A100417 -id:A100417 - cp's OEIS frontend

A100933 Complement of A100417.

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 50, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 98, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 143, 145

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2008

nonn

Cf. A100417, A141757, A141586, A141756, A141757.

A100762 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n and let P(n) = A100549(n); then a(n) = Product_{ q <= P(n) } q^e_q; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 9, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 2, 1, 24, 1, 2, 27, 4, 1, 2, 1, 32, 1, 2, 1, 36, 1, 2, 1, 8, 1, 2, 1, 4, 9, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 8, 1, 2, 1, 12, 1, 2, 9, 64, 1, 2, 1, 4, 1, 2, 1, 72, 1, 2, 3, 4, 1, 2, 1, 80, 81, 2, 1, 12, 1, 2, 1, 8, 1, 18, 1, 4, 1, 2, 1, 96, 1

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2008

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A100549, A100417, A141586, A082725.

# First load the procedure pp from A100549
# B = prod_{p <= pp(n)} p^e_p
B := proc(n) local v,f,pv; global pp; option remember;
pv := pp(n);
v := 1:
for f in op(2..-1,ifactors(n)) while f[1] <= pv do
v := v * f[1]^f[2];
end do;
return v;
end proc;

{1}~Join~Array[Function[{q, P}, Times @@ Power @@@ Select[q, First@# <= P &]] @@ {#, Prime@ PrimePi[1 + Max@ #[[All, -1]] ]} &@ FactorInteger[#] &, 96, 2] (* Michael De Vlieger, Nov 13 2018 *)

A100549(n) = if(1==n,1,prime(primepi(1+vecmax(factor(n)[,2]))));
A100762(n) = if(1==n,1,my(u = A100549(n), f=factor(n)); prod(i=1, #f~, if(f[i, 1]<=u, f[i, 1]^f[i, 2], 1))); \\ Antti Karttunen, Nov 11 2018

A100549 Let n = 2^e_2 * 3^e_ * 5^e_ * ... be the prime factorization of n; then a(n) = largest prime <= 1 + max{e_2, e_3, e_5, ...}; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 5, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 5, 5, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 5, 2, 3, 3, 3, 2, 2, 2, 3, 2

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2008

nonn

			If n = 8 = 2^3, a(n) = (largest prime <= 3+1) = 3.
If n = 480 = 2^5*3*5, a(n) = (largest prime <= 1 + max{5,1,1}) = 5.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A100762, A100417, A141586, A082725.

# if n = prod_p p^e_p, then
# pp = largest prime <= 1 + max e_p
with(numtheory):
pp := proc(n) local f,m; option remember;
if (n = 1) then
return 1;
end if;
m := 1:
for f in op(2..-1,ifactors(n)) do
if (f[2] > m) then
m := f[2]:
end if;
end do;
prevprime(m+2);
end proc;

{1}~Join~Array[Prime@PrimePi[1 + Max@FactorInteger[#][[All, -1]]] &, 105, 2] (* Michael De Vlieger, Nov 13 2018 *)

a(n) = if (n==1, 1, precprime(1 + vecmax(factor(n)[,2]~))); \\ Michel Marcus, Nov 14 2018

A141757 Even terms in A100933.

Original entry on oeis.org

50, 98, 150, 242, 250, 294, 338, 350, 490, 550, 578, 650, 686, 722, 726, 750, 850, 950, 1014, 1050, 1058, 1078, 1150, 1210, 1274, 1450, 1470, 1550, 1650, 1666, 1682, 1690, 1694, 1734, 1750, 1850, 1862, 1922, 1950, 2050, 2058, 2150, 2166, 2254, 2350, 2366

Offset: 1

David Applegate and N. J. A. Sloane, Sep 15 2008

nonn

with(numtheory):
# For A100549: if n = prod_p p^e_p, then pp = largest prime <= 1 + max e_p
pp := proc(n) local f,m; option remember;
if (n = 1) then
return 1;
end if;
m := 1:
for f in op(2..-1,ifactors(n)) do
if (f[2] > m) then
m := f[2]:
end if;
end do;
prevprime(m+2);
end proc;
# For A100762: B = prod_{p <= pp(n)} p^e_p
B := proc(n) local v,f,pv; global pp; option remember;
pv := pp(n);
v := 1:
for f in op(2..-1,ifactors(n)) while f[1] <= pv do
v := v * f[1]^f[2];
end do;
return v;
end proc;
# For A100417: Bgood = (is pp(n) = pp(B(n))), that is, is B(n) enough to establish pp(n)?
Bgood := proc(n) global pp;
`if`(pp(B(n))=pp(n),true,false);
end proc;
# For A100933 and A141757:
t0:=select(not Bgood, [$1..3000]);
t1:=[];
for n from 1 to nops(t0) do
if t0[n] mod 2 = 0 then t1:=[op(t1),t0[n]]; fi; od: t1;

cp's OEIS Frontend

A100933 Complement of A100417.

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A100762 Let n = 2^e_2 * 3^e_3 * 5^e_5 * ... be the prime factorization of n and let P(n) = A100549(n); then a(n) = Product_{ q <= P(n) } q^e_q; a(1) = 1 by convention.

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Maple

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A100549 Let n = 2^e_2 * 3^e_ * 5^e_ * ... be the prime factorization of n; then a(n) = largest prime <= 1 + max{e_2, e_3, e_5, ...}; a(1) = 1 by convention.

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Author

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Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A141757 Even terms in A100933.

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Author

Keywords

Programs

Maple