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.

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A199337 Number of highly composite numbers not divisible by n.

Original entry on oeis.org

0, 1, 3, 3, 8, 3, 14, 9, 12, 8, 26, 4, 37, 14, 8, 18, 53, 12, 67, 8, 14, 26, 86, 9, 54, 37, 40, 14, 107, 8, 122, 51, 26, 53, 14, 12, 145, 67, 37, 11, 163, 14, 180, 26, 13, 86, 202, 18, 107, 54, 53, 37, 222, 40, 26, 15, 67, 107, 252, 8, 275, 122, 16, 79, 37
Offset: 1

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Author

J. Lowell, Nov 05 2011

Keywords

Comments

The sequence is well defined since for any n there is m such that n | A002182(k) for all k >= m. This follows from eq.(54) in Ramanujan (1915): [log_p P] <= e_p <= 2*[log_p P'], where for any N = A002182(k), P = gpf(N) is the greatest prime factor, e_p = valuation(N, p) is the exponent of any p in the prime factorization, P' = nextprime(P+1) and [.] = floor: The right inequality gives N <= Product_{prime p} p^(2*[log_p P']) = A003418(P')^2, so P -> oo as N -> oo. Then the left inequality implies e_p -> oo for any p, as N -> oo. - M. F. Hasler, Jan 03 2020
Sequences A329570 and A329571 give the gpf P as above and L = A003418(P) such that all A002182(k) >= L^2 are divisible by n. - M. F. Hasler, Jan 07 2020

Examples

			a(6) = 3 because among highly composite numbers, only 1, 2, and 4 are not divisible by 6.
To illustrate the comment, we prove that n = 12 = A002182(5), respectively n = 60 = A002182(9), divide all A002182(k) >= n (whence a(12) = 5 - 1 = 4, a(30) = 9 - 1 = 8): From eq.(54) we have e_2 >= 2 and e_3 >= 1 when [log_2 P] >= 2, [log_3 P] >= 1, which is the case for P >= 5. To get gpf(N) >= 5, use the other side of the inequality, e_p <= 2*[log_p P'] with P = 3, P' = 5: This gives e_2 <= 4; e_3, e_5 <= 2; e_p = 0 for p > 5. Thus all N = a(n) > 2^4*3^2*5^2 = 3600 must have gpf(N) > 3, i.e., gpf(N) >= 5. This implies e_2 >= 2 and e_3 >= 1 and also e_5 >= 1, so we have 12 | N and 60 | N for all N = A002182(k) > 3600. The terms between 12 (resp. 60) and 3600 are also multiples of 12 (resp. 60), which completes the proof. - _M. F. Hasler_, Jan 03 2020

Links

Crossrefs

Cf. A002182, A106037.
Cf. A329570, A329571 (bounds from the Ramanujan formula).

Programs

  • Mathematica
    (* let t be terms of b002182 *) Table[Length[Select[t, Mod[#, n] > 0 &]], {n, 100}] (* T. D. Noe, Mar 18 2012 *)

Extensions

Extended by T. D. Noe, Mar 18 2012

A329570 a(n) is the least prime P such that log(P)/log(p) >= valuation(n,p) for all primes p.

Original entry on oeis.org

2, 2, 3, 5, 5, 3, 7, 11, 11, 5, 11, 5, 13, 7, 5, 17, 17, 11, 19, 5, 7, 11, 23, 11, 29, 13, 29, 7, 29, 5, 31, 37, 11, 17, 7, 11, 37, 19, 13, 11, 41, 7, 43, 11, 11, 23, 47, 17, 53, 29, 17, 13, 53, 29, 11, 11, 19, 29, 59, 5, 61, 31, 11, 67, 13, 11, 67, 17, 23, 7, 71, 11, 73, 37, 29, 19, 11
Offset: 1

Views

Author

M. F. Hasler, Jan 03 2020

Keywords

Comments

Related to the inequality (54) in Ramanujan's paper about highly composite numbers A002182, also used in A199337: This is the largest prime factor of the bound A329571(n)^2 above which all highly composite numbers are divisible by n.

Links

Crossrefs

Cf. A002182, A007918, A034699, A199337, A329571.

Programs

  • Mathematica
    a[n_] := NextPrime[Max[Power @@@ FactorInteger[n]] - 1]; a[1] = 2; Array[a, 100] (* Amiram Eldar, Jan 17 2025 *)
  • PARI
    apply( {A329570(n,f=Col(factor(max(n,2))), P=nextprime(vecmax([log(f[1])*f[2] | f<-f])))=[while( logint(P,f[1]) < f[2], P=nextprime(P+1)) | f<-f]; P}, [1..99])

Formula

a(n) = A007918(A034699(n)). - Amiram Eldar, Jan 17 2025

A330737 a(n) is the first index k in A002182 (highly composite numbers) from which onward all terms A002182(i), i >= k, are multiples of the n-th prime, a(0) = 1 by convention.

Original entry on oeis.org

1, 2, 4, 9, 15, 28, 38, 55, 71, 92, 110, 125, 146, 167, 183, 206, 225, 258, 281, 313, 339, 363, 399, 425, 453, 488, 515, 550, 585, 618, 657, 705, 739, 794, 830, 866, 902, 950, 999, 1036, 1074, 1113, 1151, 1198, 1234, 1270, 1306, 1347, 1393, 1436, 1479, 1528, 1571, 1615, 1671, 1719, 1774, 1824, 1875, 1925, 1975, 2026, 2087, 2170, 2235
Offset: 0

Views

Author

Antti Karttunen, Dec 29 2019

Keywords

Comments

Equivalently, a(n) is the first index k in A002182 from which onward all terms A002182(i), i >= k, are multiples of A002110(n), the n-th primorial number.
Question: Is this sequence well-defined for any n > 1? For all n? See also A199337.
Note that this differs from A072846 at n = 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, ...
Yes, the sequence is well defined for all n, see A199337 for proof that all A002182(k) >= A329571(n)^2 are divisible by n. - M. F. Hasler, Jan 07 2020

Examples

			a(0) = 1 as A002110(0) = 1, and A002182(1) = 1, and as all integers are divisible by 1, including all terms of A002182.
A002182(9) = 60, and because from then onward all highly composite numbers are multiples of 30 (= A002110(3) = prime(1)*prime(2)*prime(3)), we have a(3) = 9.

Links

Crossrefs

Cf. A002110, A002182, A072846, A199337, A328520, A329571.

Programs

  • PARI
    \\ v002182 contains the terms of A002182 up to some suitably big value:
A330737(n) = if(!n,1,my(x=prime(n)); forstep(k=#v002182,1,-1,if(v002182[n]%x,return(1+k))));
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