cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

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Author

Antti Karttunen, Feb 05 2022

Keywords

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

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.

Programs

  • 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));

A369959 Numbers k such that A003415(k) >= A276086(k) and gcd(k, A003415(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, 210, 214, 2310, 2313, 2315, 2317, 2318, 2342, 2346, 2370, 2526, 4622, 4830, 30030, 30031, 30033, 30034, 30035, 30038, 30041, 30043, 30046, 30054, 30061, 30062, 30066, 30069, 30074, 30094, 30098, 30102, 30242, 30245, 30247, 30249, 30254, 30270, 30274, 30282, 32342, 32345, 32347, 32350, 32354, 32374, 32553, 60062
Offset: 1

Views

Author

Antti Karttunen, Feb 07 2024

Keywords

Links

Crossrefs

Intersection of A351228 and A369962.
Subsequence of A369958.
Cf. A003415, A083345, A085731, A276086, A324198, A351251, A369960 (subsequence).

Programs

  • 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); };
isA369959(n) = { my(t=A003415(n), u=A276086(n), g=gcd(n,t), h=gcd(n,u)); ((g==h) && ((t/g) >= (u/h))); };

Formula

{k | A085731(n) == A324198(n) and A083345(k) >= A351251(k)}.

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

Views

Author

Antti Karttunen, Feb 07 2024

Keywords

Links

Crossrefs

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

Programs

  • 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); };
isA369958(n) = ((A003415(n)/gcd(n, A003415(n))) >= (A276086(n)/gcd(n, A276086(n))));

Formula

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

A369963 Numbers k for which x = gcd(k, A003415(k)) = gcd(k, A276086(k)) and x > 1, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

9, 49, 99, 117, 153, 171, 207, 242, 245, 261, 279, 333, 369, 387, 423, 477, 531, 549, 603, 639, 657, 711, 747, 801, 850, 873, 909, 927, 963, 981, 1017, 1071, 1125, 1143, 1150, 1179, 1233, 1250, 1251, 1341, 1359, 1413, 1450, 1467, 1503, 1519, 1557, 1611, 1629, 1719, 1737, 1773, 1791, 1899, 1989, 2007, 2043, 2050, 2061
Offset: 1

Views

Author

Antti Karttunen, Feb 07 2024

Keywords

Comments

Nonsquarefree terms in A369962.

Links

Crossrefs

Intersection of A013929 and A369962.
Cf. A003415, A085731, A276086, A324198, A369960 (subsequence).

Programs

  • 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); };
isA369963(n) = (!issquarefree(n) && (gcd(n,A003415(n))==gcd(n,A276086(n))));
  • PARI
    A085731(n) = { my(f=factor(n)); for(i=1, #f~, if (f[i, 2] % f[i, 1], f[i, 2]--); ); factorback(f); };
A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
isA369963(n) = if(!n,0, my(u=A085731(n)); if(u<2, 0, (u == A324198(n))));

Formula

{k | A085731(k) == A324198(k) and A085731(k) > 1}.
Showing 1-4 of 4 results.

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