A053589 -id:A053589 - cp's OEIS frontend

A099795 Least common multiple of 1, 2, 3, ..., prime(n)-1.

1, 2, 12, 60, 2520, 27720, 720720, 12252240, 232792560, 80313433200, 2329089562800, 144403552893600, 5342931457063200, 219060189739591200, 9419588158802421600, 3099044504245996706400, 164249358725037825439200, 9690712164777231700912800

Offset: 1

Ray Chandler, Oct 29 2004

nonn

Alternative definition: a(n) = Product{i = 1..(n-1)}prime(i)^e_i, where prime(i)^e_i is the greatest power of prime(i) which does not exceed prime(n). Every term is a product of prime powers, and also of primorial powers(the greatest of which is A002110(n-1); see Example and A053589). - David James Sycamore, Oct 24 2024

			For n = 7, prime(7) = 17, using the alternative definition (see Comment), a(7) = 2^4*3^2*5^1*7^1*11^1*13^1 = 16*9*5*7*11*13 = 720720 = 24*30030 = 2^2*6*30030 = A002110(1)^2*A002110(2)*A002110(6). - _David James Sycamore_, Oct 24 2024

Robert Israel, Table of n, a(n) for n = 1..342

Cf. A094998, A099794, A099796, A038610, A000040.

Cf. A002110, A007947, A053589.

[Lcm([2..p-1]): p in PrimesUpTo(70)]; // Bruno Berselli, Feb 06 2015

Primes:= select(isprime, [2,$3..100]):
seq(ilcm($2..Primes[i]-1),i=1..nops(Primes)); # Robert Israel, Jul 19 2016

LCM@@Range[#]&/@(Prime[Range[20]]-1) (* Harvey P. Dale, Jan 30 2015 *)

a(n) = (A094998(n)-1) / A099796(n).
a(n) = A038610(A000040(n)). - Anthony Browne, Jul 19 2016
Rad(a(n)) = A007947(a(n)) = A002110(n-1). - David James Sycamore, Oct 24 2024

a(18) from Bruno Berselli, Feb 06 2015

A328475 Convert the primorial base expansion of n into its prime product form, then divide by the largest primorial which divides that product: a(n) = A111701(A276086(n)).

Original entry on oeis.org

1, 1, 3, 1, 9, 3, 5, 5, 15, 1, 45, 3, 25, 25, 75, 5, 225, 15, 125, 125, 375, 25, 1125, 75, 625, 625, 1875, 125, 5625, 375, 7, 7, 21, 7, 63, 21, 35, 35, 105, 1, 315, 3, 175, 175, 525, 5, 1575, 15, 875, 875, 2625, 25, 7875, 75, 4375, 4375, 13125, 125, 39375, 375, 49, 49, 147, 49, 441, 147, 245, 245, 735, 7, 2205, 21, 1225, 1225, 3675, 35, 11025, 105

Offset: 0

Antti Karttunen, Oct 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2559
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences related to primorial base

Cf. A002110, A053589, A111701, A276086, A328476, A328399 (rgs-transform).

Cf. A143293 (indices of 1's after a(0)=1).

A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328475(n) = A111701(A276086(n));

a(n) = A111701(A276086(n)).

A328478 Divide n by the largest primorial that divides it and repeat until a fixed point is reached; a(n) is the fixed point.

Original entry on oeis.org

1, 1, 3, 1, 5, 1, 7, 1, 9, 5, 11, 1, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 1, 25, 13, 27, 7, 29, 1, 31, 1, 33, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 45, 23, 47, 1, 49, 25, 51, 13, 53, 9, 55, 7, 57, 29, 59, 1, 61, 31, 63, 1, 65, 11, 67, 17, 69, 35, 71, 1, 73, 37, 75, 19, 77, 13, 79, 5, 81, 41, 83, 7, 85, 43, 87, 11, 89

Offset: 1

Antti Karttunen, Oct 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial numbers

Cf. A007814 (gives the number of iterations to reach a fixed point), A025487 (indices of 1's).
Cf. A002110, A053589, A111701, A328479.
Cf. also A093411 for analogous sequence.

A111701[n_] := A111701[n] = Block[{m = n, k = 1}, While[IntegerQ[m/Prime[k]], m = m/Prime[k]; k++]; m];
a[n_] := a[n] = If[A111701[n] == n, n, a[A111701[n]]];
Array[a, 105] (* Jean-François Alcover, Jan 11 2022, after Robert G. Wilson v in A111701 *)

A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A328478(n) = { my(u=A111701(n)); if(u==n, return(n), return(A328478(u))); };

If A111701(n) == n, then a(n) = n, otherwise a(n) = a(A111701(n)).

a(n) = n / A328479(n).

Definition clarified by N. J. A. Sloane, Jan 19 2021

A328476 Convert the primorial base expansion of n into its prime product form, then subtract the largest primorial which divides that product: a(n) = A276151(A276086(n)).

Original entry on oeis.org

0, 0, 2, 0, 8, 12, 4, 8, 14, 0, 44, 60, 24, 48, 74, 120, 224, 420, 124, 248, 374, 720, 1124, 2220, 624, 1248, 1874, 3720, 5624, 11220, 6, 12, 20, 36, 62, 120, 34, 68, 104, 0, 314, 420, 174, 348, 524, 840, 1574, 2940, 874, 1748, 2624, 5040, 7874, 15540, 4374, 8748, 13124, 26040, 39374, 78540, 48, 96, 146, 288, 440, 876, 244, 488

Offset: 0

Antti Karttunen, Oct 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2559
Index entries for sequences related to primorial base

Cf. A002110, A053589, A276086, A276087, A276151, A326810, A328475.

Cf. A143293 (indices of other zeros after a(0)=0).

A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328476(n) = A276151(A276086(n));

a(n) = A276151(A276086(n)).

A276086(a(n)) = A276087(n) / A326810(n).

A328479 a(n) = n/A328478(n), where A328478(n) is obtained by repeatedly dividing n by the largest primorial that divides it until a fixed point is reached.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 6, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 60, 1, 2, 1, 64, 1, 6, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 30, 1, 4, 1, 2, 1, 96, 1, 2, 1, 4, 1, 6, 1, 8, 1

Offset: 1

Antti Karttunen, Oct 19 2019

nonn

a(n) is the largest term in A025487 that divides n evenly. - Hal M. Switkay, May 04 2021

Antti Karttunen, Table of n, a(n) for n = 1..30030
Index entries for sequences related to primorial numbers

Cf. A002110, A053589, A111701, A328478.

A111701[n_] := A111701[n] = Block[{m = n, k = 1}, While[IntegerQ[m/Prime[k]], m = m/Prime[k]; k++]; m];
A328478[n_] := If[A111701[n] == n, n, A328478[A111701[n]]];
A328479[n_] := n/A328478[n];
Array[A328479, 105] (* Jean-François Alcover, Jan 11 2022, after Robert G. Wilson v in A111701 *)

A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A328478(n) = { my(u=A111701(n)); if(u==n, return(n), return(A328478(u))); };
A328479(n) = (n/A328478(n));

a(n) = n / A328478(n).

A328580 a(n) is the largest primorial dividing A276086(n), where A276086 converts the primorial base expansion of n into its prime product form.

Original entry on oeis.org

1, 2, 1, 6, 1, 6, 1, 2, 1, 30, 1, 30, 1, 2, 1, 30, 1, 30, 1, 2, 1, 30, 1, 30, 1, 2, 1, 30, 1, 30, 1, 2, 1, 6, 1, 6, 1, 2, 1, 210, 1, 210, 1, 2, 1, 210, 1, 210, 1, 2, 1, 210, 1, 210, 1, 2, 1, 210, 1, 210, 1, 2, 1, 6, 1, 6, 1, 2, 1, 210, 1, 210, 1, 2, 1, 210, 1, 210, 1, 2, 1, 210, 1, 210, 1, 2, 1, 210, 1, 210

Offset: 0

Antti Karttunen, Oct 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2559
Index entries for sequences related to primorial base

Cf. A002110, A053589, A276086, A326810, A328475, A328476, A328570.

A328580(n) = { my(i=1, p=2, pr=1); while(n && (n%p), pr *= p; n = n\p; p = nextprime(1+p)); (pr); };

a(n) = A053589(A276086(n)).
a(n) = A002110(A328570(n)-1).
a(n) = A276086(n) / A328475(n).
a(n) = A276086(n) - A328476(n).
A328476(n) / a(n) = A328475(n) - 1.

A328473 a(n) = A276156(n) - A002110(A007814(n)).

Original entry on oeis.org

0, 0, 2, 0, 6, 6, 8, 0, 30, 30, 32, 30, 36, 36, 38, 0, 210, 210, 212, 210, 216, 216, 218, 210, 240, 240, 242, 240, 246, 246, 248, 0, 2310, 2310, 2312, 2310, 2316, 2316, 2318, 2310, 2340, 2340, 2342, 2340, 2346, 2346, 2348, 2310, 2520, 2520, 2522, 2520, 2526, 2526, 2528, 2520, 2550, 2550, 2552, 2550, 2556, 2556, 2558, 0, 30030

Offset: 1

Antti Karttunen, Oct 18 2019

A276156(n) converts the binary expansion of n to a number whose primorial base representation has the same digits of 0's and 1's, thus each one of its terms is a unique sum of distinct primorial numbers. This sequence is otherwise similar, but the primorial number corresponding to the least significant 1-bit of n is dropped from the sum, so the sum is not unique anymore.

Cf. A002110, A053589, A129760, A276151, A276156, A328461, A328474.

A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328473(n) = (A276156(n)-A002110(valuation(n,2)));

a(n) = A276156(A129760(n)).

a(n) = A276151(A276156(n)) = A276156(n) - A002110(A007814(n)).

A055770 Largest factorial number which divides n.

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 24, 1, 2, 1, 2, 1, 6

Offset: 1

Labos Elemer, Jul 12 2000

Largest m! which divides n.

			3! = 6 divides 12, so a(12) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, Tina Rajabi, Some Combinatorics of Factorial Base Representations, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.

Cf. A000142, A055881 (values of the m's), A055926, A055874, A073575.

Cf. also A053589.

With[{rf=Reverse[Range[7]!]},Table[SelectFirst[rf,Divisible[n,#]&],{n,120}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 05 2017 *)

A055770(n) = { my(m=1, i=2); while(!(n%m), m *= i; i++); return(m/(i-1)); } \\ Antti Karttunen, Dec 19 2018

a(n) = A000142(A055881(n)). - Antti Karttunen, Dec 19 2018

Name changed, old name moved to comments by Antti Karttunen, Dec 19 2018

A328399 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328475(i) = A328475(j) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 4, 5, 1, 6, 2, 7, 7, 8, 4, 9, 5, 10, 10, 11, 7, 12, 8, 13, 13, 14, 10, 15, 11, 16, 16, 17, 16, 18, 17, 19, 19, 20, 1, 21, 2, 22, 22, 23, 4, 24, 5, 25, 25, 26, 7, 27, 8, 28, 28, 29, 10, 30, 11, 31, 31, 32, 31, 33, 32, 34, 34, 35, 16, 36, 17, 37, 37, 38, 19, 39, 20, 40, 40, 41, 22, 42, 23, 43, 43, 44, 25, 45, 26, 46, 46, 47, 46, 48, 47

Offset: 0

Antti Karttunen, Oct 19 2019

nonn

Restricted growth sequence transform of A328475, defined as A328475(n) = A111701(A276086(n)).

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A002110, A053589, A111701, A276086, A143293 (indices of 1's after a(0)=1).

Cf. also A328477.

up_to = 32768;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328475(n) = A111701(A276086(n));
v328399 = rgs_transform(vector(up_to+1, n, A328475(n-1)));
A328399(n) = v328399[1+n];

A114562 The first occurrence of n in A111701.

Original entry on oeis.org

1, 4, 3, 8, 5, 36, 7, 16, 9, 20, 11, 72, 13, 28, 15, 32, 17, 108, 19, 40, 21, 44, 23, 144, 25, 52, 27, 56, 29, 900, 31, 64, 33, 68, 35, 216, 37, 76, 39, 80, 41, 252, 43, 88, 45, 92, 47, 288, 49, 100, 51, 104, 53, 324, 55, 112, 57, 116, 59, 1800, 61, 124, 63, 128, 65, 396, 67

Offset: 1

Robert G. Wilson v, Feb 04 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A053589, A111701.

Complement of this sequence A095300.

f[n_] := Block[{m = n, k = 1}, While[ IntegerQ[ m/Prime@k], m = m/Prime@k; k++ ]; m]; g[n_] := Block[{k = 1}, While[f@k != n, k++ ]; k]; Array[g, 67]

a(2n-1) = 2n-1; a(2n) = k*4n for some k>0, if 2n == 0 (mod 3) then k = 3, if 2n ==0 (15 mod) k = 3*5, if 2n ==0 (105 mod) k = 3*5*7, if 2n ==0 (1155 mod) k = 3*5*7*11, etc.

a(n) = n*A053589(n). - David A. Corneth, Mar 30 2021

cp's OEIS Frontend

A099795 Least common multiple of 1, 2, 3, ..., prime(n)-1.

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A328475 Convert the primorial base expansion of n into its prime product form, then divide by the largest primorial which divides that product: a(n) = A111701(A276086(n)).

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A328478 Divide n by the largest primorial that divides it and repeat until a fixed point is reached; a(n) is the fixed point.

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A328476 Convert the primorial base expansion of n into its prime product form, then subtract the largest primorial which divides that product: a(n) = A276151(A276086(n)).

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A328479 a(n) = n/A328478(n), where A328478(n) is obtained by repeatedly dividing n by the largest primorial that divides it until a fixed point is reached.

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A328580 a(n) is the largest primorial dividing A276086(n), where A276086 converts the primorial base expansion of n into its prime product form.

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A328473 a(n) = A276156(n) - A002110(A007814(n)).

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A055770 Largest factorial number which divides n.

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