A373247 -id:A373247 - cp's OEIS frontend

A373980 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A181819(n), A373247(n)], for all i, j >= 1.

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, 29, 30, 3, 31, 3, 32, 33, 34, 3, 35, 36, 37, 38, 39, 3, 40, 29, 41, 42, 43, 3, 44, 3, 45, 46, 47, 48, 49, 3, 50, 51, 52, 3, 53, 3, 54, 55, 56, 48, 57, 3, 58, 59, 60, 3, 61, 42, 62, 63, 64, 3, 65, 38, 66, 67, 68, 69, 70, 3, 71, 72

Offset: 1

Antti Karttunen, Jun 24 2024

nonn

Restricted growth sequence transform of the quadruple [A003415(n), A085731(n), A181819(n), A373247(n)].
For all i, j >= 1:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A300249(i) = A300249(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A369051(i) = A369051(j),
  a(i) = a(j) => A373250(i) = A373250(j).

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

Cf. A003415, A085731, A181819, A373247.
Cf. A101296, A300249, A305801, A369051, A373250.
Differs from A353520 and A361021 first at n=130, where a(130) = 82, while A353520(130) = A361021(130) = 96.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
Aux373980(n) = { my(d=A003415(n), s=A181819(n)); [d, s, gcd(n,d), n%s]; };
v373980 = rgs_transform(vector(up_to, n, Aux373980(n)));
A373980(n) = v373980[n];

A373981 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A107463(n), A181819(n), A373247(n)], for all i, j >= 1.

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, 29, 30, 3, 31, 3, 32, 33, 34, 3, 35, 36, 37, 38, 39, 3, 40, 29, 41, 42, 43, 3, 44, 3, 45, 46, 47, 48, 49, 3, 50, 51, 52, 3, 53, 3, 54, 46, 55, 48, 56, 3, 57, 58, 59, 3, 60, 42, 61, 62, 63, 3, 64, 38, 65, 66, 67, 68, 69, 3, 70, 71

Offset: 1

Antti Karttunen, Jun 24 2024

nonn

Restricted growth sequence transform of the triple [A107463(n), A181819(n), A373247(n)].
For all i, j >= 1:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A373250(i) = A373250(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A373976(i) = A373976(j).

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

Cf. A107463, A181819, A373247.
Cf. A101296, A305801, A373250, A373976.
Cf. also A373379, A373980.

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; };
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
A107463(n) = if(n<=1,n,if(isprime(n),1,A001414(n)));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
Aux373981(n) = { my(s=A181819(n)); [A107463(n), s, n%s]; };
v373981 = rgs_transform(vector(up_to, n, Aux373981(n)));
A373981(n) = v373981[n];

A373246 Greatest common divisor of n and its prime shadow: a(n) = gcd(n, A181819(n)).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 6, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 9, 1, 2, 1, 10, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 1

Offset: 1

Antti Karttunen, May 29 2024

nonn

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

Cf. A181819, A373247, A373249.

Cf. A325755 [numbers k for which a(k) = A181819(k)], A353566 (their characteristic function).

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A373246(n) = gcd(n, A181819(n));

a(n) = gcd(A181819(n), A373247(n)).

a(n) = A181819(n) / A373249(n).

A373250 Lexicographically earliest infinite sequence such that a(i) = a(j) => A181819(i) = A181819(j) and i mod A181819(i) = j mod A181819(j), for all i, j >= 1, where A181819 is the prime shadow of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 30 2024

nonn

Restricted growth sequence transform of the ordered pair [A181819(n), A373247(n)].
For all i, j:
  A373251(i) = A373251(j) => a(i) = a(j),
  a(i) = a(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A373246(i) = A373246(j),
  a(i) = a(j) => A373249(i) = A373249(j),
  a(i) = a(j) => A353566(i) = A353566(j).

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

Cf. A101296, A181819, A353566, A373246, A373247, A373249, A373251.

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; };
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
Aux373250(n) = [A181819(n), n%A181819(n)];
v373250 = rgs_transform(vector(up_to, n, Aux373250(n)));
A373250(n) = v373250[n];

A373251 Lexicographically earliest infinite sequence such that a(i) = a(j) => A181819(i) = A181819(j), i mod A181819(i) = j mod A181819(j), and gcd(i,A276086(i)) = gcd(j,A276086(j)), for all i, j >= 1, where A181819 is the prime shadow of n, and A276086 is the primorial base exp-function.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 30 2024

nonn

Restricted growth sequence transform of the triple [A181819(n), A373247(n), A324198(n)].
For all i, j:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A373248(i) = A373248(j),
  a(i) = a(j) => A373250(i) = A373250(j).

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

Cf. A101296, A181819, A324198, A373247, A373248, A373250.

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; };
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
Aux373251(n) = [A181819(n), n%A181819(n), A324198(n)];
v373251 = rgs_transform(vector(up_to, n, Aux373251(n)));
A373251(n) = v373251[n];

cp's OEIS Frontend

A373980 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A181819(n), A373247(n)], for all i, j >= 1.

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A373981 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A107463(n), A181819(n), A373247(n)], for all i, j >= 1.

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A373246 Greatest common divisor of n and its prime shadow: a(n) = gcd(n, A181819(n)).

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A373250 Lexicographically earliest infinite sequence such that a(i) = a(j) => A181819(i) = A181819(j) and i mod A181819(i) = j mod A181819(j), for all i, j >= 1, where A181819 is the prime shadow of n.

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A373251 Lexicographically earliest infinite sequence such that a(i) = a(j) => A181819(i) = A181819(j), i mod A181819(i) = j mod A181819(j), and gcd(i,A276086(i)) = gcd(j,A276086(j)), for all i, j >= 1, where A181819 is the prime shadow of n, and A276086 is the primorial base exp-function.

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