A351228 -id:A351228 - cp's OEIS frontend

A369666 Numbers k > 1 for which A276085(A003415(k)) == k (mod 4), where A003415 is the arithmetic derivative, and A276085 is the primorial base log-function.

6, 8, 10, 12, 15, 22, 24, 30, 34, 40, 42, 50, 56, 58, 60, 64, 65, 66, 70, 77, 78, 82, 84, 86, 104, 112, 114, 118, 120, 122, 126, 128, 130, 132, 136, 140, 141, 142, 146, 152, 154, 161, 168, 174, 180, 182, 184, 185, 188, 189, 194, 196, 201, 202, 204, 206, 209, 214, 220, 221, 222, 228, 230, 232, 236, 238, 242, 246, 250

Offset: 1

Antti Karttunen, Feb 06 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A276085, A369665 (characteristic function).
Cf. A369663 and A369664 (subsequences).
Cf. also A351228.

PARI
```
\\ See A369665.
```

A370127 Numbers k such that (A276086(k)/s)^s < k^(s-1), where A276086 is the primorial base exp-function, and s = bigomega(k).

Original entry on oeis.org

30, 32, 36, 60, 210, 212, 216, 240, 420, 2310, 2312, 2313, 2314, 2316, 2318, 2320, 2322, 2324, 2328, 2340, 2344, 2346, 2352, 2370, 2376, 2400, 2520, 2522, 2528, 2550, 2730, 4620, 4624, 4626, 4632, 4650, 4656, 4680, 4830, 4832, 4860, 6930, 30030, 30031, 30032, 30033, 30034, 30035, 30036, 30037, 30038, 30039, 30040

Offset: 1

Antti Karttunen, Feb 22 2024

nonn

Numbers k such that A276086(k) < s * k^((s-1)/s), with s = A001222(k).

For these numbers it must hold that A276086(k) < A003415(k) because (A003415(k)/s)^s >= k^(s-1) [with s = A001222(k)] holds for all k >= 2. See Ufnarovski and Åhlander, Theorem 9, point (4). In other words, this is a subsequence of A351228 \ {6}.

Antti Karttunen, Table of n, a(n) for n = 1..5204
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Setwise difference A351228 \ A370128.
Cf. A001222, A003415, A276086.
Cf. A066576 (subsequence).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA370127(n) = { my(x=A276086(n), s=bigomega(n)); ((x/s)^s < n^(s-1)); };

A351229 Numbers k for which A003415(k) >= A276086(k) > k, where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

2349, 2376, 2400, 2552, 4656, 4680, 4832, 4860, 6936, 6960, 30056, 30080, 30100, 30150, 30256, 30282, 32382, 32384, 32562, 36960, 60080, 510568, 510592, 510996, 511020, 511152, 511176, 511200, 512940, 513096, 513120, 513252, 513272, 515172, 515196, 515352, 515376, 515552, 517448, 517472, 519750, 540636, 540660, 540792

Offset: 1

Antti Karttunen, Feb 05 2022

The terms appear to come in batches dictated by their primorial base expansion (A049345), these terms having only low digit values in that base.

Index entries for sequences related to primorial base

Cf. A003415, A049345, A276086.
Intersection of A351227 and A351228.
Positions of ones in A351089.

Select[Range[550000], Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] >= m > #] &] (* Michael De Vlieger, Feb 05 2022 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA351229(n) = { my(u=A276086(n)); ((u > n) && (A003415(n) >= u)); };

A369958 Numbers k such that A003415(k)/gcd(k, A003415(k)) >= A276086(k)/gcd(k, A276086(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

6, 30, 33, 42, 63, 210, 212, 213, 214, 220, 420, 429, 462, 2310, 2312, 2313, 2314, 2315, 2316, 2317, 2318, 2319, 2320, 2325, 2330, 2340, 2342, 2343, 2344, 2345, 2346, 2355, 2370, 2373, 2379, 2380, 2520, 2522, 2526, 2530, 2535, 2552, 2730, 3003, 4620, 4622, 4623, 4626, 4628, 4630, 4654, 4680, 4830, 4836, 4862, 6930, 6942, 7150

Offset: 1

Antti Karttunen, Feb 07 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..13305 (terms up to 3*A002110(9))

Cf. A003415, A083345, A276086, A351251.
Subsequences: A002110 (after its two initial terms), A369959, A369960.
Cf. also A351228.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA369958(n) = ((A003415(n)/gcd(n, A003415(n))) >= (A276086(n)/gcd(n, A276086(n))));

{k | A083345(k) >= A351251(k)}.

A370128 Numbers k such that (A276086(k)/s)^s >= k^(s-1) and A276086(k) <= A003415(k), where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and s = bigomega(k).

Original entry on oeis.org

6, 213, 214, 2315, 2317, 2319, 2342, 2343, 2348, 2349, 2372, 2523, 2524, 2526, 2552, 4622, 4623, 4628, 4652, 6932, 6936, 6960, 30041, 30043, 30046, 30052, 30054, 30062, 30074, 30075, 30076, 30093, 30094, 30098, 30100, 30102, 30150, 30242, 30245, 30249, 30254, 30256, 30258, 30273, 30274, 30282, 32343, 32345, 32347

Offset: 1

Antti Karttunen, Feb 22 2024

nonn

Numbers k such that A003415(k) >= A276086(k) >= s * k^((s-1)/s), with s = A001222(k).

See comments in A370127.

Antti Karttunen, Table of n, a(n) for n = 1..6439

Setwise difference A351228 \ A370127.

Cf. A001222, A003415, A276086.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA370128(n) = { my(x=A276086(n), s=bigomega(n)); ((x<=A003415(n)) && ((x/s)^s >= n^(s-1))); };

A373603 The second smallest k such that A003415(k) == A276086(k) mod A002110(n), or -1 if no such k exists, where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A002110 gives the n-th primorial.

Original entry on oeis.org

2, 9, 26, 122, 1382, 21446, 204566, 9699686, 90387605

Offset: 1

Antti Karttunen, Jun 22 2024

For n > 1, the index of the next term in A373849, after its sixth term 0, that is a multiple of A002110(n), as for n >= 1, the smallest k such that A003415(k) == A276086(k) mod A002110(n) gives the sequence 1, 6, 6, 6, 6, 6, 6, 6, ..., because A003415(6) = A276086(6).
Provided that such k exists for every n (and the escape clause is not needed), then the sequence is by necessity monotonic. If it is strictly monotonic, then it implies that k=6 is the only k such that A003415(k) = A276086(k). See also comments in A351228.
Note that if we instead search for the smallest k such that A276086(k) is a multiple of A002110(n) we obtain A143293, partial sums of the primorial numbers. See also A368703.

Cf. A002110, A003415, A143293, A276086, A373849.

Cf. also A351228, A368703, A371104, A373848.

A002110(n) = prod(i=1,n,prime(i));
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A373603(n) = { my(m=A002110(n), c=2); for(i=1,oo,if(0==((A276086(i)-A003415(i))%m), c--; if(0==c, return(i)))); };

A373849 Difference between the primorial base exp-function and the arithmetic derivative.

Original entry on oeis.org

1, 2, 2, 5, 5, 17, 0, 9, 3, 24, 38, 89, 9, 49, 66, 142, 193, 449, 104, 249, 351, 740, 1112, 2249, 581, 1240, 1860, 3723, 5593, 11249, -24, 13, -59, 28, 44, 114, -25, 69, 84, 194, 247, 629, 134, 349, 477, 1011, 1550, 3149, 763, 1736, 2580, 5230, 7819, 15749, 4294, 8734, 13033, 26228, 39344, 78749, -43, 97, 114, 243

Offset: 0

Antti Karttunen, Jun 23 2024

sign

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A003415, A276086, A327858 [= gcd(a(n), A003415(n))], A351228 (indices of nonpositive terms).
Cf. A328382, A369971, A373603.
Cf. A359821 (positions of even terms), A359822 (of odd terms).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A373849(n) = (A276086(n)-A003415(n));

a(n) = A276086(n) - A003415(n).

A369656 Numbers k such that k' is odd and also k'' is odd, where k' stands for the arithmetic derivative of k, A003415(k).

Original entry on oeis.org

6, 10, 22, 27, 30, 34, 42, 50, 58, 66, 70, 78, 82, 86, 99, 105, 114, 118, 122, 125, 126, 130, 142, 146, 154, 165, 174, 182, 194, 202, 206, 207, 214, 222, 230, 231, 238, 242, 243, 246, 250, 255, 273, 274, 279, 282, 285, 286, 298, 302, 310, 318, 326, 333, 343, 345, 346, 357, 358, 366, 369, 370, 374, 382, 385, 386

Offset: 1

Antti Karttunen, Feb 05 2024

nonn

Numbers k that are in A235991 and also A003415(k) is in A235991.

Cf. A000035, A003415, A068346, A235991, A351228, A369655 (characteristic function).

PARI
```
\\ See A369655.
```

A371104 Starting from k=7, each subsequent term is the next larger k such that the ratio A276086(k)/A003415(k) is nearer to 1 than for the previous k in the sequence.

Original entry on oeis.org

7, 8, 213, 214, 2325, 2532, 4625, 30282, 32358, 32384, 60098, 570816, 572884, 575190, 9732128, 243513275

Offset: 1

Antti Karttunen, Mar 12 2024

Note that A276086(6) / A003415(6) = 5/5 = 1. If there are any x > 6, for which the ratio is 1, then the least one of them will terminate this sequence. Question: Could this sequence actually be infinite?

If it exists, a(17) > 1207959552.

			          k   A049345(k)    A276086(k)/A003415(k)  A276086(k)-A003415(k)
  ----------------------------------------------------------------------
          7,        101,       10/1        = 10,                9
          8,        110,       15/12       = 1.25,              3
        213,      10011,       66/74       = 0.89189189,       -8
        214,      10020,       99/109      = 0.90825688,      -10
       2325,     100211,     1950/1780     = 1.0955056,       170
       2532,     110200,     3575/3388     = 1.0551948,       187
       4625,     200021,     3042/2900     = 1.0489655,       142
      30282,    1011200,    32725/34181    = 0.95740324,    -1456
      32358,    1100300,    27625/26971    = 1.0242483,       654
      32384,    1101210,   116025/117696   = 0.98580241,    -1671
      60098,    2001110,    30345/30749    = 0.98686136,     -404
     570816,   12011100,  2114035/2093568  = 1.0097761,     20467
     572884,   12100020,   642447/643056   = 0.99905296,     -609
     575190,   12200000,   927979/927483   = 1.0005348,       496
    9732128,  101103110, 26152035/26148912 = 1.0001194,      3123
  243513275, 1220000021, 99685818/99683810 = 1.0000201,      2008.

Index entries for sequences related to primorial base

Cf. A003415, A049345, A276086, A351228.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
print1(7, ", "); r = A276086(7)/A003415(7); for(n=7, oo, t=A276086(n)/A003415(n); if(abs(1-t) < abs(1-r), r=t; print1(n, ", ")))

cp's OEIS Frontend

A369666 Numbers k > 1 for which A276085(A003415(k)) == k (mod 4), where A003415 is the arithmetic derivative, and A276085 is the primorial base log-function.

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A370127 Numbers k such that (A276086(k)/s)^s < k^(s-1), where A276086 is the primorial base exp-function, and s = bigomega(k).

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A351229 Numbers k for which A003415(k) >= A276086(k) > k, where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

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A369958 Numbers k such that A003415(k)/gcd(k, A003415(k)) >= A276086(k)/gcd(k, A276086(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A370128 Numbers k such that (A276086(k)/s)^s >= k^(s-1) and A276086(k) <= A003415(k), where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and s = bigomega(k).

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A373603 The second smallest k such that A003415(k) == A276086(k) mod A002110(n), or -1 if no such k exists, where A003415 is the arithmetic derivative, A276086 is the primorial base exp-function, and A002110 gives the n-th primorial.

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A373849 Difference between the primorial base exp-function and the arithmetic derivative.

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A369656 Numbers k such that k' is odd and also k'' is odd, where k' stands for the arithmetic derivative of k, A003415(k).

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A371104 Starting from k=7, each subsequent term is the next larger k such that the ratio A276086(k)/A003415(k) is nearer to 1 than for the previous k in the sequence.

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