A328521 -id:A328521 - cp's OEIS frontend

A354880 Numbers k where A133411(k) is not equal to A328521(k-1).

17, 19, 21, 51, 52, 55, 56, 57, 58, 59, 60, 61, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 127, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169

Offset: 1

J. Lowell, Jun 09 2022

nonn

			A133411(17)=5587021440. (5587021440/19 = A133411(16) = 294053760.) But A328521(16)=2205403200. (2205403200 has 16 prime factors and is A328521(16) even if not 294053760 times an integer [the quotient is 15/2].) Thus, 17 is a term of this sequence.

Cf. A133411, A328521.

A133411 Smallest highly composite number of the form k*a(n-1) where k is an integer greater than 1.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 240, 720, 5040, 10080, 20160, 221760, 665280, 8648640, 17297280, 294053760, 5587021440, 27935107200, 642507465600, 1927522396800, 13492656777600, 26985313555200, 782574093100800, 24259796886124800

Offset: 1

J. Lowell, Nov 25 2007

nonn

Conjecture: subsequence of A019505.

			6 is not in the sequence because 6 is not a multiple of 4, the previous term.

Antti Karttunen, Table of n, a(n) for n = 1..223 (computed from the 10000 term b-file of A002182 prepared from Flammenkamp's data)

Cf. A002182, A019505, A328521, A330744 (primorial deflation).

sublist_of_first_proper_multiple_terms_of(v) = { my(u=v[1], lista=List(u)); for(i=2,#v,if((v[i]>u)&&!(v[i]%u), u = v[i]; listput(lista,u))); Vec(lista); };
v133411 = sublist_of_first_proper_multiple_terms_of(v002182); \\ v002182 contains the terms of A002182.
A133411(n) = v133411[n]; \\ Antti Karttunen, Jan 10 2020

a(12)-a(24) from Donovan Johnson, Sep 09 2008

A330743 a(n) is the first term k of A329902 for which A056239(k) = n.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 40, 60, 84, 168, 336, 528, 792, 936, 1872, 2448, 3060, 4560, 4788, 8280, 15456, 23184, 29232, 31248, 62496, 74592, 124320, 137760, 144480, 157920, 315840, 356160, 559680, 623040, 644160, 966240, 1061280, 1124640, 1686960, 1734480, 2049840, 2218320, 2330640, 2499120, 4165200, 4539600, 4726800, 4820400

Offset: 0

Antti Karttunen, Jan 13 2020

nonn

Note that in contrast to A330744 this is not monotonic. The first point where a(n) > a(n+1) occurs is at a(120) = 5481774144 > a(121) = 5452302240. See also comment in A328521, whose primorial deflation this sequence is.

a(n-1) differs from A330744(n) at n = 17, 19, 21, 51, 52, 55, 56, 57, 58, 59, 60, 61, ...

Antti Karttunen, Table of n, a(n) for n = 0..4257

Primorial deflation of A328521.
Cf. A056239, A319626, A329900, A329902, A330748.
Cf. also A330744.

A330743(n) = { for(k=1,oo,if(A056239(A329902(k))==n,return(A329902(k)))); };

v329902 = readvec("a329902.txt"); \\ File for the first 779674 terms of A329902 as prepared by Michael De Vlieger.
A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1])));
A330743list() = { my(m=Map(), lista=List([]), t); for(i=1, #v329902, t = A056239(v329902[i]); if(!mapisdefined(m,t), mapput(m,t,v329902[i]))); for(n=0,oo,if(mapisdefined(m,n,&t), listput(lista,t), return(Vec(lista)))); };
v330743 = A330743list();
A330743(n) = v330743[1+n];
for(n=0,#v330743-1,write("b330743.txt", n, " ", A330743(n)));

a(n) = A329902(min{i: A056239(A329902(i))==n}).
a(n) = A329902(A330748(n)).
a(n) = A329900(A328521(n)) = A319626(A328521(n)).

A330748 Index of the smallest element in A002182 that has exactly n prime factors counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 12, 14, 19, 21, 23, 32, 37, 47, 50, 62, 70, 80, 91, 99, 105, 109, 124, 140, 143, 159, 166, 182, 198, 217, 221, 240, 253, 276, 297, 304, 327, 352, 357, 381, 398, 424, 449, 475, 485, 512, 540, 570, 584, 617, 642, 676, 704, 738, 765, 770, 805, 841, 877, 913, 937, 949, 985, 1021, 1058, 1096, 1134, 1169

Offset: 0

Antti Karttunen, suggested by M. F. Hasler, Jan 10 2020

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..4257 (computed using A. Flammenkamp's 779674-term HCN dataset; terms 0..329 from _Antti Karttunen_)

Cf. A001222, A002182, A112778, A328521, A329902, A330743.

A330748(n) = { for(k=1,#v112778,if(v112778[k]==n,return(k))); -(1/0); };

v329902 = readvec("a329902.txt"); \\ File for the first 779674 terms of A329902
A056239(n) = if(1==n,0,my(f=factor(n)); sum(i=1, #f~, f[i,2] * primepi(f[i,1])));
A330748list() = { my(m=Map(), lista=List([]), t); for(i=1, #v329902, t = A056239(v329902[i]); if(!mapisdefined(m,t), mapput(m,t,i))); for(n=0,oo,if(mapisdefined(m,n,&t), listput(lista,t), return(Vec(lista)))); };
v330748 = A330748list();
A330748(n) = v330748[1+n];
for(n=0,#v330748-1,write("b330748.txt", n, " ", A330748(n))); \\ Antti Karttunen, Jan 13 2020

a(n) = min{k: A112778(k)=n}.
A002182(a(n)) = A328521(n).
A329902(a(n)) = A330743(n).

A328522 Largest highly composite number that has n prime factors counted with multiplicity.

Original entry on oeis.org

1, 2, 6, 12, 60, 180, 1260, 2520, 27720, 83160, 1081080, 3603600, 61261200, 698377680, 3491888400, 80313433200, 240940299600, 1686582097200, 48910880818800, 1516237305382800, 3032474610765600, 112201560598327200, 3066842656354276800, 131874234223233902400, 659371171116169512000, 30990445042459967064000

Offset: 0

David A. Corneth, Jan 04 2020

nonn

These numbers are upper bounds on the largest term in A002182 that is not divisible by k for some k.

Cf. A001222, A002182, A112778, A328521.

With[{s = Take[Import["https://oeis.org/b002182.txt", "Data"][[All, -1]], 240]}, Drop[TakeWhile[#, IntegerQ], -6] &@ Table[s[[Lookup[#, n][[-1]] ]], {n, 0, Max@ Keys@ #}] &@ PositionIndex[Map[PrimeOmega, s]]] (* Michael De Vlieger, Jan 19 2020, using b-file at A002182. Caution: ensure full population of a given value of bigomega by extending scope beyond the desired number of terms. *)

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A354880 Numbers k where A133411(k) is not equal to A328521(k-1).

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A133411 Smallest highly composite number of the form k*a(n-1) where k is an integer greater than 1.

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A330743 a(n) is the first term k of A329902 for which A056239(k) = n.

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A330748 Index of the smallest element in A002182 that has exactly n prime factors counted with multiplicity.

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PARI

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A328522 Largest highly composite number that has n prime factors counted with multiplicity.

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Mathematica