A278162 -id:A278162 - cp's OEIS frontend

A278254 Least number with the prime signature of n^2; square of the least number with the prime signature of n.

1, 4, 4, 16, 4, 36, 4, 64, 16, 36, 4, 144, 4, 36, 36, 256, 4, 144, 4, 144, 36, 36, 4, 576, 16, 36, 64, 144, 4, 900, 4, 1024, 36, 36, 36, 1296, 4, 36, 36, 576, 4, 900, 4, 144, 144, 36, 4, 2304, 16, 144, 36, 144, 4, 576, 36, 576, 36, 36, 4, 3600, 4, 36, 144, 4096, 36, 900, 4, 144, 36, 900, 4, 5184, 4, 36, 144, 144, 36, 900, 4, 2304, 256, 36, 4, 3600, 36, 36, 36

Offset: 1

Antti Karttunen, Nov 19 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Bisection of A278259.

Cf. A000290, A046523, A278238, A278253, A278160, A278162.

Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[ n^2], {n, 120}] (* Michael De Vlieger, Nov 21 2016 *)

a(n)=my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i])^2 \\ Charles R Greathouse IV, Feb 03 2017

(define (A278254 n) (A000290 (A046523 n)))
(define (A278254 n) (A046523 (A000290 n)))

a(n) = A046523(n^2) = A046523(n)^2.

a(n) = A278259(2*n).

A278160 Least number with the prime signature of ((n+1)^2 - 1).

Original entry on oeis.org

2, 8, 6, 24, 6, 48, 12, 48, 12, 120, 6, 120, 30, 96, 30, 288, 6, 360, 30, 120, 30, 240, 12, 240, 72, 120, 24, 840, 6, 960, 30, 192, 210, 360, 30, 360, 30, 240, 30, 1680, 6, 840, 60, 120, 60, 480, 12, 1440, 60, 360, 30, 1080, 30, 2160, 210, 240, 30, 840, 6, 840, 60, 384, 420, 1920, 30, 840, 30, 840, 30, 5040, 6, 720, 60, 120, 420, 840, 30, 3360, 48, 480, 48

Offset: 1

Antti Karttunen, Nov 19 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A005563, A046523.
Cf. A001359 (positions of 6's).
Cf. also A278240, A278242, A278253, A278254, A278162.

Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[ (n + 1)^2 - 1], {n, 120}] (* Michael De Vlieger, Nov 21 2016 *)

(define (A278160 n) (A046523 (A005563 n)))
(define (A005563 n) (* n (+ 2 n)))

a(n) = A046523(A005563(n)) = A046523(((n+1)^2)-1).

A278260 Least number with the same prime signature as {the n-th quarter-square}+1.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 2, 6, 6, 2, 2, 2, 12, 6, 6, 2, 6, 6, 2, 6, 6, 6, 6, 2, 30, 6, 2, 2, 6, 2, 2, 30, 30, 2, 12, 8, 6, 6, 2, 2, 30, 2, 6, 12, 30, 6, 2, 2, 6, 30, 2, 6, 30, 2, 6, 6, 6, 6, 6, 12, 30, 6, 12, 6, 30, 2, 6, 6, 6, 6, 2, 6, 30, 30, 12, 2, 6, 6, 2, 6, 12, 2, 6, 6, 60, 6, 6, 6, 6, 6, 6, 30, 210, 6, 6, 6, 6, 30, 6, 2, 6, 6, 6, 6, 30, 6, 2, 2, 30, 30, 2, 6

Offset: 1

Antti Karttunen, Nov 22 2016

nonn

Great majority of the terms seem to be primorials (A002110), 9149 of the first 10000, meaning also that the corresponding terms in A033638 are squarefree (A005117).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A002110, A005117, A033638, A046523.
Cf. A278162 (bisection).
Cf. also A278254, A278259.

(define (A278260 n) (A046523 (A033638 n)))

a(n) = A046523(A033638(n)).

A278244 Least number with the prime signature of the n-th centered square number (A001844(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 6, 2, 6, 2, 6, 6, 2, 6, 2, 6, 6, 2, 6, 2, 4, 12, 2, 30, 2, 2, 6, 6, 24, 2, 2, 6, 2, 6, 2, 2, 30, 6, 6, 2, 6, 30, 2, 6, 6, 6, 12, 2, 6, 12, 2, 6, 6, 12, 6, 6, 6, 6, 12, 6, 2, 30, 6, 6, 6, 2, 30, 6, 6, 2, 2, 12, 2, 6, 6, 6, 6, 6, 60, 2, 6, 6, 2, 6, 2, 2, 30, 2, 30, 6, 2, 30, 6, 30, 6, 30, 24, 2, 6, 2, 2, 30, 2, 12, 2, 6, 30, 6, 30, 2, 2, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Nov 22 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A001844, A046523.

Cf. also A278162, A278254, A278259.

A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From Charles R Greathouse IV, Aug 17 2011
A001844(n) = 2*((n^2) + n) + 1;
A278244(n) = A046523(A001844(n));
for(n=0, 10000, write("b278244.txt", n, " ", A278244(n)));

(define (A278244 n) (A046523 (A001844 n)))

a(n) = A046523(A001844(n)).

cp's OEIS Frontend

A278254 Least number with the prime signature of n^2; square of the least number with the prime signature of n.

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A278160 Least number with the prime signature of ((n+1)^2 - 1).

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A278260 Least number with the same prime signature as {the n-th quarter-square}+1.

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A278244 Least number with the prime signature of the n-th centered square number (A001844(n)).

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