A083722 -id:A083722 - cp's OEIS frontend

A293892 Restricted growth sequence transform of A083722.

1, 1, 1, 2, 1, 3, 1, 4, 5, 6, 1, 7, 1, 8, 9, 10, 1, 11, 1, 12, 13, 14, 1, 15, 16, 17, 15, 18, 1, 19, 1, 20, 21, 22, 23, 24, 1, 25, 26, 27, 1, 28, 1, 29, 30, 31, 1, 32, 33, 34, 35, 36, 1, 37, 38, 39, 40, 41, 1, 42, 1, 43, 44, 45, 46, 47, 1, 48, 49, 50, 1, 51, 1, 52, 53, 54, 55, 56, 1, 57, 58, 59, 1, 60, 61, 62, 63, 64, 1, 65, 66, 67, 68, 69, 70, 71, 1, 72, 73

Offset: 1

Antti Karttunen, Nov 02 2017

nonn

a(n) = 1 iff n is prime or n is 1.

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A083722, A293893, A293894.

allocatemem(2^30);
up_to = 65536;
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A083722(n) = { if(n==1, return (1)); my(gpf = vecmax(factor(n)[, 1])); my(pp = 1); forprime(p=gpf+1, n, pp *= p; ); pp; } \\ This function from Michel Marcus, Jun 26 2016
write_to_bfile(1,rgs_transform(vector(up_to,n,A083722(n))),"b293892.txt");

A293894 Numbers n such that A083722(n) > 1 and A083722(n) occurs earlier in A083722.

Original entry on oeis.org

27, 125, 147, 539, 2197, 2992, 3159, 3249, 3757, 4199, 4851, 5733, 6517, 11774, 15717, 16807, 19652, 20475, 25289, 28899, 30625, 31213, 31465, 33275, 34122, 41327, 43384, 44616, 50255, 60858, 61250, 61750, 62271

Offset: 1

Antti Karttunen, Nov 02 2017

nonn

Equally, numbers n such that A293892(n) > 1 and A293892(n) <= max(A293892(1) .. A293892(n-1)).
Starts like A137800 except that term 3159 is not included in A137800, and furthermore, the latter sequence is not monotonic.
Question: Are there such runs of composites that contain three or more numbers whose largest prime factor is the same prime? In other words, is the intersection of A293893 and A293894 empty or not?

David A. Corneth, Table of n, a(n) for n = 1..101

Cf. A083722, A137800, A293892, A293893.

Flatten@ Values@ Map[Rest, Rest@ PositionIndex@ Array[Times @@ Select[Prime@ Range[#1, #1 + #2], Function[p, p <= #3]] & @@ {PrimePi@ NextPrime[FactorInteger[#][[-1, 1]]], PrimePi@ #, #} &, 10^4]] (* Michael De Vlieger, Nov 03 2017 *)

upto(n) = {my(l = List(), p = 3, res = List, c); forprime(q = 5, nextprime(n + 1), for(i = p + 1, q - 1, f = factor(i)[, 1]; listput(l, [f[#f], precprime(i), i])); p = q); listsort(l); i = 1; while(i < #l - 1, if(l[i][1] == l[i+1][1], if(l[i][2] == l[i+1][2], listput(res, l[i+1][3]))); i++); listsort(res); res} \\ David A. Corneth, Nov 03 2017

A293893 Numbers n such that A083722(n) > 1 and A083722(n) occurs later in A083722.

Original entry on oeis.org

24, 120, 140, 528, 2184, 2975, 3146, 3230, 3740, 4180, 4840, 5720, 6498, 11745, 15686, 16800, 19635, 20449, 25270, 28880, 30618, 31200, 31434, 33264, 34075, 41310, 43355, 44590, 50232, 60835, 61236, 61731, 62234

Offset: 1

Antti Karttunen, Nov 02 2017

nonn

Equally, numbers n such that A293892(n) > 1 and A293892(n) occurs later in A293892.
Starts like A137799 except that term 3146 is not included in A137799, and furthermore, the latter sequence is not monotonic.
See also comments in A293894.

Cf. A083722, A137799, A293892, A293894.

A083720 Product of the primes less than the greatest prime factor of n but not dividing n.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 30, 1, 2, 3, 210, 1, 2310, 15, 2, 1, 30030, 1, 510510, 3, 10, 105, 9699690, 1, 6, 1155, 2, 15, 223092870, 1, 6469693230, 1, 70, 15015, 6, 1, 200560490130, 255255, 770, 3, 7420738134810, 5, 304250263527210, 105, 2, 4849845

Offset: 1

Reinhard Zumkeller, May 04 2003

a(n) is squarefree, and all squarefree numbers appear infinitely often. a(m) = a(n) if and only if rad(m) = rad(n), where rad is A007947. - Charles R Greathouse IV, Apr 09 2024

Rad(n*a(n)) = A002110(A000720(A006530(n))) is the smallest primorial number divisible by rad(n). - David James Sycamore, May 15 2024

Michael De Vlieger, Table of n, a(n) for n = 1..2376

Cf. A079067, A083722.

See the formula section for the relationships with A000040, A002110, A006530, A007947, A049084.

Array[Times @@ Complement[Prime@ Range@ PrimePi@ Last[#], #] &[FactorInteger[#][[All, 1]]] &, 46] (* Michael De Vlieger, Apr 09 2024 *)

a(n) = A002110(A049084(A006530(n)))/A007947(n).
a(A000040(k)) = A002110(k-1).
a(n) = 1 iff n = m*A002110(k) and A006530(m) <= A000040(k).

Edited by Peter Munn, Apr 09 2024

A083721 Number of primes greater than the greatest prime factor of n but not greater than n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 3, 0, 2, 3, 5, 0, 5, 0, 5, 4, 3, 0, 7, 6, 3, 7, 5, 0, 7, 0, 10, 6, 4, 7, 9, 0, 4, 6, 9, 0, 9, 0, 9, 11, 5, 0, 13, 11, 12, 8, 9, 0, 14, 11, 12, 8, 6, 0, 14, 0, 7, 14, 17, 12, 13, 0, 12, 10, 15, 0, 18, 0, 9, 18, 13, 16, 15, 0, 19, 20, 9, 0, 19, 16, 9, 13, 18, 0, 21, 18

Offset: 1

Reinhard Zumkeller, May 04 2003

nonn

a(n) = A000720(n) - A000720(A006530(n));

a(n) > 0 if and only if n is composite.

Cf. A083722, A079067.

a(n)=if(n>3,my(f=factor(n)[,1]);primepi(n)-primepi(f[#f]),0) \\ Charles R Greathouse IV, Feb 21 2013

cp's OEIS Frontend

A293892 Restricted growth sequence transform of A083722.

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A293894 Numbers n such that A083722(n) > 1 and A083722(n) occurs earlier in A083722.

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A293893 Numbers n such that A083722(n) > 1 and A083722(n) occurs later in A083722.

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A083720 Product of the primes less than the greatest prime factor of n but not dividing n.

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A083721 Number of primes greater than the greatest prime factor of n but not greater than n.

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PARI