A370127 -id:A370127

Search: a370127 -id:a370127

Displaying 1-3 of 3 results found.

page 1

A351228

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

+10
20

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

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

PROG

(PARI)

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));

CROSSREFS

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.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Feb 05 2022

STATUS

approved

A328118

Numbers k for which A276086(k) <= A002620(k), where A276086 is the primorial base exp-function and A002620(k) = floor(k^2/4).

+10
4

6, 7, 8, 12, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 60, 61, 62, 63, 64, 65, 66, 67, 68, 72, 90, 91, 92, 93, 96, 120, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 234, 235, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

PROG

(PARI)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

A002620(n) = ((n^2)>>2);

isA328118(n) = (A276086(n) <= A002620(n));

CROSSREFS

Cf. A002620, A276086, A328119 (complement).

Subsequences: A328110 (after its two initial terms), A351228, A370127.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 08 2019

STATUS

approved

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).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

See comments in A370127.

LINKS

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

PROG

(PARI)

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))); };

CROSSREFS

Setwise difference A351228 \ A370127.

Cf. A001222, A003415, A276086.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 22 2024

STATUS

approved

page 1

Search completed in 0.006 seconds