A380476 -id:A380476 - cp's OEIS frontend

A380530 Positions of records in A380528.

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

Offset: 1

Antti Karttunen, Feb 09 2025

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

Index entries for sequences related to primorial base.

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

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

Antti Karttunen, Feb 03 2025

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.

			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.

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.

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

cp's OEIS Frontend

A380530 Positions of records in A380528.

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