A278254 -id:A278254 - cp's OEIS frontend

A278253 Least number with the prime signature of the n-th triangular number.

1, 2, 6, 6, 6, 6, 12, 36, 12, 6, 30, 30, 6, 30, 120, 24, 12, 12, 30, 210, 30, 6, 60, 180, 12, 24, 120, 30, 30, 30, 48, 240, 30, 30, 420, 60, 6, 30, 420, 60, 30, 30, 30, 420, 60, 6, 120, 360, 36, 60, 210, 30, 24, 120, 420, 420, 30, 6, 210, 210, 6, 60, 1440, 480, 210, 30, 30, 210, 210, 30, 180, 180, 6, 60, 420, 210, 210, 30, 120, 2160, 48, 6, 210, 2310, 30, 30

Offset: 1

Antti Karttunen, Nov 19 2016

nonn

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

Cf. A000217, A046523, A278254, A278256.

Diagonal 2 of triangle A278218.

Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[ (n^2 - n)/2], {n, 2, 87}] (* Michael De Vlieger, Nov 21 2016 *)

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

(define (A278253 n) (A046523 (A000217 n)))

a(n) = A046523(A000217(n)).

A278162 Least number with the prime signature of n^2 + 1.

Original entry on oeis.org

1, 2, 2, 6, 2, 6, 2, 12, 6, 6, 2, 6, 6, 30, 2, 6, 2, 30, 12, 6, 2, 30, 6, 30, 2, 6, 2, 30, 6, 6, 6, 30, 12, 30, 6, 6, 2, 30, 12, 6, 2, 12, 6, 60, 6, 6, 6, 210, 6, 6, 6, 6, 6, 30, 2, 30, 2, 120, 6, 6, 6, 6, 6, 30, 6, 6, 2, 30, 24, 6, 12, 6, 30, 210, 2, 30, 6, 30, 6, 6, 6, 30, 12, 210, 2, 6, 6, 30, 6, 30, 2, 30, 6, 60, 2, 6, 6, 30, 30, 60, 6, 6, 6, 30, 6, 30, 6

Offset: 0

Antti Karttunen, Nov 19 2016

nonn

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

Bisection of A278260.
Cf. A002522, A046523.
Cf. A005574 (positions of 2's), A085722 (of 6's).
Cf. also A278160, A278244, A278254.

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

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
A278162(n) = A046523((n^2)+1);
for(n=0, 10000, write("b278162.txt", n, " ", A278162(n)));

(define (A278162 n) (A046523 (+ 1 (* n n))))

a(n) = A046523(A002522(n)) = A046523((n^2)+1).

a(n) = A278260(2*n).

A278259 Least number with the prime signature of the n-th quarter-square, a(1) = 0.

Original entry on oeis.org

0, 1, 2, 4, 6, 4, 12, 16, 12, 4, 30, 36, 30, 4, 24, 64, 72, 16, 60, 36, 30, 4, 60, 144, 60, 4, 30, 36, 210, 36, 240, 256, 48, 4, 60, 144, 60, 4, 60, 144, 420, 36, 210, 36, 30, 4, 120, 576, 360, 16, 60, 36, 120, 64, 360, 144, 60, 4, 210, 900, 210, 4, 96, 1024, 480, 36, 210, 36, 210, 36, 1260, 1296, 180, 4, 30, 36, 210, 36, 840, 576, 120, 4, 210, 900, 210, 4, 60

Offset: 1

Antti Karttunen, Nov 22 2016

nonn

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

Cf. A002620, A046523.
Cf. A278254, A278256 (bisections).
Cf. also A278260.

(define (A278259 n) (if (<= n 1) 0 (A046523 (A002620 n))))

a(1) = 0; for n > 1, a(n) = A046523(A002620(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).

A278238 a(n) = A278233(A000695(n)) = A278233(n)^2.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 17 2016

nonn

Cf. A048720, A000695, A278233, A278239, A278254.

(define (A278238 n) (A278233 (A000695 n)))

a(n) = A278233(A000695(n)).

a(n) = A278233(n)^2.

A278255 Least number with the prime signature of the n-th pentagonal number.

Original entry on oeis.org

1, 2, 12, 6, 6, 6, 30, 12, 12, 6, 48, 210, 6, 6, 210, 24, 12, 12, 60, 30, 30, 30, 30, 60, 12, 30, 1080, 30, 6, 30, 30, 240, 60, 6, 420, 60, 30, 6, 210, 420, 6, 120, 192, 30, 60, 6, 210, 840, 12, 12, 420, 210, 6, 120, 210, 60, 210, 6, 120, 210, 30, 30, 420, 96, 30, 30, 180, 210, 30, 210, 30, 1260, 6, 30, 5040, 30, 210, 30, 30, 120, 144, 60, 60, 210, 30, 6, 2310

Offset: 1

Antti Karttunen, Nov 21 2016

nonn

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

Cf. A000326, A046523.

Cf. also A278246, A278247, A278251, A278253, A278254.

(define (A278255 n) (A046523 (A000326 n)))

a(n) = A046523(A000326(n)).

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

A278253 Least number with the prime signature of the n-th triangular number.

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A278162 Least number with the prime signature of n^2 + 1.

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A278259 Least number with the prime signature of the n-th quarter-square, a(1) = 0.

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

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A278238 a(n) = A278233(A000695(n)) = A278233(n)^2.

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A278255 Least number with the prime signature of the n-th pentagonal number.

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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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