A291758 -id:A291758

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A295300

Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)].

+10
15

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 44, 49, 50, 51, 44, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 58, 62, 65, 66, 67, 68, 69, 70, 58, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 80

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Restricted growth sequence transform of A291752.

For all i, j:

a(i) = a(j) => A291751(i) = A291751(j),

a(i) = a(j) => A326199(i) = A326199(j) => A294877(i) = A294877(j),

a(i) = a(j) => A322021(i) = A322021(j),

a(i) = a(j) => A295888(i) = A295888(j),

a(i) = a(j) => A296090(i) = A296090(j).

LINKS

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

PROG

(PARI)

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; };

A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523

A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));

A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));

Aux295300(n) = (1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n));

v295300 = rgs_transform(vector(up_to, n, Aux295300(n)));

A295300(n) = v295300[n];

CROSSREFS

Cf. A003557, A046523, A048250, A101296, A286360, A291750, A291751, A291752, A291757, A291758, A295888, A296090, A322021, A326199.

Cf. also A305801, A305900, A323368, A324400.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 19 2017

EXTENSIONS

Name changed and the comments section added by Antti Karttunen, Jul 13 2019

STATUS

approved

A291757

a(n) = (1/2)*(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3*A046523(n)).

+10
7

1, 2, 2, 12, 2, 16, 2, 59, 18, 16, 2, 80, 2, 16, 16, 261, 2, 94, 2, 80, 16, 16, 2, 355, 33, 16, 129, 80, 2, 436, 2, 1097, 16, 16, 16, 826, 2, 16, 16, 355, 2, 436, 2, 80, 94, 16, 2, 1493, 52, 125, 16, 80, 2, 505, 16, 355, 16, 16, 2, 1832, 2, 16, 94, 4497, 16, 436, 2, 80, 16, 436, 2, 3415, 2, 16, 125, 80, 16, 436, 2, 1493, 888, 16, 2, 1832, 16, 16, 16, 355, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16385

FORMULA

a(n) = (1/2)*(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3*A046523(n)).

PROG

(PARI)

A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523

A291757(n) = (1/2)*(2 + ((A003557(n)+A046523(n))^2) - A003557(n) - 3*A046523(n));

CROSSREFS

Cf. A000027, A003557, A046523, A291750, A291752, A291756, A291758, A294877 (rgs-transform), A319347.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Sep 10 2017

EXTENSIONS

Name changed by Antti Karttunen, Nov 28 2018

STATUS

approved

A322021

Lexicographically earliest such sequence a that a(i) = a(j) => A046523(i) = A046523(j) and A048250(i) = A048250(j), for all i, j.

+10
5

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 26, 42, 43, 44, 45, 18, 42, 46, 47, 22, 42, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 54, 58, 61, 62, 63, 64, 26, 65, 54, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 52, 78, 79, 80, 81, 75, 82, 83, 26

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Restricted growth sequence transform of A291758, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291758(i) = A291758(j) <=> A046523(i) = A046523(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.

For all i, j:

A295300(i) = A295300(j) => a(i) = a(j),

a(i) = a(j) => A304411(i) = A304411(j),

a(i) = a(j) => A304412(i) = A304412(j).

LINKS

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

PROG

(PARI)

up_to = 65537;

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523

A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));

v322021 = rgs_transform(vector(up_to, n, [A046523(n), A048250(n)]));