A373986 -id:A373986 - cp's OEIS frontend

A373988 Lexicographically earliest infinite sequence such that a(i) = a(j) => A373986(i) = A373986(j), for all i, j >= 1.

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

Offset: 1

Antti Karttunen, Jun 25 2024

nonn

Restricted growth sequence transform of A373986.

For all i, j >= 1: A305800(i) = A305800(j) => a(i) = a(j).

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

Cf. A108951, A373158, A373986.

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; };
A373986(n) = { my(f=factor(n),m=1,s=0); for(i=1, #f~, my(x=prod(i=1,primepi(f[i, 1]),prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); s/gcd(m,s); };
v373988 = rgs_transform(vector(up_to, n, A373986(n)));
A373988(n) = v373988[n];

A373158 Fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

0, 2, 6, 4, 30, 8, 210, 6, 12, 32, 2310, 10, 30030, 212, 36, 8, 510510, 14, 9699690, 34, 216, 2312, 223092870, 12, 60, 30032, 18, 214, 6469693230, 38, 200560490130, 10, 2316, 510512, 240, 16, 7420738134810, 9699692, 30036, 36, 304250263527210, 218, 13082761331670030, 2314, 42, 223092872, 614889782588491410, 14

Offset: 1

Antti Karttunen, May 27 2024

nonn

Completely additive with a(p^e) = e * A002110(A000720(p)).

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to primorial numbers

Cf. also A000040, A000720, A002110, A003961, A034386, A060389, A108951, A276085, A276154, A373984 [= A108951(n)-a(n)], A373985 [= gcd(a(n), A108951(n))], A373986, A373987.

A373158(n) = { my(f=factor(n)); sum(i=1, #f~, f[i, 2]*prod(i=1,primepi(f[i, 1]),prime(i))); }; \\ corrected Jun 25 2024

From Antti Karttunen, Jun 25 2024, Oct 28 2024: (Start)
a(n) = A276085(A003961(n)).
For n >= 1, a(A000040(n)) = A002110(n), a(A002110(n)) = A060389(n).
(End)

Data [first incorrect term was at a(8)] and the faulty PARI-program corrected by Antti Karttunen, Jun 25 2024

A373985 a(n) = gcd(A108951(n), A373158(n)), where A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 2, 6, 4, 30, 4, 210, 2, 12, 4, 2310, 2, 30030, 4, 36, 8, 510510, 2, 9699690, 2, 36, 4, 223092870, 12, 60, 4, 18, 2, 6469693230, 2, 200560490130, 2, 12, 4, 60, 16, 7420738134810, 4, 12, 12, 304250263527210, 2, 13082761331670030, 2, 6, 4, 614889782588491410, 2, 420, 2, 36, 2, 32589158477190044730, 4, 180, 24, 36, 4

Offset: 1

Antti Karttunen, Jun 25 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences related to primorial numbers

Cf. A000040, A002110, A034386, A108951, A373158, A373984, A373986, A373987.

A373985(n) = { my(f=factor(n),m=1,s=0); for(i=1, #f~, my(x=prod(i=1,primepi(f[i, 1]),prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); gcd(m,s); };

a(n) = gcd(A373158(n), A373984(n)).
a(n) = A108951(n) / A373987(n).
For n >= 2, a(n) = A373158(n) / A373986(n).
For n >= 1, a(A000040(n)) = A002110(n).

A373987 Denominator of A373158(n) / A108951(n), where A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 4, 3, 15, 1, 12, 1, 105, 5, 2, 1, 36, 1, 60, 35, 1155, 1, 4, 15, 15015, 12, 420, 1, 180, 1, 16, 1155, 255255, 105, 9, 1, 4849845, 15015, 20, 1, 1260, 1, 4620, 180, 111546435, 1, 48, 105, 900, 85085, 60060, 1, 108, 385, 70, 1616615, 3234846615, 1, 18, 1, 100280245065, 1260, 16, 5005, 13860, 1, 1021020

Offset: 1

Antti Karttunen, Jun 25 2024

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences related to primorial numbers

Cf. A108951, A373158, A373985, A373986 (numerators).

A373987(n) = { my(f=factor(n),m=1,s=0); for(i=1, #f~, my(x=prod(i=1,primepi(f[i, 1]),prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); m/gcd(m,s); };

A373987(n) = denominator(A373158(n)/A108951(n));

a(n) = A108951(n) / A373985(n) = A108951(n) / gcd(A108951(n), A373158(n)).

cp's OEIS Frontend

A373988 Lexicographically earliest infinite sequence such that a(i) = a(j) => A373986(i) = A373986(j), for all i, j >= 1.

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A373158 Fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

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A373985 a(n) = gcd(A108951(n), A373158(n)), where A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

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A373987 Denominator of A373158(n) / A108951(n), where A108951 is fully multiplicative and A373158 is fully additive with a(p) = p# for prime p, where x# is the primorial A034386(x).

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