A328113 -id:A328113 - cp's OEIS frontend

A327963 If A327928(n) > 0, a(n) = A276086(n), otherwise a(n) = A003415(n).

0, 1, 1, 9, 1, 5, 1, 15, 6, 7, 1, 25, 1, 9, 150, 225, 1, 21, 1, 375, 10, 13, 1, 625, 10, 15, 3750, 5625, 1, 31, 1, 21, 14, 19, 126, 35, 1, 21, 210, 315, 1, 41, 1, 525, 39, 25, 1, 875, 14, 45, 5250, 7875, 1, 4375, 8750, 13125, 22, 31, 1, 49, 1, 33, 51, 441, 18, 61, 1, 735, 26, 59, 1, 1225, 1, 39, 55, 11025, 18, 71, 1, 18375, 36750, 43, 1, 30625, 22, 45

Offset: 1

Antti Karttunen, Oct 07 2019

nonn

After zero, sequence contains only terms of A048103.

Antti Karttunen, Table of n, a(n) for n = 1..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A048103, A129251, A276085, A276086, A327859, A327928, A327929, A327964, A327965, A328097, A328098, A328099, A328113 (rgs-transform).

Cf. also A285330.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327963(n) = if(1==n,0,my(u=A003415(n)); if(!A129251(u),u,A276086(n)));

If A327928(n) > 0 [when n is one of the terms in A327929], then a(n) = A276086(n), otherwise a(n) = A003415(n).

For n > 1, a(n) = A276086(A327964(n)).

A327964 a(1) = 1, and for n > 1, a(n) = A276085(A327963(n)).

Original entry on oeis.org

1, 0, 0, 4, 0, 6, 0, 8, 3, 30, 0, 12, 0, 4, 15, 16, 0, 32, 0, 20, 7, 2310, 0, 24, 7, 8, 27, 28, 0, 6469693230, 0, 32, 31, 510510, 35, 36, 0, 32, 39, 40, 0, 7420738134810, 0, 44, 2312, 12, 0, 48, 31, 10, 51, 52, 0, 54, 55, 56, 211, 6469693230, 0, 60, 0, 212, 30032, 64, 5, 1922760350154212639070, 0, 68, 2311, 32589158477190044730, 0, 72, 0, 2312, 216, 76, 5

Offset: 1

Antti Karttunen, Oct 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A002110, A003415, A129251, A276085, A276086, A327963, A328113.

Cf. A327929 (a subsequence of fixed points).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327963(n) = if(1==n,0,my(u=A003415(n)); if(!A129251(u),u,A276086(n)));
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A327964(n) = if(1==n,n,A276085(A327963(n)));

a(1) = 1, and for n > 1, a(n) = A276085(A327963(n)).
a(p) = 0 for all primes p.
a(A327929(n)) = A327929(n) for all n. [But note that there are also other fixed points.]

A328315 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328099(i) = A328099(j) for all i, j.

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

Offset: 1

Antti Karttunen, Oct 14 2019

nonn

Restricted growth sequence transform of A328099, defined as A328099(n) = min(A003415(n), A276086(n)).

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

Cf. A003415, A276086, A328099, A328113.

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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A328099(n) = min(A003415(n),A276086(n));
v328315 = rgs_transform(vector(up_to, n, A328099(n)));
A328315(n) = v328315[n];

cp's OEIS Frontend

A327963 If A327928(n) > 0, a(n) = A276086(n), otherwise a(n) = A003415(n).

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A327964 a(1) = 1, and for n > 1, a(n) = A276085(A327963(n)).

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A328315 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328099(i) = A328099(j) for all i, j.

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