A305900 -id:A305900 - cp's OEIS frontend

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.

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];

A373594 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n<=3) = n, f(p) = 0 for primes p > 3, and for composite n, f(n) = [A007814(n), A065339(n), A083025(n), A373591(n), A373592(n)].

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 13 2024

nonn

Restricted growth sequence transform of the function f given in the definition.
Note that for composite n, f(n) can be defined in general as a quintuple vector [v(n), w(n), x(n), y(n), z(n)], where v, w, x, y and z are any five of these six sequences: A007814, A007949, A065339, A083025, A373591, A373592. This follows because A007814(n) + A065339(n) + A083025(n) = A007949(n) + A373591(n) + A373592(n) = A001222(n), so the omitted sixth element can be always worked out from the remaining five.
For all i, j > 1:
  A305900(i) = A305900(j) => a(i) = a(j) => A373595(i) = A373595(j).

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

Cf. A007814, A007949, A065339, A083025, A373591, A373592.

Cf. also A305900, A373595.

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; };
A007814(n) = valuation(n,2);
A065339(n) = sum(i=1, #n=factor(n)~, (3==n[1, i]%4)*n[2, i]);
A083025(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%4)*n[2, i]);
A373591(n) = sum(i=1, #n=factor(n)~, (1==n[1, i]%3)*n[2, i]);
A373592(n) = sum(i=1, #n=factor(n)~, (2==n[1, i]%3)*n[2, i]);
Aux373594(n) = if(n<=3, n, if(isprime(n), 0, [A007814(n), A083025(n), A065339(n), A373591(n), A373592(n)]));
v373594 = rgs_transform(vector(up_to, n, Aux373594(n)));
A373594(n) = v373594[n];

A374040 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A007814(n), A007949(n)], for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 01 2024

nonn

Restricted growth sequence transform of the quadruple [A003415(n), A085731(n), A007814(n), A007949(n)].
For all i, j >= 1:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A322026(i) = A322026(j),
  a(i) = a(j) => A369051(i) = A369051(j) => A083345(i) = A083345(j),
  a(i) = a(j) => b(i) = b(j), where b can be any of the sequences listed at the crossrefs-section, under "some of the other matched sequences".

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

Cf. A003415, A007814, A007949, A085731.
Some of the other matched sequences (see comments): A083345, A359430, A369001, A369004, A369643, A369658, A373143, A373474, A373483.
Cf. also A322026, A353521, A369051, A373268, A372573, A374131 for similar and related constructions.
Differs from A305900 first at n=77, where a(77) = 50, while A305900(77) = 59.

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]));
Aux374040(n) = { my(d=A003415(n)); [d, gcd(n,d), valuation(n,2), valuation(n,3)]; };
v374040 = rgs_transform(vector(up_to, n, Aux374040(n)));
A374040(n) = v374040[n];

A374211 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), with f(1) = 1, and for n > 1, f(n) = [A278226(A328768(n)), A374212(n), A374213(n)], where A328768 is the first primorial based variant of the arithmetic derivative, and A374212 and A374213 are its 2- and 3-adic valuations.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 30 2024

nonn

Restricted growth sequence transform of the function f given in the definition.
For all i, j >= 1:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A152822(i) = A152822(j),
  a(i) = a(j) => A373982(i) = A373982(j) => A328771(i) = A328771(j),
  a(i) = a(j) => A373991(i) = A373991(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A278226, A328768, A374212, A374213.
Cf. A305900, A152822, A328771, A373982, A373991.
Cf. also A374131.

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; };
A002110(n) = prod(i=1,n,prime(i));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A278226(n) = A046523(A276086(n));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
Aux374211(n) = if(1==n, n, my(u=A328768(n)); [A278226(u), valuation(u, 2), valuation(u, 3)]);
v374211 = rgs_transform(vector(up_to, n, Aux374211(n)));
A374211(n) = v374211[n];

A374478 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A364255(i) = A364255(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 07 2024

nonn

Restricted growth sequence transform of the ordered pair [A348717(n), A364255(n)].
For all i, j >= 1:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A305891(i) = A305891(j),
  a(i) = a(j) => A374477(i) = A374477(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A163511, A348717, A364255.
Cf. also A305891, A305897, A364297, A364949, A365392, A365423, A374477.
Differs from A374040 first at n=77, where a(77) = 59, while A374040(77) = 50.
Differs from A305900 first at n=95, where a(95) = 39, while A305900(95) = 74.

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; };
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A364255(n) = gcd(n, A163511(n));
Aux374478(n) = [A348717(n), A364255(n)];
v374478 = rgs_transform(vector(up_to, n, Aux374478(n)));
A374478(n) = v374478[n];

A379000 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n is prime > 5, with f(n) = n for all other n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

For all i, j:
  a(i) = a(j) => A305900(i) = A305900(j) => A305801(i) = A305801(j) => A305800(i) = A305800(j),
  a(i) = a(j) => A379001(i) = A379001(j) => A379002(i) = A379002(j),
  a(i) = a(j) => A379005(i) = A379005(j).

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

Cf. A000720.

Cf. A112765, A305800, A305801, A305900, A379001, A379002, A379005.

A379000(n) = if(n<=7, n, if(isprime(n), 7, 4+n-primepi(n)));

For n < 7, a(n) = n, for primes > 5, a(n) = 7, and for composite n > 7, a(n) = 4+n-A000720(n).

cp's OEIS Frontend

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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A373594 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(n<=3) = n, f(p) = 0 for primes p > 3, and for composite n, f(n) = [A007814(n), A065339(n), A083025(n), A373591(n), A373592(n)].

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A374040 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A007814(n), A007949(n)], for all i, j >= 1.

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A374478 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A364255(i) = A364255(j), for all i, j >= 1.

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A379000 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n is prime > 5, with f(n) = n for all other n.

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