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 1-6 of 6 results.

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.

A129252 Smallest prime factor p of n such that p^p is a divisor of n, a(n)=1 if no such factor exists.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 07 2007

Keywords

Examples

			For n = 108 = 2^2 * 3^3, it is 2 that is the smallest prime factor p satisfying p^p | 108, thus a(108) = 2.

Links

Crossrefs

Cf. A020639, A048103 (positions of 1's), A008578, A051674, A129251, A359550, A368333, A380528.
Differs from A327936 for the first time at n=108.

Programs

  • Mathematica
    Array[If[IntegerQ@ #, #, 1] &@ First@ SelectFirst[FactorInteger[#], #1 <= #2 & @@ # &] &, 120] (* Michael De Vlieger, Oct 01 2019 *)
  • PARI
    A129252(n) = { my(f = factor(n)); for(k=1, #f~, if(f[k, 2]>=f[k, 1], return(f[k, 1]))); (1); }; \\ Antti Karttunen, Oct 01 2019
  • PARI
    A129252(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(p)); if(pp > n, return(1))); }; \\ Antti Karttunen, Feb 09 2025

Formula

a(n) = 1 iff A129251(n) = 0.
a(A048103(n)) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/p^p) + Sum_{p prime} ((1/p^(p-1)) * Product_{primes q < p} (1-1/q^q)) = 1.30648526015949409005... . - Amiram Eldar, Nov 07 2022

Extensions

Data section extended to a(120) by Antti Karttunen, Oct 01 2019

A380470 Numbers k that are squarefree, but A380459(k) is not in A048103.

Original entry on oeis.org

10, 15, 21, 22, 30, 33, 34, 35, 39, 42, 46, 51, 55, 57, 58, 65, 66, 69, 70, 77, 78, 82, 85, 87, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 123, 129, 130, 133, 138, 141, 142, 143, 145, 154, 155, 159, 161, 165, 166, 170, 174, 177, 178, 182, 183, 185, 187, 190, 195, 201, 202, 203, 205, 209, 210, 213
Offset: 1

Views

Author

Antti Karttunen, Feb 02 2025

Keywords

Comments

See also comments in A380530.

Examples

			From _Antti Karttunen_, May 09 2025: (Start)
10 = 2*5 is a term, as it is squarefree, and A380459(10) = 54 = 2 * 3^3, thus the prime factor 3 overflows, i.e., has an exponent at least as large as that prime.
5117046 = 2*3*11*31*41*61 is a term, as it is squarefree, and A380459(5117046) = 2 * 3^2 * 5^4 * 7^11 * 11^22 * 13^36 * 17^31 * 19^8, thus the least prime factor which overflows is 7 [= A380528(5117046)].
31203546 = 2*3*11*31*101*151 is a term, as it is squarefree, and A380459(31203546) = 2 * 3^2 * 5^4 * 7^6 * 11^28 * 13^37 * 17^56 * 19^18 * 23^2, thus the least prime factor which overflows is 11 [= A380528(31203546)].
(End)

Links

Crossrefs

Setwise difference A005117 \ A380468.
Cf. A380469 (characteristic function), A380528, A380530.

Programs

A380475 a(n) is the least term in A380468 that has exactly n prime factors.

Original entry on oeis.org

1, 2, 6, 186, 4686
Offset: 0

Views

Author

Antti Karttunen, Feb 03 2025

Keywords

Comments

If it exists, a(5) > 2^35. - Antti Karttunen, Feb 19 2025
Conjecture: Sequence is finite and a(4) is the last term. This is equivalent to the claim that no arithmetic derivative (A003415) of a product of five or more distinct primes, (i.e., the value of the (n-1)-st elementary symmetric polynomial formed from those n distinct primes) can be formed as a carry-free sum of those n summands in primorial base (A049345). See also A380476 and A380528, A380530.

Examples

			186 = 2*3*31 and A276086(186/2) = 2058 = 2 * 3 * 7^3, A276086(186/3) = 3 * 7^2, A276086(186/31) = 5, whose product =  2^1 * 3^2 * 5^1 * 7^5 = 1512630 = A380459(186), and as all the exponents are less than the corresponding primes, the product is in A048103, and because there are no any smaller number with three prime factors satisfying the same condition (of A380468), 186 is the term a(3) of this sequence. Note that A049345(A003415(186)) = 5121, where the digits are the exponents in the product read from the largest to the smallest prime factor.
See also the example in A380476 about 4686.

Links

Crossrefs

Cf. A001221, A001222, A003415, A048103, A049345, A276086, A380459, A380468, A380476 (terms of A380468 with more than three prime factors).
Cf. A047247, A047257.
Cf. also A317836, A358235, A380525, A380528, A380530.

Formula

a(n) = Min_{k in A380468} for which A001221(k) = n.

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.

A380529 Smallest prime p such that p^p is a divisor of A380459(A005117(n)), or 1 if no such factor exists, where A380459(n) = Product_{d|n} A276086(n/d)^A349394(d) and A005117 lists the squarefree numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 3, 1, 2, 3, 2, 1, 1, 2, 1, 5, 1, 3, 1, 2, 1, 2, 2, 3, 1, 1, 1, 2, 3, 1, 2, 3, 1, 1, 1, 2, 5, 1, 3, 1, 2, 1, 2, 1, 2, 2, 3, 2, 1, 1, 3, 1, 2, 3, 1, 1, 3, 2, 1, 5, 2, 3, 2, 1, 2, 1, 2, 3, 1, 2, 1, 1, 3, 1, 2, 3, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 2, 1, 2, 3, 1, 3, 1
Offset: 1

Views

Author

Antti Karttunen, Feb 09 2025

Keywords

Links

Crossrefs

Cf. A005117, A129252, A276086, A349394, A380459, A380468, A380470, A380528.

Programs

  • PARI
    for(n=1,oo,if(issquarefree(n),print1(A380528(n),", ")));

Formula

a(n) = A380528(A005117(n)) = A129252(A380459(A005117(n))).
Showing 1-6 of 6 results.

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