A354933 -id:A354933 - cp's OEIS frontend

A354995 a(n) = A354933(n) - A034699(n).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8

Offset: 1

Antti Karttunen, Jun 18 2022

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A034699, A354933.

Cf. A051283 (apparently gives the positions of nonzero terms). Cf. also A354996.

a[n_] := SelectFirst[Divisors[n], # >= n/# && CoprimeQ[#, n/#] &] - Max[Power @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 18 2022 *)

A034699(n) = if(1==n, n, my(f=factor(n)); vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2])));
A354933(n) = fordiv(n,d,if((d>=(n/d)) && 1==gcd(d,n/d), return(d)));
A354995(n) = (A354933(n) - A034699(n));

A051283 Numbers k such that if one writes k = Product p_i^e_i (p_i primes) and P = max p_i^e_i, then k/P > P.

Original entry on oeis.org

30, 60, 70, 84, 90, 105, 120, 126, 132, 140, 154, 165, 168, 180, 182, 195, 198, 210, 220, 231, 234, 252, 260, 264, 273, 280, 286, 306, 308, 312, 315, 330, 336, 340, 357, 360, 364, 374, 380, 385, 390, 396, 399, 408, 418, 420, 429, 440, 442, 455, 456, 462

Offset: 1

Leroy Quet

Numbers k such that A346418(k) > 1 (conjectured). - Amiram Eldar, Jul 16 2021

Numbers k for which A354933(k) > A034699(k), i.e., where A354995(k) > 0 (conjectured).- Antti Karttunen, Jun 19 2022

			120 = 2^3*3^1*5^1, P = 2^3 = 8. 120 is included because 120/8 = 15 > 8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A034699, A080170, A110612, A346418, A354933, A354995, A354996 (characteristic function).

a051283 n = a051283_list !! (n-1)
a051283_list = filter (\x -> (a034699 x) ^ 2 < x) [1..]
-- Reinhard Zumkeller, May 30 2013

ok[n_] := n > Max[Power @@@ FactorInteger[n]]^2; Select[Range[465], ok] (* Jean-François Alcover, Apr 11 2011 *)

lista(nn) = for(k=2, nn, f=factor(k); if(k>vecmax(vector(#f[, 1], i, f[i, 1]^f[i, 2]))^2, print1(k, ", "))); \\ Jinyuan Wang, Feb 28 2020

a(n) = A080170(n) + 1 (conjectured). - Ralf Stephan, Feb 20 2004

More terms from James Sellers, Dec 11 1999

A052128 a(1) = 1; for n > 1, a(n) is the largest divisor of n that is coprime to a larger divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 3, 1, 2, 1, 4, 1, 5, 1, 1, 3, 2, 5, 4, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 2, 5, 7, 3, 2, 1, 5, 1, 2, 7, 1, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 7, 5, 2, 3, 8, 1, 9, 7, 4, 3, 2, 5, 3, 1, 2, 9, 4, 1, 6

Offset: 1

James Sellers, Jan 21 2000

nonn

Least k > 0 such that the resultant of the k-th cyclotomic polynomial and the n-th cyclotomic polynomial is not 1. - Benoit Cloitre, Oct 13 2002
From Jianing Song, Sep 28 2022: (Start)
a(n) is the largest divisor d of n such that d <= sqrt(n) and that gcd(d,n/d) = 1.
Proof: write n = Product_{1<=i<=r} (p_i)^(e_i), let d be the largest divisor of n such that d <= sqrt(n) and that gcd(d,n/d) = 1. Obviously we have a(n) >= d. Suppose that a(n) = Product_{1<=i<=s} (p_i)^(m_i) for s <= r, 1 <= m_i <= e_i, then the larger divisor to which a(n) is coprime is a divisor of Product_{s+1<=i<=r} (p_i)^(e_i), so by definition we have a(n) <= min{Product_{1<=i<=s} (p_i)^(e_i), Product_{s+1<=i<=r} (p_i)^(e_i)} <= d. Thus a(n) = d. (End)

			a(6) = 6 / 3^1 = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A054372, A076388, A354933.

Table[best = 1; d = Divisors[n]; While[Length[d] > 1, e = d[[1]]; d = Rest[d]; If[Min[GCD[e, d]] == 1, best = e]]; best, {n, 102}] (* T. D. Noe, Aug 23 2013 *)

a(n) = my(i, j, d = divisors(n)); forstep (i = #d-1, 1, -1, for (j = i+1, #d, if (gcd(d[i], d[j]) == 1, return (d[i])))); 1 \\ Michel Marcus, Aug 22 2013

a(n)=my(f=factor(n),v=[1]); for(i=1,#f~,v=concat(v, f[i,1]^f[i,2] *v)); v=vecsort(v); forstep(i=#v\2,2,-1,for(j=i+1,#v-1, if(gcd(v[i],v[j])==1,return(v[i])))); 1 \\ Charles R Greathouse IV, Aug 22 2013

A052128(n) = fordiv(n,d,if((d>=(n/d)) && 1==gcd(d,n/d), return(n/d))); \\ Antti Karttunen, Jun 16 2022

a(n) = n / A354933(n) = A354933(n) - A076388(n). - Antti Karttunen, Jun 16 2022

Terms corrected by Charles R Greathouse IV, Aug 22 2013

Definition rewritten by Jianing Song, Sep 28 2022

A076388 a(n) = minimum of y-x such that x <= y, x*y = n and gcd(x,y)=1.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 7, 8, 3, 10, 1, 12, 5, 2, 15, 16, 7, 18, 1, 4, 9, 22, 5, 24, 11, 26, 3, 28, 1, 30, 31, 8, 15, 2, 5, 36, 17, 10, 3, 40, 1, 42, 7, 4, 21, 46, 13, 48, 23, 14, 9, 52, 25, 6, 1, 16, 27, 58, 7, 60, 29, 2, 63, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 22, 15, 4, 7, 78, 11, 80, 39

Offset: 1

T. D. Noe, Oct 11 2002

If n is a prime or a power of a prime, a(n) = n-1. Similar to A056737, which does not have the gcd(x,y)=1 condition.

			a(12) = 1 because of the possible (x,y) pairs, (1,12), (2,6), (3,4), the pair (3,4) yields the minimum difference and satisfies gcd(x,y)=1.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (terms 2..10000 from Ivan Neretin)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034699, A052128, A056737, A354933.

Differs from |A354988(n)| for the first time at n=60, where a(60) = 7, while A354988(60) = -11.

nMax = 100; Table[dvs = Divisors[n]; i = 1; j = 1; While[n/dvs[[i]] > dvs[[i]], If[GCD[n/dvs[[i]], dvs[[i]]] == 1, j = i]; i++ ]; n/dvs[[j]] - dvs[[j]], {n, 2, nMax}]

A076388(n) = fordiv(n,d,if((d>=(n/d)) && 1==gcd(d,n/d), return(d-(n/d)))); \\ Antti Karttunen, Jun 16 2022

a(n) = A354933(n) - A052128(n). - Corrected by Antti Karttunen, Jun 16 2022

Definition formally changed from x < y to x <= y, to accommodate the prepended term a(1)=0 - Antti Karttunen, Jun 16 2022

cp's OEIS Frontend

A354995 a(n) = A354933(n) - A034699(n).

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A051283 Numbers k such that if one writes k = Product p_i^e_i (p_i primes) and P = max p_i^e_i, then k/P > P.

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A052128 a(1) = 1; for n > 1, a(n) is the largest divisor of n that is coprime to a larger divisor of n.

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A076388 a(n) = minimum of y-x such that x <= y, x*y = n and gcd(x,y)=1.

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