A327978 -id:A327978 - 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
```
\\ See the attached PARI-program
```

A341518 Numbers k such that the primorial base representation of their arithmetic derivative does not contain digits larger than 1.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 23, 28, 29, 30, 31, 37, 41, 43, 45, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 83, 87, 89, 97, 101, 103, 107, 108, 109, 112, 113, 127, 131, 136, 137, 139, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 189, 191, 193, 197, 198, 199, 203, 209, 210, 211, 212, 217

Offset: 1

Antti Karttunen, Feb 28 2021

nonn

Numbers k for which A328390(k) <= 1, numbers k such that A003415(k) is in A276156.

Numbers k such that A327859(k) = A276086(A003415(k)) is squarefree.

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

Cf. A003415, A005117, A049345, A276086, A276156, A327859, A328114, A328390.
Positions of nonzero terms in A341517.
Subsequences: A000040, A327978, A328232, A369647 (terms k where A051903(k) obtains novel values).
Cf. also A327969.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
ismaxprimobasedigit_at_most(n,k) = { my(s=0, p=2); while(n, if((n%p)>k, return(0)); n = n\p; p = nextprime(1+p)); (1); };
isA341518(n) = ismaxprimobasedigit_at_most(A003415(n),1); \\ Antti Karttunen, Feb 03 2024

For all n > 2, A328390(a(n)) = A328114(A003415(a(n))) = 1.

A328232 Numbers whose arithmetic derivative (A003415) is a primorial number, including cases where it is the first primorial, A002110(0) = 1.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 161, 163, 167, 173, 179, 181, 191, 193, 197, 199, 209, 211, 221, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317

Offset: 1

Antti Karttunen, Oct 09 2019

nonn

Numbers n such that A327859(n) = A276086(A003415(n)) is a prime.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial numbers

Cf. A002110, A003415, A024451 (arith. deriv. of primorials), A068346, A276086, A327859, A328233.
Union of A000040 and A327978 (gives the composite terms).
Differs from A189710 for the first time by having term a(39) = 161, which is not included in A189710, while A189710(44) = 185 is the first term in latter that is not included here.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327859(n) = A276086(A003415(n));
isA328232(n) = isprime(A327859(n));

A377992 Irregular triangle giving on row n all antiderivatives of A024451(n), for n >= 2.

Original entry on oeis.org

6, 30, 58, 210, 435, 507, 2310, 8435, 21827, 29233, 30030, 39030, 62762, 69914, 76442, 78874, 510510, 1342785, 1958673, 9699690, 28235362, 223092870, 975351895, 1527890095, 1885679383, 2189118743, 2329696457, 2338611863, 3485765789, 4586671213, 5453593183, 5472849253, 5674340053, 8071055747, 8931775397, 9332889127

Offset: 2

Antti Karttunen, Nov 19 2024

Row n lists in ascending order all numbers k whose arithmetic derivative k' [A003415(k)] is equal to A024451(n) = A003415(A002110(n)). For A024451(1) = 1, there is an infinite number of integers k for which A003415(k) = 1 (namely, all the primes), therefore the rows start from index n=2, with each having A377993(n) terms. Note that as a whole, this sequence is not monotonic, for example, the last term on row 9, 1171314743479 is larger than the first term of row 10, 6469693230.

Because A024451 is a subsequence of A048103, this is also. And if all terms of A024451 are squarefree as is conjectured, then all terms of this sequence are cubefree (A004709).

			The initial rows of the triangle:
Row n  terms
   2   6;
   3   30, 58;
   4   210, 435, 507;
   5   2310, 8435, 21827, 29233;
   6   30030, 39030, 62762, 69914, 76442, 78874;
   7   510510, 1342785, 1958673;
   8   9699690, 28235362;
   9   223092870, 975351895, 1527890095, ..., , 1167539981207, 1171314743479; (row 9 has 330 terms that are given separately in A378209)
  10   6469693230, 27623935255, 37262208055;
  11   200560490130, 345634019382, 440192669882;
  etc.
The only terms that occur on row 4 are k = 210, 435, 507 ( = 2*3*5*7, 3*5*29, 3 * 13^2) as they are only numbers for which A003415(k) = 247 = A024451(4) = A003415(A002110(4)), as we have (2*3*5*7)' = (3*5)'*(2*7) + (2*7)'*3*5 = (8*14) + (9*15) = (3*5*29)' = (3*5)'*29 + (3*5)*29' = (8*29 + 15*1) = (3 * 13 * 13)' = (3*13)'*13 + (3*13)*13' = 16*13 + 3*13*1 = 19*13 = 247.
 Note that 507 is so far the only known term in this triangle that is not squarefree (in A005117).

Cf. A003415, A005117, A024451, A377993 (row lengths).
Subsequence of A048103, conjectured also to be a subsequence of A004709.
Cf. A002110 (left edge, from its term a(2)=6 onward), A378209 (row 9).
Cf. also A327978, A366890, A369240, A377987.

A369641 Composite numbers k such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

Original entry on oeis.org

9, 10, 14, 15, 16, 28, 30, 45, 58, 62, 74, 87, 108, 112, 136, 155, 161, 189, 198, 203, 209, 210, 212, 217, 221, 225, 236, 244, 246, 247, 282, 290, 299, 323, 361, 374, 399, 422, 435, 478, 482, 507, 717, 1055, 1205, 1477, 1480, 1631, 1673, 1687, 1940, 2132, 2189, 2212, 2308, 2356, 2519, 2524, 2561, 2587, 2655, 2766

Offset: 1

Antti Karttunen, Jan 31 2024

nonn

Composite terms of A341518, i.e., composite numbers k such that A327859(k) = A276086(A003415(k)) is squarefree number, or equally, k' is in A276156.

Antti Karttunen, Table of n, a(n) for n = 1..20054 (terms less than 2^30)

Setwise difference A341518 \ A158611.
Cf. A003415, A276086, A276156, A327859, A341517, A369640 (characteristic function).
Cf. A327978, A328243, A369642 (subsequences).

PARI
```
\\ See A369640
```

A368702 Numbers whose second arithmetic derivative (A068346) is a primorial number (A002110) > 1.

Original entry on oeis.org

14, 186, 258, 322, 338, 3318, 3962, 5334, 6106, 7674, 8970, 9186, 9978, 10930, 11994, 12154, 12614, 12970, 13218, 13234, 14626, 15226, 15914, 16378, 17122, 18226, 18658, 19058, 19874, 20194, 20962, 21082, 21106, 21218, 44718, 49358, 57346, 58354, 75442, 76162, 81802, 87814, 95114, 102794, 113505, 114918, 130802

Offset: 1

Antti Karttunen, Jan 09 2024

nonn

Numbers k for which A003415(k) is one of the terms of A327978.

Cf. A002110, A003415, A068346, A327978.

Subsequence of A046100 and of A328244.

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]));
A068346(n) = A003415(A003415(n));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
isA368702flat(n) = { my(u=A068346(n)); ((u>1)&&(1==A276150(u))); };
k=0; for(n=1, A002620(A002110(7)), if(isA368702flat(n), k++; write("b368702.txt", k, " ", 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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A341518 Numbers k such that the primorial base representation of their arithmetic derivative does not contain digits larger than 1.

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A328232 Numbers whose arithmetic derivative (A003415) is a primorial number, including cases where it is the first primorial, A002110(0) = 1.

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A377992 Irregular triangle giving on row n all antiderivatives of A024451(n), for n >= 2.

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A369641 Composite numbers k such that k' is a sum of distinct primorial numbers, where k' stands for the arithmetic derivative of k, A003415.

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PARI

A368702 Numbers whose second arithmetic derivative (A068346) is a primorial number (A002110) > 1.

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PARI