A051451 -id:A051451 - cp's OEIS frontend

A056835 a(n) = lcm(s(1),...,s(n)) where {s(n)} = A024619 and a(0) = 1.

1, 6, 30, 60, 420, 420, 1260, 1260, 1260, 13860, 27720, 360360, 360360, 360360, 360360, 6126120, 6126120, 6126120, 116396280, 116396280, 116396280, 116396280, 116396280, 116396280, 2677114440, 5354228880, 26771144400

Offset: 0

Labos Elemer, Aug 30 2000

nonn

			Overlaps with but is different from A003418 and A051451. E.g., 840 or 720720 are not here, but 1260 and 13860 are here, not there.

Robert Israel, Table of n, a(n) for n = 0..3930

Cf. A024619, A003418, A051451. Range of values is A056836. All members belong to A025487.

N:= 100: # to get a(0) to a(N)
A[0]:= 1:
m:= 0:
for s from 2 while m < N do
if nops(numtheory:-factorset(s)) > 1 then
  m:= m+1;
  A[m]:= ilcm(A[m-1],s);
fi
od:
seq(A[i],i=0..N); # Robert Israel, Dec 20 2015

s = Select[Range@ 50, ! PrimePowerQ@ # &]; Table[LCM @@ Take[s, n], {n, Length@ s}] (* Michael De Vlieger, Dec 20 2015 *)

Link typos corrected, cross-reference added by Matthew Vandermast, Nov 13 2010

A056836 Distinct values of sequence obtained when LCM is applied to initial segments of sequence A024619 union {1}.

Original entry on oeis.org

1, 6, 30, 60, 420, 1260, 13860, 27720, 360360, 6126120, 116396280, 2677114440, 5354228880, 26771144400, 80313433200, 2329089562800, 72201776446800, 2671465728531600, 109530094869795600, 4709794079401210800

Offset: 0

Labos Elemer, Aug 30 2000

nonn

A003418, A051451, A000961, A024619.

Distinct values of A056835. Subsequence of A025487.

Link typo corrected, cross-references added by Matthew Vandermast, Nov 15 2010

A064354 Distinct values arising in the sequence of the least common multiples of the first n composite numbers.

Original entry on oeis.org

1, 4, 12, 24, 72, 360, 2520, 5040, 55440, 277200, 3603600, 10810800, 21621600, 367567200, 6983776800, 160626866400, 1124388064800, 32607253879200, 1010824870255200, 2021649740510400, 74801040398884800, 224403121196654400

Offset: 1

Labos Elemer, Sep 25 2001

nonn

			Different from A003418 and A051451.

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A002808, A003418, A051451, A025543.

t=t=Select[ Range[ 1, 200 ], !PrimeQ[ # ] & ] Union[ Table[ Apply[ LCM, Table[ Part[ t, w ], {w, 1, s} ] ], {s, 1, 100} ] ]
Join[{1},With[{cmps=Select[Range[100],CompositeQ]},Table[LCM@@Take[cmps,n],{n,Length[cmps]}]]]//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 06 2018 *)

{ n=0; b=1; for (m=1, 10^9, if (m>1, if (isprime(m) || (b=lcm(a, m)) == a, next)); write("b064354.txt", n++, " ", a=b); if (n==100, break) ) } \\ Harry J. Smith, Sep 12 2009

a(22) from Harry J. Smith, Sep 12 2009

Definition clarified by Harvey P. Dale, Dec 06 2018

A216152 A205957(n) where n is a nonprime number.

Original entry on oeis.org

1, 2, 12, 48, 144, 1440, 34560, 483840, 7257600, 58060800, 3135283200, 125411328000, 2633637888000, 57940033536000, 5562243219456000, 27811216097280000, 723091618529280000, 6507824566763520000, 364438175738757120000, 327994358164881408000000

Offset: 1

Peter Luschny, Sep 02 2012

nonn

The distinct values of A205957. Partial products of A216153.

a(1),...,a(10) are highly totient numbers (A097942) and products of distinct factorials (A058295). The author conjectures that this is true in general.

Peter Luschny, The von Mangoldt Transformation.

Cf. A051451.

A205957[n_] := Exp[-Sum[MoebiusMu[p] Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]];
Table[A205957[n], {n, 0, 30}] // DeleteDuplicates (* Jean-François Alcover, Jul 08 2019 *)

# sorted(list(set([A205957(n) for n in (0..31)])))
def A216152_list(n) :
    C = filter(lambda k: not is_prime(k), (1..n))
    return [A205957(c) for c in C]
A216152_list(31)

a(n) = A205957(A018252(n)).

A242298 Once a number in this sequence is divisible by all numbers 1 to m, subsequent terms are constrained to have the same property; choose the smallest permissible number that is greater than the previous term.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 300, 360, 420, 840, 1680, 2520, 5040, 7560, 10080, 12600, 15120, 17640, 20160, 22680, 25200, 27720, 55440, 83160, 110880, 138600, 166320, 194040, 221760, 249480, 277200, 304920, 332640, 360360, 720720, 1441440

Offset: 1

J. Lowell, May 10 2014

nonn

All terms from A095848 belong to this sequence.

			After 6, none of 7,8,9,10 or 11 are in the sequence since they are not divisible by 1,2 and 3 as 6 is. 12 is a term, but is now divisible by 1,2,3 and 4, adding a new constraint on subsequent terms.
After 24, 30 is not in the sequence because 24 is divisible by all numbers from 1 to 4 and 30 is not divisible by 4. But 36, which is divisible by all of 1 through 4, is a term.
As an irregular table, the n-th row consists of all numbers divisible by A051451(n) but not by A051451(n+1). - _Tom Edgar_, May 22 2014

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A003418, A051451, A080765, A368777.

consecd(a) = {d = divisors(a); for (i=2, #d, if (d[i] - d[i-1] > 1, return(i-1));); return(a);}
findnext(a) = {nconsd = consecd(a); na = a + 1; while (consecd(na) < nconsd, na++); na;}
lista(nn) = {a = 1; print1(a, ", "); for (n=1, nn, a = findnext(a); print1(a, ", "););} \\ Michel Marcus, May 11 2014

first(n) = {
	my(res = vector(n), step = 1, oldm = 1, newm = 1);
	res[1] = 1;
	for(i = 2, n,
		while(res[i-1] % (newm+1) == 0,
			newm++;
		);
		if(newm > oldm,
			step = lcm([step, lcm([oldm..newm])]);
			oldm = newm
		);
		res[i] = res[i-1]+step
	);
	res
} \\ David A. Corneth, Jan 28 2024

a(1) = 1. If n > 1, a(n) = a(n-1) + A368777(a(n-1)). - Hal M. Switkay, Jan 26 2024

More terms from Michel Marcus, May 11 2014

A248906 Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

Original entry on oeis.org

0, 1, 2, 5, 8, 3, 16, 37, 66, 9, 128, 7, 256, 17, 10, 549, 1024, 67, 2048, 13, 18, 129, 4096, 39, 8200, 257, 16450, 21, 32768, 11, 65536, 131621, 130, 1025, 24, 71, 262144, 2049, 258, 45, 524288, 19, 1048576, 133, 74, 4097, 2097152, 551, 4194320, 8201

Offset: 1

Franklin T. Adams-Watters, Mar 06 2015

nonn

			The prime power divisors of 12 are 2, 3, and 4. These are indices 1, 2, and 3 in the list of prime powers, so a(12) = 2^(1-1) + 2^(2-1) + 2^(3-1) = 7.

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

Cf. A246655, A065515, A034729, A000961.

Cf. A095874, A210208, A073093, A000079.

a248906 = sum . map ((2 ^) . subtract 2 . a095874) . tail . a210208_row
-- Reinhard Zumkeller, Mar 07 2015

al(n) = my(r=vector(n),pps=[p| p <- [1..n], isprimepower(p)],p2); for(k=1,#pps,p2=2^(k-1);forstep(j=pps[k],n,pps[k],r[j]+=p2));r

Additive with a(p^k) = Sum_{j=1..k} 2^(A065515(p^j)-1).
a(A051451(k)) = 2^k - 1.
a(n) = Sum_{k=1..A073093(n)} 2^(A095874(A210208(n,k))-2). - Reinhard Zumkeller, Mar 07 2015

A368777 a(n) is the largest divisor of n that is a term of the sequence A003418, the least common multiple of the first k natural numbers.

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 60, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 12

Offset: 1

Hal M. Switkay, Jan 11 2024

The graph of this sequence gives it the appearance of a ruler-like function. If n is odd, a(n) = 1. If n is even and not a multiple of 6, a(n) = 2. If n is a multiple of 6 but not of 12, a(n) = 6, and so on.

			a(18) = 6 as 18 is divisible by lcm(1, 2, 3) = 6 but not by lcm(1, 2, 3, 4) = 12. so 6 is the largest divisor of 18 that is a term of A003418. - _David A. Corneth_, Jan 28 2024

Hal M. Switkay, Table of n, a(n) for n = 1..5040

Cf. A003418, A051451, A027750, A055874.

seq[max_] := Module[{lcms = Table[LCM @@ Range[k], {k, max}]}, Table[Max[Select[Divisors[k], MemberQ[lcms, #] &]], {k, 1, max}]]; seq[100] (* Amiram Eldar, Jan 12 2024 *)

a(n) = for(i = 2, n, if(n%i != 0, return(lcm([1..i-1])))); n \\ David A. Corneth, Jan 27 2024

a(n) = A003418(A055874(n))

A001041 a(0)=12; thereafter a(n) = 12 times the product of the first n primes.

Original entry on oeis.org

12, 24, 72, 360, 2520, 27720, 360360, 6126120, 116396280, 2677114440, 77636318760, 2406725881560, 89048857617720, 3651003162326520, 156993135980040360, 7378677391061896920, 391069901726280536760, 23073124201850551668840

Offset: 0

Bob Marshall, 1811 California St., Eureka CA 95501

Bob Marshall calls these the "holotomic numbers" and opines: "This sequence, aptly designated as the Holotomic Sequence, represents the minimal plural quantities that accommodate the maximum amount of consecutive factors." Thus this can be seen as a version of A051451. - Charles R Greathouse IV, Jan 12 2012

If we were to take the maximum number of consecutive factors as the definition, the sequence would be prefixed with 1, 2, 6. - Franklin T. Adams-Watters, Jun 13 2015

T. D. Noe, Table of n, a(n) for n = 0..100
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
Index to divisibility sequences

Equals 12 * A002110.

[12] cat [12*&*[NthPrime(i):i in [1..n]]:n in [1..17]]; // Marius A. Burtea, Feb 11 2020

12*FoldList[ #1*Prime[ #2 ]&, 1, Range[ 20 ] ]

a(n)=12*prod(i=1,n,prime(i)); \\ Charles R Greathouse IV, Jan 12 2012

More terms from James Sellers, Sep 19 2000

A053608 Numbers x = LCM(1,2,...,k) such that x^2 + 1 is prime.

Original entry on oeis.org

1, 2, 6, 420, 360360, 718766754945489455304472257065075294400

Offset: 1

Labos Elemer, Feb 09 2000

The next term has k > 10^4, if it exists. - Amiram Eldar, Aug 23 2024

Cf. A003418, A049536, A046029, A051418, A051451, A051739, A053609.

Select[FoldList[LCM, 1, Select[Range[100], PrimePowerQ]], PrimeQ[#^2 + 1] &] (* Amiram Eldar, Aug 23 2024 *)

A053609 Primes of form x^2+1 where x = LCM(1,2,...,k) for some k.

Original entry on oeis.org

2, 5, 37, 176401, 129859329601, 516625648014869290354797521879383114125823989794742396526049715541246671360001

Offset: 1

Labos Elemer, Feb 09 2000

The next term has k > 10^4, if it exists. - Amiram Eldar, Aug 23 2024

Cf. A049536, A046029, A051739, A051418, A051451, A053608.

Select[FoldList[LCM, 1, Select[Range[100], PrimePowerQ]]^2 + 1, PrimeQ] (* Amiram Eldar, Aug 23 2024 *)

a(n) = A053608(A053608(n)) = A053608(n)^2 + 1. - Amiram Eldar, Aug 23 2024

cp's OEIS Frontend

A056835 a(n) = lcm(s(1),...,s(n)) where {s(n)} = A024619 and a(0) = 1.

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A056836 Distinct values of sequence obtained when LCM is applied to initial segments of sequence A024619 union {1}.

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A064354 Distinct values arising in the sequence of the least common multiples of the first n composite numbers.

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Mathematica

PARI

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A216152 A205957(n) where n is a nonprime number.

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A242298 Once a number in this sequence is divisible by all numbers 1 to m, subsequent terms are constrained to have the same property; choose the smallest permissible number that is greater than the previous term.

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A248906 Binary representation of prime power divisors of n: Sum_{p^k | n} 2^(A065515(p^k)-1).

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Haskell

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A368777 a(n) is the largest divisor of n that is a term of the sequence A003418, the least common multiple of the first k natural numbers.

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A001041 a(0)=12; thereafter a(n) = 12 times the product of the first n primes.

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