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 1-3 of 3 results.

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.

A323405 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 13 2019

Keywords

Comments

For all i, j:
A305801(i) = A305801(j) => a(i) = a(j),
a(i) = a(j) => A323371(i) = A323371(j),
a(i) = a(j) => A247074(i) = A247074(j).

Links

Crossrefs

Cf. A000010, A003557, A023900, A063994, A247074, A305801, A323371, A323404.
Differs from A323370 for the first time at n=78, where a(78) = 58, while A323370(78) = 52.
Cf. also A323374.

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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
Aux323405(n) = if((n>2)&&isprime(n),0,[A003557(n), A023900(n), A063994(n)]);
v323405 = rgs_transform(vector(up_to, n, Aux323405(n)));
A323405(n) = v323405[n];

A323373 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A049559(i) = A049559(j) and A160595(i) = A160595(j).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 13 2019

Keywords

Comments

Restricted growth sequence transform of [A049559(n), A160595(n)].
For all i, j: a(i) = a(j) => A000010(i) = A000010(j).
For all i, j > 1:
a(i) = a(j) => A323374(i) = A323374(j).

Links

Crossrefs

Cf. A000010, A049559, A160595, A323374.
Cf. also A323404.

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; };
A049559(n) = gcd(eulerphi(n), n-1);
A160595(n) = if(1==n, n, numerator(eulerphi(n)/(n-1)));
v323373 = rgs_transform(vector(up_to, n, [A049559(n), A160595(n)]));
A323373(n) = v323373[n];
Showing 1-3 of 3 results.

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