, 576, 768, 1460, 2048, 1280, 1376, 1076, 4096, 1888, 8192, 976, 3172, 2816, 16384, 2612, 2568, 2316, 3392, 2496, 32768
Offset: 1

Views

Author

Pontus von Brömssen, Mar 20 2023

Keywords

Comments

Also, a(n) is the number of paths from (0, ..., 0) to P in which each step adds a nonnegative integer to each coordinate (and a positive number to at least one coordinate), where P is the partition with Heinz number n.

Examples

			The partition with Heinz number 6 is (1,2), and p^1*q^2 has 8 ordered factorizations, where p and q are distinct primes, so a(6) = 8. With p = 3 and q = 2, for example, we have the 8 = A074206(12) factorizations 12 = 2*6 = 3*4 = 4*3 = 6*2 = 2*2*3 = 2*3*2 = 3*2*2.

Crossrefs

Cf. A074206, A181821, A361666.

Formula

a(n) = A074206(A181821(n)).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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