A373246 -id:A373246 - cp's OEIS frontend

A373247 a(n) = n mod A181819(n), where A181819(n) is the prime shadow of n.

0, 0, 1, 1, 1, 2, 1, 3, 0, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 2, 1, 4, 1, 2, 2, 4, 1, 6, 1, 10, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 3, 4, 1, 4, 3, 6, 1, 2, 1, 0, 1, 2, 3, 12, 1, 2, 1, 2, 1, 6, 1, 12, 1, 2, 3, 4, 1, 6, 1, 10, 4, 2, 1, 0, 1, 2, 3, 8, 1, 6, 3, 2, 1, 2, 3, 8, 1, 2, 3, 1, 1, 6, 1, 4, 1

Offset: 1

Antti Karttunen, May 29 2024

nonn

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

Cf. A181819, A373246.

Cf. A325755 (positions of 0's), A353566 (their characteristic function).

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A373247(n) = (n%A181819(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];

A373249 a(n) = A181819(n) / gcd(n, A181819(n)), where A181819(n) is the prime shadow of n.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 2, 5, 1, 2, 2, 1, 2, 2, 4, 7, 2, 1, 2, 3, 4, 2, 2, 5, 3, 2, 5, 3, 2, 4, 2, 11, 4, 2, 4, 1, 2, 2, 4, 1, 2, 4, 2, 3, 2, 2, 2, 7, 3, 3, 4, 3, 2, 5, 4, 5, 4, 2, 2, 1, 2, 2, 2, 13, 4, 4, 2, 3, 4, 4, 2, 5, 2, 2, 2, 3, 4, 4, 2, 7, 7, 2, 2, 1, 4, 2, 4, 5, 2, 2, 4, 3, 4, 2, 4, 11, 2, 3, 2, 9, 2, 4, 2, 5, 8

Offset: 1

Antti Karttunen, May 30 2024

nonn

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

Cf. A181819, A373246.

Cf. A325755 (positions of 1's), A353566 (their characteristic function).

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A373249(n) = { my(s=A181819(n)); s/gcd(n, s); };

a(n) = A181819(n) / A373246(n) = A181819(n) / gcd(n, A181819(n)).

A373248 Lexicographically earliest infinite sequence such that a(i) = a(j) => gcd(i,A181819(i)) = gcd(j,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, 1, 1, 2, 1, 1, 4, 5, 1, 6, 1, 2, 7, 1, 1, 6, 1, 5, 3, 2, 1, 2, 8, 2, 3, 2, 1, 2, 1, 1, 3, 2, 9, 10, 1, 2, 3, 11, 1, 12, 1, 2, 13, 2, 1, 2, 9, 14, 3, 2, 1, 2, 15, 12, 3, 2, 1, 16, 1, 2, 17, 1, 1, 2, 1, 2, 3, 18, 1, 19, 1, 2, 20, 2, 9, 2, 1, 5, 3, 2, 1, 21, 15, 2, 3, 2, 1, 6, 9, 2, 3, 2, 1, 2, 1, 22, 4, 15

Offset: 1

Antti Karttunen, May 29 2024

nonn

Restricted growth sequence transform of the ordered pair [A324198(n), A373246(n)].

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

Cf. A181819, A276086, A324198, A373246.

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; };
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); };
Aux373248(n) = [gcd(n, A181819(n)), A324198(n)];
v373248 = rgs_transform(vector(up_to, n, Aux373248(n)));
A373248(n) = v373248[n];

cp's OEIS Frontend

A373247 a(n) = n mod A181819(n), where A181819(n) is the prime shadow of 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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A373249 a(n) = A181819(n) / gcd(n, A181819(n)), where A181819(n) is the prime shadow of n.

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A373248 Lexicographically earliest infinite sequence such that a(i) = a(j) => gcd(i,A181819(i)) = gcd(j,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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Author

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Comments

Links

Crossrefs

Programs

PARI