A300226 -id:A300226 - cp's OEIS frontend

A300248 Filter sequence combining A046523(n) and A078898(n), the prime signature of n and the number of times the smallest prime factor of n is the smallest prime factor for numbers <= n.

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

Offset: 1

Antti Karttunen, Mar 03 2018

nonn

Restricted growth sequence transform of P(A046523(n), A078898(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

			a(10) = a(65) (= 6) because A078898(10) = A078898(65) = 5 (both numbers occur in column 5 of A083221) and because both have the same prime-signature (both are nonsquare semiprimes).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046523, A078898.

Cf. also A300226, A300229, A300247.

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])); }
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From A046523
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); };
Aux300248(n) = if(1==n,0,(1/2)*(2 + ((A078898(n)+A046523(n))^2) - A078898(n) - 3*A046523(n)));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300248(n))),"b300248.txt");

A323234 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, and for n > 1, f(n) = ordered pair [A053645(n), A079944(n-2)], where A053645(n) gives n without its most significant bit, while A079944(n-2) gives the second most significant bit of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 08 2019

Also the restricted growth sequence transform of function f(1) = 0, f(n) = [A053645(n), A278222(n)] for n > 1.
For all i, j:
  a(i) = a(j) => A286622(i) = A286622(j),
  a(i) = a(j) => A323235(i) = A323235(j),
  a(i) = a(j) => A323236(i) = A323236(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A053645, A079944, A278222, A286622, A323235, A323236.

Cf. also A300226 (an analogous filter sequence for prime factorization).

up_to = 65537;
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; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A079944off0(n) = (1==binary(2+n)[2]);
A323234aux(n) = if(1==n,0,[A053645(n), A079944off0(n-2)]);
v323234 = rgs_transform(vector(up_to,n,A323234aux(n)));
A323234(n) = v323234[n];

A350068 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A350063(i) = A350063(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 4, 4, 8, 2, 7, 2, 7, 6, 9, 2, 10, 3, 9, 5, 11, 2, 12, 2, 13, 4, 9, 4, 14, 2, 4, 9, 15, 2, 16, 2, 7, 7, 17, 2, 18, 3, 19, 9, 7, 2, 10, 6, 10, 9, 20, 2, 21, 2, 9, 7, 22, 4, 12, 2, 11, 4, 16, 2, 23, 2, 9, 7, 11, 4, 12, 2, 18, 8, 9, 2, 24, 9, 25, 17, 26, 2, 24, 6, 11, 20, 27, 9

Offset: 1

Antti Karttunen, Jan 29 2022

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A350063(n)].

For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A046523, A156552, A322993, A350063.
Cf. A000040 (positions of 2's), A001248 (of 3's).
Cf. also A101296, A300226, A305897, A350065.

up_to = 3003;
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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A350063(n) = if(1==n,0,A046523(A000265(A156552(n))));
Aux350068(n) = [A046523(n),A350063(n)];
v350068 = rgs_transform(vector(up_to, n, Aux350068(n)));
A350068(n) = v350068[n];

A300246 Filter sequence combining A046523(n) and A078899(n), the prime signature of n and the number of times the greatest prime factor of n is the greatest prime factor for numbers <= n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 8, 9, 2, 10, 2, 7, 8, 4, 2, 11, 12, 4, 13, 7, 2, 14, 2, 15, 8, 4, 16, 17, 2, 4, 8, 18, 2, 14, 2, 7, 19, 4, 2, 20, 21, 22, 8, 7, 2, 23, 16, 24, 8, 4, 2, 25, 2, 4, 22, 26, 16, 14, 2, 7, 8, 27, 2, 28, 2, 4, 29, 7, 30, 14, 2, 31, 32, 4, 2, 33, 16, 4, 8, 24, 2, 34, 30, 7, 8, 4, 16, 35, 2, 36, 22, 37, 2, 14, 2, 24, 38

Offset: 1

Antti Karttunen, Mar 09 2018

nonn

Restricted growth sequence transform of P(A046523(n), A078899(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

			a(30) = a(42) (= 14) because A078899(30) = A078899(42) = 6 and both numbers are products of three distinct primes, thus have the same prime signature.
a(35) = a(55) = a(65) (= 16) because A078899(35) = A078899(55) = A078899(65) = 5 and because all three are nonsquare semiprimes.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046523, A078899.
Cf. also A300247, A300248.
Differs from A300226 for the first time at n=40, where a(40) = 18.

up_to = 65537;
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])); }
A006530(n) = if(1==n, n, vecmax(factor(n)[, 1]));
A078899(n) = { if(n<=1,n, my(gpf=A006530(n),k=1,m=n/gpf); while(m>1,if(A006530(m)<=gpf,k++); m--); (k)); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux300246(n) = if(1==n,0,(1/2)*(2 + ((A078899(n)+A046523(n))^2) - A078899(n) - 3*A046523(n)));
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300246(n))),"b300246.txt");

A339874 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A052126(n) for n > 1, and f(1) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 25 2020

nonn

For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A001222(i) = A001222(j),
  a(i) = a(j) => A322826(i) = A322826(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A052126, A322826.

Cf. also A305800, A300226, A334107, A334108.

up_to = 65537;
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; };
A052126(n) = if(1==n,n,(n/vecmax(factor(n)[, 1])));
Aux339874(n) = if(1==n,0,A052126(n));
v339874 = rgs_transform(vector(up_to, n, Aux339874(n)));
A339874(n) = v339874[n];

a(1) = 1; for n > 1, a(n) = 1 + A322826(n).

cp's OEIS Frontend

A300248 Filter sequence combining A046523(n) and A078898(n), the prime signature of n and the number of times the smallest prime factor of n is the smallest prime factor for numbers <= n.

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A323234 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, and for n > 1, f(n) = ordered pair [A053645(n), A079944(n-2)], where A053645(n) gives n without its most significant bit, while A079944(n-2) gives the second most significant bit of n.

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A350068 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A350063(i) = A350063(j), for all i, j >= 1.

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A300246 Filter sequence combining A046523(n) and A078899(n), the prime signature of n and the number of times the greatest prime factor of n is the greatest prime factor for numbers <= n.

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A339874 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A052126(n) for n > 1, and f(1) = 0.

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