A369000 -id:A369000 - cp's OEIS frontend

A366890 Irregular triangle, wherein row n lists in ascending order all numbers k whose arithmetic derivative k' is equal to the n-th primorial, A002110(n), and that have more than two prime factors with multiplicity. Rows of length zero are simply omitted, i.e., when A369000(n) = 0.

1547371, 79332523, 1102527599503, 25336943536819, 25962012375103, 25970380120783, 66702554987143, 526285951027003, 927949814519899, 7777707036642079, 9584173681667203, 13082430772438171, 22101822021783739, 4958985803436403, 32006922970429003, 32076018550175863, 49806227168831659, 84682266449971639, 97995266657958403

Offset: 1

Antti Karttunen, Jan 09 2024

For n > 0, numbers k such that A003415(k) = A002110(n) and A001222(k) > 2.
Sequence as a whole is not listed in ascending order, even though each batch of solutions for each n for which A369000(n) > 0 are. For example, we have a(14) < a(13) because A003415(22101822021783739) = A002110(12), while A003415(4958985803436403) = A002110(13). See the examples.
Question: Are there any common terms with A036785, that is, with A368697?

			For rows n=1..6, 9 & 10 nothing is listed, as those rows are empty.
Row for n=7 has just one term: 1547371 (= 7^2 * 23 * 1373). Note that A003415(1547371) = 510510 = A002110(7).
Row for n=8 has just one term: 79332523 (= 17^2 * 277 * 991).
Row for n=11 has two terms:
  1102527599503 (= 11^2 * 11071 * 823033),
  25336943536819 (= 157 * 743 * 5749 * 37781).
Row for n=12 has nine terms:
  25962012375103 (= 7^2 * 8597 * 61630451),
  25970380120783 (= 7^2 * 41387 * 12806141),
  66702554987143 (= 19^2 * 167 * 1106416889),
  526285951027003 (= 73 * 3919 * 7013 * 262313),
  927949814519899 (= 269 * 271 * 1697 * 7501033),
  7777707036642079 (= 2203 * 2791 * 7349 * 172127),
  9584173681667203 (= 2131 * 5953 * 7901 * 95621),
  13082430772438171 (= 3109 * 5861 * 24421 * 29399),
  22101822021783739 (= 8783 * 11777 * 13921 * 15349).
Row for n=13 has 18 terms, and begins with:
  4958985803436403 (= 37^2 * 137 * 26440450451),
and ends with:
  3206697143570677543 (= 36899 * 41983 * 45233 * 45763).
Note that A003415(3206697143570677543) = 304250263527210 = A002110(13).

Antti Karttunen, Table of n, a(n) for n = 1..31
Antti Karttunen, PARI program

When the whole sequence is sorted into ascending order, equal to A327978 without any semiprime solutions (solutions in A001358), and also a subsequence of following sequences: A004709, A327862, A328234.
Cf. A001222, A002110, A003415, A116979, A351029, A368703.
Cf. also A036785, A368697, A328243, A369240.

PARI
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\\ See the attached PARI-program
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A351029 Number of integers whose arithmetic derivative is equal to the n-th primorial.

Original entry on oeis.org

0, 1, 3, 19, 114, 905, 9494, 124181, 2044847, 43755729, 1043468388, 30309948250

Offset: 1

Antti Karttunen, Feb 01 2022

Number of integers k such that A003415(k) = A002110(n).
a(7) = A116979(7) + 1 since 1547371'=510510 and 1547371=7^2*23*1373 and every other example has only two prime factors. a(8) > A116979(8) because there is at least one term k in A327978 for which A003415(k) = 9699690 = A002110(8), which is not semiprime, that k being 79332523 = 17^2 * 277 * 991. - Edited by Craig J. Beisel, Sep 13 2022 and Antti Karttunen, Jan 05 2023
Most such k are semiprimes, i.e., are "Goldbachian solutions", counted by A116979. The non-semiprime solutions (A366890) form a very tiny minority, and are counted by A369000. - Antti Karttunen, Jan 19 2024

			a(1) = 0 because there are no such k that A003415(k) = 2 = A002110(1).
a(2) = 1 because there is only one number, 9, such that A003415(9) = A002110(2) = 6.
a(3) = 3 because there are exactly three numbers, k = 161, 209, 221, for which A003415(k) = A002110(3) = 30. (See A327978). These are all semiprime solutions, generated by the partitions of 30 into 2 primes: 30 = 7 + 23 = 11 + 19 = 13 + 17, and we have 7*23 = 161; 11*19 = 209; 13*17 = 221.

Cf. A002110, A002620, A003415, A099302, A099303, A116979, A327978, A366890 (nonsemiprime solutions), A368703 (the least of solutions), A368704 (the largest of solutions), A369000.

Cf. also A369239.

A002110(n) = prod(i=1,n,prime(i));
A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A351029(n) = { my(g=A002110(n)); sum(k=1,A002620(g),A003415(k)==g); }; \\ Very naive and slow. See comments in A327978.

A351029(n) = {v=prod(j=1,n,prime(j)); c=0; for(k=2, v^2/4, d=0; m=factor(k); for(i=1, matsize(m)[1], d+=(m[i,2]/m[i,1])*k; if(d>v, break;); ); if(d==v, c=c+1; ); ); c;} \\ Craig J. Beisel, Sep 13 2022

a(n) = Sum_{k=1..A002620(A002110(n))} [A003415(k) = A002110(n)], where [ ] is the Iverson bracket.

a(n) = A116979(n) + A369000(n). - Antti Karttunen, Jan 19 2024

a(7) from Craig J. Beisel, Sep 13 2022

a(8)..a(12) [the last based on the value of A116979(12)] from Antti Karttunen, Jan 09 2024

A116979 Number of distinct representations of primorials as the sum of two primes.

Original entry on oeis.org

0, 0, 1, 3, 19, 114, 905, 9493, 124180, 2044847, 43755729, 1043468386, 30309948241

Offset: 0

Jonathan Vos Post, Apr 01 2006

Related to Goldbach's conjecture. Let g(2n) = A002375(n). The primorials produce maximal values of the function g in the following sense: the basic shape of the function g is k*x/log(x)^2 and each primorial requires a larger value of k than the previous one. - T. D. Noe, Apr 28 2006

Relates also to a more generic problem of how many numbers there are such that their arithmetic derivative is equal to the n-th primorial number. See A351029. - Antti Karttunen, Jan 17 2024

			a(2) = 1 because 2nd primorial = 6 = 3 + 3 uniquely.
a(3) = 3 because 3rd primorial = 30 = 7 + 23 = 11 + 19 = 13 + 17.
a(4) = 19 because 4th primorial = 210 = 11 + 199 = 13 + 197 = 17 + 193 = 19 + 191 = 29 + 181 = 31 + 179 = 37 + 173 = 43 + 167 = 47 + 163 = 53 + 157 = 59 + 151 = 61 + 149 = 71 + 139 = 73 + 137 = 79 + 131 = 83 + 127 = 97 + 113 = 101 + 109 = 103 + 107.

Eric Weisstein's World of Mathematics, Primorial.
Index entries for sequences related to Goldbach conjecture

Cf. A000040, A002110, A351029, A369000.

Cf. A002375 (number of decompositions of 2n into unordered sums of two odd primes).

n=1; Join[{0,0}, Table[n=n*Prime[k]; cnt=0; Do[If[PrimeQ[2n-Prime[i]],cnt++ ], {i,2,PrimePi[n]}]; cnt, {k,2,10}]] (* T. D. Noe, Apr 28 2006 *)

a(n) = #{p(i) + p(j) = A002110(n) for p(k) = A000040(k) and i >= j}.

a(n) = A351029(n) - A369000(n). - Antti Karttunen, Jan 17 2024

More terms from T. D. Noe, Apr 28 2006

a(11)-a(12) from Donovan Johnson, Dec 19 2009

A369239 Number of integers whose arithmetic derivative is larger than 1 and equal to the n-th partial sum of primorial numbers.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 1, 27, 0, 319, 1

Offset: 1

Antti Karttunen, Jan 18 2024

Note how there are generally less solutions for even n than for odd n. This is explained by the fact that A143293(2n) == 1 (mod 4) and A143293(2n+1) == 3 (mod 4) and the arithmetic derivative A003415 of a product of any three odd primes (A046316) is always of the form 4k+3, therefore the solution set counted by a(2n) does not have any solutions from A046316 that contribute the majority of the solutions counted by a(2n+1). See also A369055.
a(13) >= 1 as there are solutions like 5744093403180469, 12538540924097819, etc., probably thousands or even more in total.
a(14) >= 1 [see examples].

			a(12) = 1 as there is a unique solution k such that k' = A143293(12) = 7628001653829, that k being 318745032938881 = 71*173*307*1259*67139. It's also the first solution with more than four prime factors.
a(14) >= 1, because as A143293(14)-2 = 13394639596851069-2 = 13394639596851067 is a prime, we have at least one solution, with A003415(2*13394639596851067) = A003415(26789279193702134) = 2+13394639596851067 = A143293(14).
For more examples, see A369240.

Antti Karttunen, PARI program for computing terms of A351029, A369000, A369239 and related sequences.

Cf. A003415, A046316, A143293, A328243, A369055, A369240.

Cf. also A351029, A369000.

PARI
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\\ See the attached program.
```

A368703 a(n) is the least integer k whose arithmetic derivative is equal to the n-th primorial, or 0 if no such k exists.

Original entry on oeis.org

2, 0, 9, 161, 2189, 29861, 510221, 1547371, 79332523, 9592991561, 265257420749, 1102527599503

Offset: 0

Antti Karttunen, Jan 16 2024

a(n) = the smallest integer k for which A003415(k) = A002110(n), and 0 if no such k exists.
If there are non-Goldbachian solutions (A366890) for some n, i.e., if A369000(n) > 0, then the smallest of them appears here as a value of a(n).
a(12) <= 25962012375103, a(13) <= 4958985803436403, a(14) <= 32442711864461575, a(15) <= 11758779158543465383. - David A. Corneth, Jan 17 2024

			a(0) = 2 as the least number k such that A003415(k) = A002110(0) = 1 is 2.
a(1) = 0 as there is no number k such that A003415(k) = A002110(1) = 2.
a(7) = 1547371 as it is the least number k such that A003415(k) = A002110(7) = 510510. See also A366890.

Antti Karttunen, PARI program
Index entries for sequences related to primorial numbers

Cf. A002110, A003415, A351029, A368704, A366890, A369000.

Cf. also A369243.

a(n) <= A368704(n).

For n<>1, A003415(a(n)) = A002110(n).

A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (pq + pr + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1

Offset: 0

Antti Karttunen, Jan 22 2024

Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum of the form 3*(p+q) + p*q, where p and q are odd primes.
Question: Will there be an eventual growth spurt for this sequence? Even though all solutions must be multiples of 3 (but not of 9), because A006862(n) == 1 (mod 3), for n > 1, and the solutions belong to a set listed by A369461.
Similar sequence A369242 grows more vigorously because A033312(n) == -1 (mod 3) for n >= 3, thus allowing non-multiples of 3 as solutions. See comments in A369252.

			a(4) = 1 as there exists a natural number 399 = 3 * 7 * 19, whose arithmetic derivative (indicated with 399', see A003415) is computed as ((3*7) + (3*19) + (7*19)) = 211 = 1 + prime(4)# = A006862(4), and because 399 is the unique term in A046316 that satisfies the condition.
a(17) >= 1 because there exists (at least one) solution k = 4903038892893242229501 = 3 * 17 * 96138017507710631951 with A003415(k) = 1+A002110(17).
For other cases, see examples in A369246.

Cf. A002110, A003415, A006862, A046316, A369054, A369246 (the solutions), A369252, A369461.

Cf. also A116979, A369000, A369239 for similar counts, also A369241, A369242 and A369247.

\\ Needs also program from A369054.
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = A369054(A002110(n)+1);

\\ Optimized version of above, employs the fact that solutions must all be multiples of 3. Outputs also terms for A369246.
search_for_3k1_cases(n) = if(3!=(n%4),0, my(p = 5, q, c=0); while(1, q = (n-(3*p)) / (3+p); if(q < p, return(c), if(1==denominator(q) && isprime(q),c++; write("b369246_by_search_order_to.txt", n, " ", 3*p*q))); p = nextprime(1+p)));
A002110(n) = prod(i=1,n,prime(i));
A369245(n) = search_for_3k1_cases(A002110(n)+1);

a(n) = A369054(A006862(n)).

cp's OEIS Frontend

A366890 Irregular triangle, wherein row n lists in ascending order all numbers k whose arithmetic derivative k' is equal to the n-th primorial, A002110(n), and that have more than two prime factors with multiplicity. Rows of length zero are simply omitted, i.e., when A369000(n) = 0.

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A351029 Number of integers whose arithmetic derivative is equal to the n-th primorial.

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A116979 Number of distinct representations of primorials as the sum of two primes.

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Mathematica

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A369239 Number of integers whose arithmetic derivative is larger than 1 and equal to the n-th partial sum of primorial numbers.

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A368703 a(n) is the least integer k whose arithmetic derivative is equal to the n-th primorial, or 0 if no such k exists.

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A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).

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PARI

PARI

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A369245 Number of representations of the n-th Euclid number, A002110(n) + 1, as a sum (pq + pr + q*r) with three odd primes p <= q <= r. (Definition implies that p=3 and q > 3).