A294877 -id:A294877 - cp's OEIS frontend

A291751 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

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, 32, 33, 34, 22, 35, 36, 37, 38, 26, 28, 39, 40, 41, 26, 42, 29, 43, 26, 44, 45, 46, 32, 47, 48, 35, 49, 50, 51, 52, 53, 54, 35, 52, 26, 55, 56, 57, 58, 59, 35, 60, 45, 61, 62, 63, 51, 64, 65, 66, 67, 68, 46, 69, 70, 47, 71

Offset: 1

Antti Karttunen, Sep 06 2017

nonn

Restricted growth sequence transform of A291750, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291750(i) = A291750(j) <=> A003557(i) = A003557(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.

Sigma (A000203) and psi (A001615) are functions of this sequence. See comments in A291750 for the reason. For example, to find the value of A001615(n) when we know just a(n), but without knowing n, let m be the least i for which a(i) = a(n); then A001615(n) = A003991(A291750(m)) = A003557(m) * A048250(m).

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

Cf. A001615, A003557, A048250, A291750, A291752, A294877, A295300, A295886, A295887, A295888, A319698, A322021.

Differs from A286603 for the first time at n = 25, where a(25) = 21, while A286603(25) = 14.

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]); \\ From A003557
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
v291751 = rgs_transform(vector(65537,n,A291750(n)));
A291751(n) = v291751[n];

Name changed and comments added by Antti Karttunen, Nov 24 2018

A295300 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)].

Original entry on oeis.org

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, 65, 66, 67, 68, 69, 70, 58, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 80

Offset: 1

Antti Karttunen, Nov 19 2017

nonn

Restricted growth sequence transform of A291752.
For all i, j:
  a(i) = a(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A326199(i) = A326199(j) => A294877(i) = A294877(j),
  a(i) = a(j) => A322021(i) = A322021(j),
  a(i) = a(j) => A295888(i) = A295888(j),
  a(i) = a(j) => A296090(i) = A296090(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A003557, A046523, A048250, A101296, A286360, A291750, A291751, A291752, A291757, A291758, A295888, A296090, A322021, A326199.

Cf. also A305801, A305900, A323368, A324400.

up_to = 100000;
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]); \\ From A003557
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)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux295300(n) = (1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n));
v295300 = rgs_transform(vector(up_to,n,Aux295300(n)));
A295300(n) = v295300[n];

Name changed and the comments section added by Antti Karttunen, Jul 13 2019

A291757 a(n) = (1/2)(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3A046523(n)).

Original entry on oeis.org

1, 2, 2, 12, 2, 16, 2, 59, 18, 16, 2, 80, 2, 16, 16, 261, 2, 94, 2, 80, 16, 16, 2, 355, 33, 16, 129, 80, 2, 436, 2, 1097, 16, 16, 16, 826, 2, 16, 16, 355, 2, 436, 2, 80, 94, 16, 2, 1493, 52, 125, 16, 80, 2, 505, 16, 355, 16, 16, 2, 1832, 2, 16, 94, 4497, 16, 436, 2, 80, 16, 436, 2, 3415, 2, 16, 125, 80, 16, 436, 2, 1493, 888, 16, 2, 1832, 16, 16, 16, 355, 2

Offset: 1

Antti Karttunen, Sep 10 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A000027, A003557, A046523, A291750, A291752, A291756, A291758, A294877 (rgs-transform), A319347.

A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A291757(n) = (1/2)*(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3*A046523(n));

a(n) = (1/2)*(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3*A046523(n)).

Name changed by Antti Karttunen, Nov 28 2018

A294876 a(n) = Product_{d|n, d>1} prime(gcd(d,n/d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 18, 10, 8, 2, 72, 2, 8, 8, 126, 2, 200, 2, 72, 8, 8, 2, 648, 22, 8, 50, 72, 2, 128, 2, 882, 8, 8, 8, 23400, 2, 8, 8, 648, 2, 128, 2, 72, 200, 8, 2, 31752, 34, 968, 8, 72, 2, 5000, 8, 648, 8, 8, 2, 10368, 2, 8, 200, 16758, 8, 128, 2, 72, 8, 128, 2, 2737800, 2, 8, 968, 72, 8, 128, 2, 31752, 1150, 8, 2, 10368, 8, 8, 8, 648, 2, 80000, 8, 72

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A294877 (rgs-version of this filter).
Cf. A000040, A000188, A034444, A055155.
Cf. also A293442, A293514, A293524.

A294876[n_] := Product[Prime[GCD[d, n/d]], {d, Rest[Divisors[n]]}];
Array[A294876, 100] (* Paolo Xausa, Feb 22 2024 *)

A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };

a(n) = Product_{d|n, d>1} A000040(gcd(d,n/d)).
Other identities. For all n >= 1:
1+A007814(a(n)) = A034444(n).
1+A056239(a(n)) = A055155(n).
For n > 1, A061395(a(n)) = A000188(n).

A351260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 06 2022

Restricted growth sequence transform of the triplet [A003415(n), A003557(n), A046523(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A294877(i) = A294877(j),
  a(i) = a(j) => A300249(i) = A300249(j),
  a(i) = a(j) => A344025(i) = A344025(j).

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

Cf. A003415, A003557, A046523.
Cf. also A305800, A294877, A300249, A344025.
Differs from A300235, A305895 and A327931 for the first time at n=105, where a(105) = 56, while A300235(105) = A305895(105) = A327931(105) = 75.
Differs from A300249 for the first time at n=425, where a(425) = 299, while A300249(425) = 198.

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; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A003557(n) = (n/factorback(factorint(n)[, 1]));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux351260(n) = [A003415(n), A003557(n), A046523(n)];
v351260 = rgs_transform(vector(up_to,n,Aux351260(n)));
A351260(n) = v351260[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];

A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(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, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 19, 20, 3, 21, 3, 22, 23, 24, 23, 25, 3, 26, 27, 28, 3, 29, 3, 30, 31, 32, 3, 33, 34, 35, 32, 36, 3, 37, 32, 38, 39, 40, 3, 41, 3, 42, 43, 44, 45, 46, 3, 47, 42, 46, 3, 48, 3, 49, 50, 51, 42, 52, 3, 53, 54, 55, 3, 56, 57, 58, 59, 60, 3, 61, 62, 63, 64, 65, 59, 66, 3, 67, 68, 69, 3, 70, 3

Offset: 1

Antti Karttunen, Jul 13 2019

nonn

For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A294877(i) = A294877(j).

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

Cf. A003557, A046523, A048250, A294877, A295300, A305801.

Differs from A323401 for the first time at n = 382 where a(382) = 253, while A323401(382) = 140.

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]); \\ From A003557
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)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux326199(n) = if((n>2)&&isprime(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
v326199 = rgs_transform(vector(up_to,n,Aux326199(n)));
A326199(n) = v326199[n];

cp's OEIS Frontend

A291751 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

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A295300 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)].

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A291757 a(n) = (1/2)*(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3*A046523(n)).

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A294876 a(n) = Product_{d|n, d>1} prime(gcd(d,n/d)).

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A351260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j >= 1.

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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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A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(n) = 0 for odd primes.

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A291757 a(n) = (1/2)(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3A046523(n)).