A369970 -id:A369970 - cp's OEIS frontend

A351228 Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

6, 30, 32, 36, 60, 210, 212, 213, 214, 216, 240, 420, 2310, 2312, 2313, 2314, 2315, 2316, 2317, 2318, 2319, 2320, 2322, 2324, 2328, 2340, 2342, 2343, 2344, 2346, 2348, 2349, 2352, 2370, 2372, 2376, 2400, 2520, 2522, 2523, 2524, 2526, 2528, 2550, 2552, 2730, 4620, 4622, 4623, 4624, 4626, 4628, 4632, 4650, 4652, 4656

Offset: 1

Antti Karttunen, Feb 05 2022

Conjecture: Apart from the initial 6, the rest of terms are the numbers k for which A003415(k) > A276086(k), thus giving the positions of zeros in A351232. In other words, it seems that only k=6 satisfies A003415(k) = A276086(k). See also comments in A351088.

Antti Karttunen, Table of n, a(n) for n = 1..11643 (terms up to 3*A002110(9))
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Index entries for sequences related to primorial base

Cf. A003415, A024451, A048103, A060735, A235991, A276085, A276086, A327862, A351088, A351089, A351232, A369656, A369663, A369664, A371104.
Union of A370127 and A370128.
Subsequence of A328118.
Subsequences: A351229, A369959, A369960, A369970 (after its two initial terms).
Cf. also A369650.

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); };
isA351228(n) = (A003415(n)>=A276086(n));

A369971 a(n) = A003415(n) mod A276086(n), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 0, 1, 1, 4, 1, 0, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, 15, 27, 32, 1, 3, 1, 17, 14, 19, 12, 25, 1, 21, 16, 68, 1, 41, 1, 48, 39, 25, 1, 112, 14, 45, 20, 56, 1, 81, 16, 92, 22, 31, 1, 43, 1, 33, 51, 192, 18, 61, 1, 72, 26, 59, 1, 156, 1, 39, 55, 80, 18, 71, 1, 176, 108, 43, 1, 124, 22

Offset: 0

Antti Karttunen, Feb 07 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A276086, A369970 (positions of 0's).

Cf. also A327858 [= gcd(a(n), A276086(n))], A328382, A342014.

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); };
A369971(n) = (A003415(n)%A276086(n));

A369972 Numbers k such that (prime(k)#)' is a multiple of prime(1+k), where prime(k)# means the k-th primorial, A002110(k), and ' stands for taking the arithmetic derivative, A003415.

Original entry on oeis.org

0, 2, 7, 14, 21, 28, 261202

Offset: 1

Antti Karttunen, Feb 07 2024

Numbers k for which A024451(k) is a multiple of A000040(1+k).

			7 is included because the primorial prime(7)# = A002110(7) = 510510 has as its arithmetic derivative 510510' = A024451(7) = 716167 = 19*37693, which is divisible by the next larger prime not present in the primorial, in this case by prime(8) = 19.

Index entries for sequences related to primorial numbers

Cf. A000040, A000720, A024451, A293457 (corresponding primes), A369970, A369973 (corresponding primorials).

Cf. also A109628.

A024451(n) = numerator(sum(i=1, n, 1/prime(i)));
isA369972(n) = !(A024451(n)%prime(1+n));

a(n) = A000720(A293457(n)) - 1.

Found a(7) by computing it as A000720(A293457(7))-1. - Antti Karttunen, Feb 08 2024

A369973 Primorials whose arithmetic derivative is divisible by the next larger prime not present in that primorial.

Original entry on oeis.org

1, 6, 510510, 13082761331670030, 40729680599249024150621323470, 2566376117594999414479597815340071648394470

Offset: 1

Antti Karttunen, Feb 07 2024

nonn

Primorials A002110(k) such that A003415(A002110(k)) [= A024451(k)] is a multiple of A000040(1+k).

a(7) = A002110(261202), which is too large to include here, or even in a b-file.

			The zeroth primorial, 1 = A002110(0), is included, because its arithmetic derivative 1' = A024451(0) = 0 is divisible by the next larger prime not present in the primorial, in this case by prime(1) = 2.
The primorial 510510 = prime(7)# is included, because its arithmetic derivative 510510' = A024451(7) = 716167 = 19*37693 is divisible by the next larger prime not present in the primorial, in this case by prime(8) = 19.

Index entries for sequences related to primorial numbers

Cf. A000040, A002110, A003415, A024451, A293457 (the corresponding primes), A369972.

Subsequence of A369970.

A002110(n) = prod(i=1,n,prime(i));
A024451(n) = numerator(sum(i=1, n, 1/prime(i)));
isA369972(n) = !(A024451(n)%prime(1+n));
for(n=0,2^10,if(isA369972(n),print1(A002110(n),", ")));

a(n) = A002110(A369972(n)).

A370115 Numbers k for which k is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 10, 15, 161, 2189, 5005, 27030, 29861, 510221, 223092341

Offset: 1

Antti Karttunen, Feb 11 2024

nonn

Question: Is the squarefreeness a necessary condition for the nonzero terms of this sequence?
Many of the terms occur also in A368703, because the arithmetic derivative of those terms is one of the primorial numbers, A002110.
If it exists, a(13) > 1241513984.

Positions of 1's in A370117, positions of 0's in A370120.
Intersection of A048103 and A369650 is a subsequence of this sequence. See the comments in latter.
Cf. A002110, A003415, A276086, A327859, A368703.
Cf. also A369970, A370114.

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); };
isA370115(n) = !(n%A276086(A003415(n)));

cp's OEIS Frontend

A351228 Numbers k for which A003415(k) >= A276086(k), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

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A369971 a(n) = A003415(n) mod A276086(n), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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A369972 Numbers k such that (prime(k)#)' is a multiple of prime(1+k), where prime(k)# means the k-th primorial, A002110(k), and ' stands for taking the arithmetic derivative, A003415.

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A369973 Primorials whose arithmetic derivative is divisible by the next larger prime not present in that primorial.

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A370115 Numbers k for which k is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

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