A380526 -id:A380526 - cp's OEIS frontend

A380525 Squarefree numbers k such that for all factorizations of k as xy, the sum (x y') + (x' * y) is carryless when the addition is done in the primorial base, A049345. Here n' stands for A003415(n), the arithmetic derivative of n.

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, 70, 71, 73, 74, 79, 83, 86, 89, 97, 101, 103, 107, 109, 113, 122, 127, 131, 134, 137, 139, 146, 149, 151, 154, 157, 158, 163, 167, 173, 179, 181, 186, 190, 191, 193, 194, 195, 197, 199, 206, 211, 218, 223, 227, 229, 233

Offset: 1

Antti Karttunen, Feb 04 2025

nonn

A380468 is a subsequence. This differs from it by containing also the terms 70, 154, 190, 195, 455, 574, 645, 1054, 1085, ...

			For n=70, there are four factorizations into two factors: 1*70, 2*35, 5*14, 7*10, and thus, applying the formula (x' * y) + (x * y') we obtain
  0*70 + 1*70' = A003415(70) = 59, and A049345(59) = 1421.
  1*35 + 2*35' = 35 + 2*12, i.e., 1021 + 400 in primorial base, (giving 1421)
  1*14 + 5*14' = 14 + 5*9, i.e., 210 + 1211 in primorial base,
  1*10 + 7*10' = 10 + 7*7, i.e., 120 + 1301 in primorial base,
and as all these sums are carryless, 70 is included in this sequence.
For n = 1518 = 2*3*11*23, we obtain eight factorizations into two factors:
x*y:    | 1*1518   2*759   3*506   6*253  11*138   22*69   23*66   33*46
--------+----------------------------------------------------------------
x' * y  |      0   34111   22410   60021    4300   41411    2100   30210 (in primorial base)
x  * y' |  66421   32310   44011    6400   62121   25010   64321   36211
--------+----------------------------------------------------------------
Sum     |  66421   66421   66421   66421   66421   66421   66421   66421 = A049345(A003415(1518)), and as all these sums are carryless, 1581 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base.

Intersection of A005117 and A358673.
Cf. A003415, A049345, A380524 (characteristic function).
Subsequences: A380468, A380526 (terms with at least 5 prime factors).

PARI
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is_A380525 = A380524;
```

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

Antti Karttunen, Feb 04 2025

nonn

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.

			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

Antti Karttunen, Table of n, a(n) for n = 1..433; all terms <= 2^35
Index entries for sequences related to primorial base.

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.

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.

cp's OEIS Frontend

A380525 Squarefree numbers k such that for all factorizations of k as xy, the sum (x y') + (x' * y) is carryless when the addition is done in the primorial base, A049345. Here n' stands for A003415(n), the arithmetic derivative of n.

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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.

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cp's OEIS Frontend

A380525 Squarefree numbers k such that for all factorizations of k as x*y, the sum (x * y') + (x' * y) is carryless when the addition is done in the primorial base, A049345. Here n' stands for A003415(n), the arithmetic derivative of n.

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PARI

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.

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PARI

A380525 Squarefree numbers k such that for all factorizations of k as xy, the sum (x y') + (x' * y) is carryless when the addition is done in the primorial base, A049345. Here n' stands for A003415(n), the arithmetic derivative of n.