A278253 -id:A278253 - cp's OEIS frontend

A069904 Number of prime factors of n-th triangular number (with multiplicity).

0, 1, 2, 2, 2, 2, 3, 4, 3, 2, 3, 3, 2, 3, 5, 4, 3, 3, 3, 4, 3, 2, 4, 5, 3, 4, 5, 3, 3, 3, 5, 6, 3, 3, 5, 4, 2, 3, 5, 4, 3, 3, 3, 5, 4, 2, 5, 6, 4, 4, 4, 3, 4, 5, 5, 5, 3, 2, 4, 4, 2, 4, 8, 7, 4, 3, 3, 4, 4, 3, 5, 5, 2, 4, 5, 4, 4, 3, 5, 8, 5, 2, 4, 5, 3, 3, 5, 4, 4, 5

Offset: 1

Reinhard Zumkeller, Apr 10 2002

nonn

			A000217(8) = 8*(8+1)/2 = 36 = 2*2*3*3, therefore a(8) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A069901, A069902, A069903, A092405, A101745, A108815, A114435, A114436, A114437, A164977, A240527, A240528, A240529, A278253.

Array[Plus@@Last/@FactorInteger[ #*(#+1)/2]&,33] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2010 *)

A069904(n) = bigomega((n*(n+1))/2); \\ Antti Karttunen, Oct 07 2017

a(n) = A001222(A000217(n)).
From Antti Karttunen, Oct 07 2017: (Start)
a(n) = (A001222(n)+A001222(n+1))-1.
a(n) = A001222(A278253(n)). (End)
From Alois P. Heinz, Aug 05 2019: (Start)
a(n) = 2 <=> n in { A164977 }.
a(n) = 3 <=> n in { A108815 }.
a(n) = 4 <=> n in { A114435 }.
a(n) = 5 <=> n in { A114436 }.
a(n) = 6 <=> n in { A114437 }.
a(n) = 7 <=> n in { A240527 }.
a(n) = 8 <=> n in { A240528 }.
a(n) = 9 <=> n in { A240529 }.
a(n) = 10 <=> n im { A101745 }. (End)

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

Original entry on oeis.org

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).

A278246 a(n) = least number with the same prime signature as n*(n+3)/2.

Original entry on oeis.org

2, 2, 4, 6, 12, 8, 6, 12, 24, 6, 6, 60, 24, 6, 24, 24, 30, 24, 6, 30, 180, 12, 6, 144, 60, 6, 48, 30, 48, 60, 6, 240, 120, 6, 30, 120, 60, 6, 60, 60, 30, 120, 6, 30, 1080, 12, 12, 360, 60, 12, 48, 210, 60, 48, 30, 60, 420, 6, 6, 840, 96, 30, 120, 96, 210, 60, 30, 30, 360, 30, 6, 1800, 30, 30, 180, 30, 840, 96, 6, 120, 480, 30, 6, 420, 420, 6, 120, 420, 30, 120

Offset: 1

Antti Karttunen, Nov 21 2016

nonn

For n > 2, 6 <= a(n) <= n*(n+3)/2. The upper bound occurs for n = 1, 45, 165, 525, 672, 1152, and no others up to 10^9. (Probably this occurs only finitely many times.) - Charles R Greathouse IV, Nov 23 2016

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

Cf. A000096, A046523, A278247, A278251, A278253, A278255.

A046523(n)=my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i])
a(n)=A046523(n*(n+3)/2) \\ Charles R Greathouse IV, Nov 21 2016

(define (A278246 n) (A046523 (A000096 n)))

a(n) = A046523(A000096(n)).

A278247 a(n) = least number with the same prime signature as n + (n+1)^2.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 6, 2, 6, 2, 2, 6, 6, 2, 2, 2, 6, 6, 2, 6, 2, 6, 2, 6, 2, 2, 6, 6, 6, 2, 8, 6, 2, 2, 6, 2, 12, 6, 2, 2, 2, 30, 2, 6, 2, 6, 30, 2, 2, 2, 2, 6, 6, 2, 2, 6, 30, 6, 2, 2, 2, 6, 2, 6, 2, 6, 6, 6, 6, 6, 2, 6, 6, 6, 30, 6, 6, 2, 12, 2, 2, 6, 6, 2, 6, 6, 30, 2, 2, 6, 2, 6, 6, 6, 2, 2, 30, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 2, 6, 30, 6, 6

Offset: 0

Antti Karttunen, Nov 21 2016

nonn

Almost all terms seem to be primorials, 4848 of the first 5000, in other words, most terms of A028387 (in the same range) are squarefree.

Antti Karttunen, Table of n, a(n) for n = 0..5000

Cf. A028387, A046523.

Cf. also A278246, A278251, A278253, A278255.

(define (A278247 n) (A046523 (A028387 n)))

a(n) = A046523(A028387(n)).

A278251 Least number with the prime signature of the n-th central polygonal number.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 21 2016

nonn

Almost all terms seem to be primorials, 4871 of the first 5212, in other words, most terms of A002061 (in the same range) are squarefree.

Antti Karttunen, Table of n, a(n) for n = 0..5211

Cf. A002061, A046523.

Cf. also A278246, A278247, A278253.

(define (A278251 n) (A046523 (A002061 n)))

a(n) = A046523(A002061(n)).

A278218 Triangle read by rows: T(n,k) = Least number with the prime signature of binomial(n,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 6, 4, 1, 1, 2, 6, 6, 2, 1, 1, 6, 6, 12, 6, 6, 1, 1, 2, 6, 6, 6, 6, 2, 1, 1, 8, 12, 24, 30, 24, 12, 8, 1, 1, 4, 36, 60, 60, 60, 60, 36, 4, 1, 1, 6, 12, 120, 210, 180, 210, 120, 12, 6, 1, 1, 2, 6, 30, 210, 210, 210, 210, 30, 6, 2, 1, 1, 12, 30, 60, 60, 360, 420, 360, 60, 60, 30, 12, 1

Offset: 0

Antti Karttunen, Nov 19 2016

			The triangle begins as:
                                 1
                              1,    1
                            1,   2,   1
                         1,   2,    2,   1
                       1,   4,   6,   4,   1
                    1,   2,   6,    6,   2,  1
                  1,   6,   6,  12,   6,   6,  1
                1,  2,   6,   6,    6,   6,  2,  1
              1,  8,  12,  24,  30,  24,  12,  8,  1
            1,  4, 36,  60,  60,   60,  60, 36,  4,  1
          1,  6, 12, 120, 210, 180, 210, 120, 12,  6,  1
        1,  2,  6, 30, 210, 210,  210, 210, 30,  6,  2,  1
      1, 12, 30, 60,  60, 360, 420, 360,  60, 60, 30, 12,  1
    1,  2, 30, 30, 30,  60, 420,  420,  60, 30, 30, 30,  2,  1
  1,  6,  6, 60, 30, 210, 210, 840, 210, 210, 30, 60,  6,  6,  1
    etc.

Cf. A007318, A046523.

Diagonals: A000012, A046523, A278253, A278252.

Table[Times @@ MapIndexed[(Prime@ First@ #2)^#1 &, #] &@ If[Length@ # == 1 && #[[1, 1]] == 1, {0}, Reverse@ Sort@ #[[All, -1]]] &@ FactorInteger[ Binomial[n, k]], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 21 2016 *)

(define (A278218 n) (A046523 (A007318 n)))

T(n,k) = A046523(C(n,k)).

a(n) = A046523(A007318(n)). [When viewed as a one-dimensional sequence.]

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)).

A278256 Least number with the same prime signature as the n-th oblong number (A002378).

Original entry on oeis.org

2, 6, 12, 12, 30, 30, 24, 72, 60, 30, 60, 60, 30, 210, 240, 48, 60, 60, 60, 420, 210, 30, 120, 360, 60, 120, 360, 60, 210, 210, 96, 480, 210, 210, 1260, 180, 30, 210, 840, 120, 210, 210, 60, 1260, 420, 30, 240, 720, 180, 420, 420, 60, 120, 840, 840, 840, 210, 30, 420, 420, 30, 420, 2880, 960, 2310, 210, 60, 420, 2310, 210, 360, 360, 30, 420, 1260, 420, 2310

Offset: 1

Antti Karttunen, Nov 22 2016

nonn

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

Odd bisection of A278259.
Cf. A002378, A046523.
Cf. also A278253.

(define (A278256 n) (A046523 (A002378 n)))

a(n) = A046523(A002378(n)).

a(n) = A278259((2*n) + 1).

cp's OEIS Frontend

A069904 Number of prime factors of n-th triangular number (with multiplicity).

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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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A278246 a(n) = least number with the same prime signature as n*(n+3)/2.

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A278247 a(n) = least number with the same prime signature as n + (n+1)^2.

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A278251 Least number with the prime signature of the n-th central polygonal number.

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A278218 Triangle read by rows: T(n,k) = Least number with the prime signature of binomial(n,k).

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

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