cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A289626 Restricted growth sequence transform of A289625, related to the structure of multiplicative group of integers modulo n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 5, 4, 3, 6, 5, 7, 4, 8, 8, 9, 4, 10, 8, 11, 6, 12, 13, 14, 7, 10, 11, 15, 8, 16, 17, 18, 9, 19, 11, 20, 10, 19, 21, 22, 11, 23, 18, 19, 12, 24, 21, 23, 14, 25, 19, 26, 10, 27, 28, 29, 15, 30, 21, 31, 16, 32, 25, 33, 18, 34, 25, 35, 19, 36, 28, 37, 20, 27, 29, 38, 19, 39, 40, 41, 22, 42, 28, 43, 23, 44, 45, 46, 19, 47, 35, 38, 24, 48, 49
Offset: 1

Views

Author

Antti Karttunen, Jul 18 2017

Keywords

Links

Crossrefs

Cf. A289625, A101296.
Cf. A000010, A002322, A034380, A046072, A289624 (some of the matching sequences).

Programs

  • PARI
    rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
write_to_bfile(1,rgs_transform(vector(16384,n,A289625(n))),"b289626_upto16384.txt");

A322592 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A289625(n) for any other number.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 2, 4, 5, 6, 2, 4, 2, 5, 7, 7, 2, 5, 2, 7, 8, 9, 2, 10, 11, 12, 13, 8, 2, 7, 2, 14, 15, 16, 17, 8, 2, 13, 17, 18, 2, 8, 2, 15, 17, 19, 2, 18, 20, 11, 21, 17, 2, 13, 22, 23, 24, 25, 2, 18, 2, 26, 27, 21, 28, 15, 2, 21, 29, 17, 2, 23, 2, 30, 22, 24, 31, 17, 2, 32, 33, 34, 2, 23, 35, 20, 36, 37, 2, 17, 38, 29, 31, 39, 40, 41, 2, 20, 31, 22, 2, 21, 2, 42, 42
Offset: 1

Views

Author

Antti Karttunen, Dec 18 2018

Keywords

Comments

For all i, j:
a(i) = a(j) => A034380(i) = A034380(j),
a(i) = a(j) => A104194(i) = A104194(j),
a(i) = a(j) => A290084(i) = A290084(j).

Links

Crossrefs

Cf. A289625, A289626, A322587, A322588, A322589, A322591.

Programs

  • PARI
    default(parisizemax,2^31);
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; };
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
Aux322592(n) = if((n>2)&&isprime(n),0,A289625(n));
v322592 = rgs_transform(vector(up_to, n, Aux322592(n)));
A322592(n) = v322592[n];
for(n=1,up_to,write("b322592.txt", n, " ", A322592(n)));

A290076 Restricted growth sequence transform of A289625(A005940(1+n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 5, 4, 3, 6, 5, 7, 4, 8, 6, 9, 4, 10, 6, 11, 6, 11, 12, 13, 7, 14, 10, 15, 8, 16, 17, 18, 9, 19, 10, 14, 10, 20, 21, 22, 11, 23, 21, 24, 11, 25, 21, 26, 13, 27, 14, 28, 14, 24, 29, 30, 15, 31, 32, 33, 16, 34, 35, 36, 18, 11, 19, 37, 19, 22, 29, 38, 14, 39, 29, 40, 20, 41, 42, 24, 22, 43, 23, 44, 23, 45, 46, 47, 24, 44, 23, 48, 25
Offset: 0

Views

Author

Antti Karttunen, Jul 19 2017

Keywords

Comments

For all i, j: a(i) = a(j) => A290077(i) = A290077(j).

Links

Crossrefs

Cf. A005940, A289626, A290077, A290082.

Programs

  • PARI
    allocatemem(2^31);
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
write_to_bfile(0,rgs_transform(vector(8193,n,A289625(A005940((1+n)-1)))),"b290076_upto8192.txt");

A296080 Restricted growth sequence transform of A289625(1+phi(n)), where phi = A000010, Euler totient function.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 05 2017

Keywords

Links

Crossrefs

Cf. A000010, A039649, A289625, A289626, A296078.
Cf. A039650.

Programs

  • PARI
    allocatemem(2^30);
up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A289625(1+eulerphi(n)))),"b296080.txt");

A303755 Ordinal transform of A289625.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 2, 2, 1, 4, 1, 1, 1, 2, 1, 3, 1, 3, 2, 1, 1, 4, 1, 2, 3, 2, 1, 2, 2, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 5, 1, 2, 1, 2, 2, 2, 1, 6, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 7, 1, 2, 2, 2, 1, 1, 1, 4, 3, 3, 1, 4, 1, 1, 2
Offset: 1

Views

Author

Antti Karttunen, Apr 30 2018

Keywords

Comments

Equally, ordinal transform of A289626.

Links

Crossrefs

Cf. A289625, A289626.
Cf. also A081373, A303756.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
v303755 = ordinal_transform(vector(up_to,n,A289625(n)));
A303755(n) = v303755[n];

A290082 Restricted growth sequence transform of A289625(A003961(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 20 2017

Keywords

Comments

For all i, j: a(i) = a(j) => A003972(i) = A003972(j).

Links

Crossrefs

Cf. A003961, A003972, A289625, A290076.

Programs

  • PARI
    rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ This function from Michel Marcus
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
write_to_bfile(1,rgs_transform(vector(16384,n,A289625(A003961(n)))),"b290082_upto16384.txt");

A318892 Filter sequence combining the prime signature of n (A046523) with A289625.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 35, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 44, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 68, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 75, 90
Offset: 1

Views

Author

Antti Karttunen, Sep 16 2018

Keywords

Comments

Restricted growth sequence transform of A289628.
For all i, j: a(i) = a(j) => A318893(i) = A318893(j).

Links

Crossrefs

Cf. A046523, A289625, A289628, A318839, A318893.

Programs

  • PARI
    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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
A318892aux(n) = [A046523(n), A289625(n)];
v318892 = rgs_transform(vector(up_to,n,A318892aux(n)));
A318892(n) = v318892[n];

A329895 Lexicographically earliest infinite sequence such that a(i) = a(j) => A219175(i) = A219175(j) and A289625(i) = A289625(j) for all i, j.

Original entry on oeis.org

1, 1, 2, 3, 4, 3, 5, 6, 7, 8, 9, 6, 10, 11, 12, 13, 14, 15, 16, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 32, 38, 39, 40, 41, 42, 36, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 36, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 49, 68, 69, 70, 71, 72, 73, 74, 75, 65, 76, 77, 78, 79, 80, 70, 81, 82, 83
Offset: 1

Views

Author

Antti Karttunen, Dec 07 2019

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A219175(n), A289625(n)].
For all i, j:
a(i) = a(j) => A327979(i) = A327979(j).

Links

Crossrefs

Cf. A002322, A219175, A289625, A327979, A329896.

Programs

  • PARI
    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; };
A219175(n) = (n%lcm(znstar(n)[2]));
A289625(n) = { my(m=1,p=2,v=znstar(n)[2]); for(i=1,length(v),m *= p^v[i]; p = nextprime(p+1)); (m); };
Aux329895(n) = [A219175(n),A289625(n)];
v329895 = rgs_transform(vector(up_to, n, Aux329895(n)));
A329895(n) = v329895[n];

A072411 LCM of exponents in prime factorization of n, a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 3
Offset: 1

Views

Author

Labos Elemer, Jun 17 2002

Keywords

Comments

The sums of the first 10^k terms, for k = 1, 2, ..., are 14, 168, 1779, 17959, 180665, 1808044, 18084622, 180856637, 1808585068, 18085891506, ... . Apparently, the asymptotic mean of this sequence is limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1.8085... . - Amiram Eldar, Sep 10 2022

Examples

			n = 288 = 2*2*2*2*2*3*3; lcm(5,2) = 10; Product(5,2) = 10, max(5,2) = 5;
n = 180 = 2*2*3*3*5; lcm(2,2,1) = 2; Product(2,2,1) = 4; max(2,2,1) = 2; it deviates both from maximum of exponents (A051903, for the first time at n=72), and product of exponents (A005361, for the first time at n=36).
For n = 36 = 2*2*3*3 = 2^2 * 3^2 we have a(36) = lcm(2,2) = 2.
For n = 72 = 2*2*2*3*3 = 2^3 * 3^2 we have a(72) = lcm(2,3) = 6.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(2,4) = 4.
For n = 360 = 2^3 * 3^2 * 5^1 we have a(360) = lcm(1,2,3) = 6.

Links

Crossrefs

Cf. A028234, A067029, A072412-A072414, A273058, A284569, A290103.
Similar sequences: A001222 (sum of exponents), A005361 (product), A051903 (maximal exponent), A051904 (minimal exponent), A052409 (gcd of exponents), A267115 (bitwise-and), A267116 (bitwise-or), A268387 (bitwise-xor).
Cf. also A055092, A060131.
Differs from A290107 for the first time at n=144.
After the initial term, differs from A157754 for the first time at n=360.

Programs

  • Mathematica
    Table[LCM @@ Last /@ FactorInteger[n], {n, 2, 100}] (* Ray Chandler, Jan 24 2006 *)
  • PARI
    a(n) = lcm(factor(n)[,2]); \\ Michel Marcus, Mar 25 2017
  • Python
    from sympy import lcm, factorint
def a(n):
    l=[]
    f=factorint(n)
    for i in f: l+=[f[i],]
    return lcm(l)
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 25 2017

Formula

a(1) = 1; for n > 1, a(n) = lcm(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 09 2016
From Antti Karttunen, Aug 22 2017: (Start)
a(n) = A284569(A156552(n)).
a(n) = A290103(A181819(n)).
a(A289625(n)) = A002322(n).
a(A290095(n)) = A055092(n).
a(A275725(n)) = A060131(n).
a(A260443(n)) = A277326(n).
a(A283477(n)) = A284002(n). (End)

Extensions

a(1) = 1 prepended and the data section filled up to 120 terms by Antti Karttunen, Aug 09 2016

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 26 2018

Keywords

Comments

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
A305801(i) = A305801(j) => a(i) = a(j),
a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

Links

Crossrefs

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
A322809 (A004526, unless an odd prime) [This sequence],
A322589 (A007913, unless an odd prime),
A322591 (A007947, unless an odd prime),
A322805 (A252463, unless an odd prime),
A323082 (A300840, unless an odd prime),
A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
A322988 (A322990, unless a prime power > 2),
A323078 (A097246, unless an odd prime),
A322808 (A097246, unless a squarefree number > 2),
A322816 (A048675, unless an odd prime),
A322807 (A285330, unless an odd prime),
A322814 (A286621, unless an odd prime),
A322824 (A242424, unless an odd prime),
A322973 (A006370, unless an odd prime),
A322974 (A049820, unless n > 1 and n is in A046642),
A323079 (A060681, unless an odd prime),
A322587 (A295887, unless an odd prime),
A322588 (A291751, unless an odd prime),
A322592 (A289625, unless an odd prime),
A323369 (A323368, unless an odd prime),
A323371 (A295886, unless an odd prime),
A323374 (A323373, unless an odd prime),
A323401 (A323372, unless an odd prime),
A323405 (A323404, unless an odd prime).

Programs

  • PARI
    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; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

Formula

a(n) = A323161(n+1) - 1.
Showing 1-10 of 14 results. Next

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