cp's OEIS Frontend

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.

%I A330737 #26 Feb 16 2025 08:33:59
%S A330737 1,2,4,9,15,28,38,55,71,92,110,125,146,167,183,206,225,258,281,313,
%T A330737 339,363,399,425,453,488,515,550,585,618,657,705,739,794,830,866,902,
%U A330737 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
%N 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.
%C A330737 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.
%C A330737 Question: Is this sequence well-defined for any n > 1? For all n? See also A199337.
%C A330737 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, ...
%C A330737 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
%H A330737 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly Composite Number</a>.
%H A330737 Wikipedia, <a href="http://www.wikipedia.org/wiki/Highly_composite_number">Highly composite number</a>.
%e A330737 a(0) = 1 as A002110(0) = 1, and A002182(1) = 1, and as all integers are divisible by 1, including all terms of A002182.
%e A330737 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.
%o A330737 (PARI)
%o A330737 \\ v002182 contains the terms of A002182 up to some suitably big value:
%o A330737 A330737(n) = if(!n,1,my(x=prime(n)); forstep(k=#v002182,1,-1,if(v002182[n]%x,return(1+k))));
%Y A330737 Cf. A002110, A002182, A072846, A199337, A328520, A329571.
%K A330737 nonn
%O A330737 0,2
%A A330737 _Antti Karttunen_, Dec 29 2019