A291751 -id:A291751 - cp's OEIS frontend

A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

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

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Restricted growth sequence transform of ordered pair [A003557(n), A323363(n)].
For all i, j:
  a(i) = a(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A323364(i) = A323364(j).

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

Cf. A001615, A003557, A291750, A291751, A323363, A323364.

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; };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
v323363 = DirInverse(vector(up_to,n,A001615(n)));
A323363(n) = v323363[n];
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
v323372 = rgs_transform(vector(up_to, n, [A003557(n), A323363(n)]));
A323372(n) = v323372[n];

A319698 Filter sequence combining A003557(n) [n divided by largest squarefree divisor of n] with A319697(n) [sum of even squarefree divisors of n].

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 31 2018

nonn

Restricted growth sequence transform of ordered pair [A003557(n), A319697(n)].

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

Cf. A003557, A319697.

Cf also A291750, A291751.

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] = f[i, 2]-1); factorback(f); }; \\ From A003557
A319697(n) = sumdiv(n, d, (!(d%2))*issquarefree(d)*d);
v319698 = rgs_transform(vector(up_to,n,[A003557(n),A319697(n)]));
A319698(n) = v319698[n];

A322021 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(i) = A046523(j) and A048250(i) = A048250(j), for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 29 2018

nonn

Restricted growth sequence transform of A291758, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291758(i) = A291758(j) <=> A046523(i) = A046523(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.
For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  a(i) = a(j) => A304411(i) = A304411(j),
  a(i) = a(j) => A304412(i) = A304412(j).

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

Cf. A046523, A048250, A291751, A291758, A294877, A295300, A322022.

Cf. also A304411, A304412.

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
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
v322021 = rgs_transform(vector(up_to, n, [A046523(n), A048250(n)]));
A322021(n) = v322021[n];

A323238 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291750(n) for all n, except for odd numbers n > 1, f(n) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

For all i, j:
  A319701(i) = A319701(j) => a(i) = a(j),
  a(i) = a(j) => A146076(i) = A146076(j),
  a(i) = a(j) => A319697(i) = A319697(j).

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

Cf. A003557, A048250, A146076, A291750, A291751, A319698, A319697, A319701, A322588, A323237, A323241.

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] = f[i, 2]-1); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
Aux323238(n) = if((n>1)&&(n%2),0,(1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n)));
v323238 = rgs_transform(vector(up_to, n, Aux323238(n)));
A323238(n) = v323238[n];

A369260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A349162(i) = A349162(j), for all i, j >= 1.

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, 15, 23, 16, 24, 25, 26, 27, 28, 21, 29, 30, 31, 30, 32, 33, 34, 26, 35, 36, 37, 38, 39, 40, 28, 30, 41, 42, 43, 44, 45, 46, 47, 44, 48, 49, 50, 51, 52, 53, 37, 54, 55, 56, 57, 58, 59, 60, 57, 44, 61, 62, 63, 41, 64, 60, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 65, 79, 57

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the ordered pair [A342671(n), A349162(n)], or equally, of the pair [A000203(n), A342671(n)], or equally, of the pair [A000203(n), A349162(n)].
For all i, j >= 1:
  A369259(i) = A369259(j) => a(i) = a(j) => A286603(i) = A286603(j).

Cf. A000203, A003961, A342671, A348993, A349162.

Cf. also A286603, A291751, A369259, A369261.

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; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
Aux369260(n) = { my(u=A342671(n)); [u, sigma(n)/u]; };
v369260 = rgs_transform(vector(up_to, n, Aux369260(n)));
A369260(n) = v369260[n];

A325381 Lexicographically earliest sequence such that a(i) = a(j) => A048250(i) = A048250(j) and A126795(i) = A126795(j) for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 08 2019

Restricted growth sequence transform of the ordered pair [A048250(n), A126795(n)].

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

Cf. A048250, A126795, A291751, A325382, A325383.

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; };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A126795(n) = gcd(n,A048250(n));
v325381 = rgs_transform(vector(up_to,n,[A048250(n),A126795(n)]));
A325381(n) = v325381[n];

A351441 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j) and A351450(i) = A351450(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 12 2022

nonn

Restricted growth sequence transform of the ordered pair [A003557(n), A351450(n)].
For all i, j >= 1:
   a(i) = a(j) => A326042(i) = A326042(j),
   a(i) = a(j) => A351455(i) = A351455(j).

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

Cf. A003557, A003961, A048250, A064989, A326042, A351450, A351455.

Cf. also A291751, A351461.

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) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A351450(n) = A064989(A048250(A003961(n)));
Aux351441(n) = [A003557(n), A351450(n)];
v351441 = rgs_transform(vector(up_to, n, Aux351441(n)));
A351441(n) = v351441[n];

A369261 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324644(i) = A324644(j) and A369445(i) = A369445(j), for all i, j >= 1.

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, 15, 23, 16, 20, 24, 25, 26, 27, 21, 28, 29, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 29, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 35, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 62, 53, 51, 70, 71, 72, 58, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 63

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the ordered pair [A324644(n), A369445(n)], or equally, of the pair [A000203(n), A324644(n)], or equally, of the pair [A000203(n), A369445(n)].

For all i, j >= 1: a(i) = a(j) => A286603(i) = A286603(j).

Cf. A000203, A276086, A324644, A369445.

Cf. also A286603, A291751, A369260 and A353523, A369047, A369051, A369062.

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; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324644(n) = gcd(sigma(n),A276086(n));
Aux369261(n) = { my(u=A324644(n)); [u, sigma(n)/u]; };
v369261 = rgs_transform(vector(up_to, n, Aux369261(n)));
A369261(n) = v369261[n];

A369259 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j), A048250(i) = A048250(j) and A342671(i) = A342671(j), for all i, j >= 1.

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, 15, 23, 24, 25, 26, 27, 28, 29, 21, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 29, 31, 43, 44, 45, 46, 47, 48, 49, 46, 50, 51, 52, 53, 54, 55, 39, 56, 57, 58, 59, 60, 61, 62, 59, 46, 63, 64, 65, 66, 67, 62, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 59

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the triplet [A003557(j), A048250(i), A342671(n)].
For all i, j >= 1:
  a(i) = a(j) => A323368(i) = A323368(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A369260(i) = A369260(j) => A286603(i) = A286603(j).

Cf. A000203, A001615, A003557, A003961, A048250, A286603, A291751, A369260.

Differs from related A296089 and A323368 for the first time at n=79, where a(79) = 69, while A296089(79) = A323368(79) = 51.

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) = (n/factorback(factor(n)[, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A342671(n) = gcd(sigma(n), A003961(n));
Aux369259(n) = [A003557(n), A048250(n), A342671(n)];
v369259 = rgs_transform(vector(up_to, n, Aux369259(n)));
A369259(n) = v369259[n];

A374485 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350388(i) = A350388(j) and A351569(i) = A351569(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 06 2024

nonn

Restricted growth sequence transform of the ordered pair [A350388(n), A351569(n)].

For all i, j >= 1: a(i) = a(j) => A000203(i) = A000203(j).

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

Cf. A000203, A350388, A350389, A351569.

Cf. also A286603, A291751.

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; };
A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351569(n) = sigma(A350389(n));
Aux374485(n) = [A350388(n), A351569(n)];
v374485 = rgs_transform(vector(up_to, n, Aux374485(n)));
A374485(n) = v374485[n];

cp's OEIS Frontend

A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

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A319698 Filter sequence combining A003557(n) [n divided by largest squarefree divisor of n] with A319697(n) [sum of even squarefree divisors of n].

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A322021 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(i) = A046523(j) and A048250(i) = A048250(j), for all i, j.

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A323238 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291750(n) for all n, except for odd numbers n > 1, f(n) = 0.

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A369260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A349162(i) = A349162(j), for all i, j >= 1.

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A325381 Lexicographically earliest sequence such that a(i) = a(j) => A048250(i) = A048250(j) and A126795(i) = A126795(j) for all i, j.

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A351441 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j) and A351450(i) = A351450(j) for all i, j >= 1.

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A369261 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324644(i) = A324644(j) and A369445(i) = A369445(j), for all i, j >= 1.

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A369259 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j), A048250(i) = A048250(j) and A342671(i) = A342671(j), for all i, j >= 1.

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