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 1-3 of 3 results.

A380468 Numbers k such that A380459(k) has no divisors of the form p^p, for any prime p.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 23, 26, 29, 31, 37, 38, 41, 43, 47, 53, 59, 61, 62, 67, 71, 73, 74, 79, 83, 86, 89, 97, 101, 103, 107, 109, 113, 122, 127, 131, 134, 137, 139, 146, 149, 151, 157, 158, 163, 167, 173, 179, 181, 186, 191, 193, 194, 197, 199, 206, 211, 218, 223, 227, 229, 233, 239, 241, 251, 254
Offset: 1

Views

Author

Antti Karttunen, Feb 01 2025

Keywords

Comments

Proof that this is a subsequence of squarefree numbers (A005117): Let's write A380459(n) = Product_{d|n} A276086(n/d)^A349394(d). Then suppose that we have a prime p such that p^e || n, with e > 1 (the maximal exponent e for which p^e divides n). We set d = p^e, and for that d, factor A276086(n/(p^e))^A349394(p^e) = A276086(n/(p^e))^(p^(e-1)) is contributed to the product A380459(n). But n/(p^e) is not divisible by p, so A020639(A276086(n/(p^e))) <= p as either p or some lesser prime q < p is the first prime missing from the factorization of n/(p^e) [see comments in A276086], so in the former case there will be a factor p^(p^(e-1)) [thus also p^p], and in the latter case a factor q^(p^(e-1)) [thus also q^q as p^(e-1) > q] present in the product, so in any such case the product will not be in A048103, and therefore each term must be squarefree.
The least terms k for which A001222(k) = 0, 1, 2, ..., are given in A380475.

Links

Crossrefs

Cf. A048103, A276086, A359550, A380459, A380467 (characteristic function), A380475.
Setwise difference A005117 \ A380470.
Subsequences: A380474, A380478 (nonprime terms).

Programs

A380530 Positions of records in A380528.

Original entry on oeis.org

1, 4, 10, 42, 366, 3246, 37266, 631266, 11563926, 271591926
Offset: 1

Views

Author

Antti Karttunen, Feb 09 2025

Keywords

Comments

Questions: Are all terms squarefree after 4, and do all terms end (in base ten) with digit 6 from the fifth term onward? Is the sequence infinite? Are there any terms with more than four prime factors? See also the conjecture given in A380475, and A380476.
For n = 1..10, A380528(a(n)) = A008578(n). If a(11) exists, it is > 2^30.
Note that for squarefree n with exactly 5 prime factors (A046387), it might be possible that primes obtained as A380528(A046387(.)) has no upper limit. First k in A046387 such that A380528(k) is a prime 2 .. 17 are: 15015, 2730, 2310, 83886, 1551066, 71559186, 1245223986. See also the last two examples in A380470 for two similar cases with exactly 6 prime factors. - Antti Karttunen, May 08 2025

Links

Crossrefs

Cf. A008578 (conjectured to give the record values), A046387, A380528, A380459, A380468, A380470, A380475, A380476.

A380476 Numbers k with at least 4 prime factors such that A380459(k) is in A048103, i.e., has no divisors of the form p^p.

Original entry on oeis.org

4686, 32406, 184866, 209166, 388086, 1099626, 1714866, 2111406, 2166846, 2356206, 3081606, 3303366, 6445806, 11366106, 21621606, 23022366, 39824466, 39826986, 42882846, 43197846, 46043826, 58216686, 61265886, 63603546, 66496506, 66611166, 87941706, 88968246, 92086746, 97117026, 101108706, 103367886, 118743306, 119658066
Offset: 1

Views

Author

Antti Karttunen, Feb 04 2025

Keywords

Comments

Numbers m with four or more distinct prime factors such that their arithmetic derivative (A003415) can be formed as a carryless (or "carry-free") sum (in the primorial base, A049345) of the respective summands. See the example.
The terms are all squarefree and even (see A380468 and A380478 to find out why). Moreover, they are all multiples of six, because A380459(n) = Product_{d|n} A276086(n/d)^A349394(d) applied to a product of 2*p*q*r, with p, q, r three odd primes > 3 would yield three subproducts which would be multiples of 3 (consider A047247), so the 3-adic valuation of the whole product would be >= 3; hence the second smallest prime factor must be 3. For a similar reason, with terms that are product of four primes, the two remaining prime factors are either both of the form 6m+1 (A002476), or they are both of the form 6m-1 (A007528).
It is conjectured that there are no terms with more than four prime factors. See A380475 and A380528, A380530, also A380526.

Examples

			4686 = 2*3*11*71 and taking subproducts of three primes at time, we obtain 2*3*11 = 66, 2*3*71 = 426, 2*11*71 = 1562, 3*11*71 = 2343. Then A380459(4686) = A276086(66) * A276086(426) * A276086(1562) * A276086(2343) = 1622849599205985150 = 2^1 * 3^2 * 5^2 * 7^6 * 11^9 * 13^1, and because all the exponents are less than the corresponding primes, the product is in A048103.
Considering the primorial base expansions of the same summands (subproducts), we obtain
    2100  = A049345(66)
   20100  = A049345(426)
   73010  = A049345(1562)
  101011  = A049345(2343)
  ------
  196221  = A049345(A003415(4686)), with the summands adding together cleanly without any carries.
Note how the primorial base digits at the bottom are the exponents in the product A380459(4686) given above, read from the largest to the smallest prime factor

Links

Crossrefs

Intersection of A033987 and A380468.
Subsequence of A005117, A358673, A380478.
Conjectured to be a subsequence of A046386.
Cf. A003415, A001222, A048103, A276086, A349394, A380459, A380467.
Cf. A002476, A007528, A047247, A047257.
Cf. also A317836, A358235, A380475, A380526, A380528, A380530.

Programs

  • PARI
    is_A380476(n) = (issquarefree(n) && (omega(n)>=4) && A380467(n)); \\ Note that issquarefree here is just an optimization as A380467(n) = 1 implies squarefreeness of n.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.