A018255 -id:A018255 - cp's OEIS frontend

A161715 a(n) = (50n^7 - 1197n^6 + 11333n^5 - 53655n^4 + 132125n^3 - 156828n^2 + 73212*n + 5040)/5040.

1, 2, 3, 5, 6, 10, 15, 30, 171, 886, 3359, 10143, 26072, 59502, 123931, 240048, 438261, 761754, 1270123, 2043641, 3188202, 4840994, 7176951, 10416034, 14831391, 20758446, 28604967, 38862163, 52116860, 69064806, 90525155, 117456180

Offset: 0

Reinhard Zumkeller, Jun 17 2009

{a(k): 0 <= k < 8} = divisors of 30:

a(n) = A027750(A006218(29) + k + 1), 0 <= k < A000005(30).

			Differences of divisors of 30 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     3     5     6    10    15    30
     1     1     2     1     4     5    15
        0     1    -1     3     1    10
           1    -2     4    -2     9
             -3     6    -6    11
                 9   -12    17
                  -21    29
                      50

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Reinhard Zumkeller, Enumerations of Divisors
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A000124, A000125, A000127, A002522, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713.

Cf. A018255, A161700, A161856. - Reinhard Zumkeller, Jun 21 2009

[(50*n^7 - 1197*n^6 + 11333*n^5 - 53655*n^4 + 132125*n^3 - 156828*n^2 + 73212*n + 5040)/5040: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011

CoefficientList[Series[(1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8, {x, 0, 50}], x] (* G. C. Greubel, Jul 16 2017 *)

x='x+O('x^50); Vec((1 -6*x +15*x^2 -19*x^3 +8*x^4 +18*x^5 -51*x^6 +84*x^7) /(-1+x)^8) \\ G. C. Greubel, Jul 16 2017

A161710_list, m = [1], [50, -321, 864, -1249, 1024, -452, 85, 1]
for _ in range(1,10**2):
    for i in range(7):
        m[i+1]+= m[i]
    A161710_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

a(n) = C(n,0) + C(n,1) + C(n,3) - 3*C(n,4) + 9*C(n,5) - 21*C(n,6) + 50*C(n,7).
G.f.: (1-6*x+15*x^2-19*x^3+8*x^4+18*x^5-51*x^6+84*x^7)/(-1+x)^8. - R. J. Mathar, Jun 18 2009
a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8). - Wesley Ivan Hurt, Apr 26 2021

A087005 Divisors of 2310.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310

Offset: 1

Reinhard Zumkeller, Jul 29 2003

2310 = 11# = A002110(5);
divisors of 2310 are squarefree (A005117) and 11-smooth;
a(A000005(2310)) = a(32) = 2310 is the last term.

Index entries for sequences related to divisors of numbers
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Divisor

Cf. A018255, A018336, A087006, A087007, A087008.

squarefrees[s_List] := Block[{a = Subsets[s]}, SortBy[Table[
Product[a[[n, m]], {m, 1, Length[a[[n]]]}], {n, 1, Length[a]}], Abs[#] &]]
A087005 = squarefrees[Prime[Range[5]]](* Fred Daniel Kline, Jan 25 2015 *)
Divisors[2310] (* Harvey P. Dale, Jun 27 2020 *)

divisors(2310) \\ Charles R Greathouse IV, Jun 21 2017

A087006 Divisors of 30030.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42, 55, 65, 66, 70, 77, 78, 91, 105, 110, 130, 143, 154, 165, 182, 195, 210, 231, 273, 286, 330, 385, 390, 429, 455, 462, 546, 715, 770, 858, 910, 1001, 1155, 1365, 1430, 2002, 2145, 2310, 2730

Offset: 1

Reinhard Zumkeller, Jul 29 2003

30030 = 13# = A002110(6);
divisors of 30030 are squarefree (A005117) and 13-smooth;
a(A000005(30030)) = a(64) = 30030 is the last term.

Nathaniel Johnston, Table of n, a(n) for n = 1..64 (full sequence)
Index entries for sequences related to divisors of numbers
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Divisor

Cf. A018255, A018336, A087005, A087007, A087008.

a=30030;Divisors[a] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2009 *)

PARI
```
divisors(30030)
```

A087007 Divisors of 510510.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 21, 22, 26, 30, 33, 34, 35, 39, 42, 51, 55, 65, 66, 70, 77, 78, 85, 91, 102, 105, 110, 119, 130, 143, 154, 165, 170, 182, 187, 195, 210, 221, 231, 238, 255, 273, 286, 330, 357, 374, 385, 390, 429, 442, 455, 462, 510, 546, 561

Offset: 1

Reinhard Zumkeller, Jul 29 2003

510510 = 17# = A002110(7);
divisors of 510510 are squarefree (A005117) and 17-smooth;
a(A000005(510510)) = a(128) = 510510 is the last term.

R. Zumkeller, Table of n, a(n) for n = 1..128
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Divisor
Index entries for sequences related to divisors of numbers

Cf. A018255, A018336, A087005, A087006, A087008.

a=510510;Divisors[a] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2009 *)

PARI
```
divisors(510510)
```

A087008 Divisors of 9699690.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 51, 55, 57, 65, 66, 70, 77, 78, 85, 91, 95, 102, 105, 110, 114, 119, 130, 133, 143, 154, 165, 170, 182, 187, 190, 195, 209, 210, 221, 231, 238, 247, 255, 266, 273, 285, 286, 323, 330

Offset: 1

Reinhard Zumkeller, Jul 29 2003

9699690 = 19# = A002110(8);
divisors of 9699690 are squarefree (A005117) and 19-smooth;
a(A000005(9699690)) = a(256) = 9699690 is the last term.

R. Zumkeller, Table of n, a(n) for n = 1..256
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Divisor
Index entries for sequences related to divisors of numbers

Cf. A018255, A018336, A087005, A087006, A087007.

a=9699690;Divisors[a] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2009 *)

divisors(9699690) \\ Charles R Greathouse IV, Apr 25 2011

A241032 Sum of n-th powers of divisors of 30.

Original entry on oeis.org

8, 72, 1300, 31752, 872644, 25170552, 741453700, 22051219752, 658764967684, 19722455410392, 591076720682500, 17723450167663752, 531571748759349124, 15945186209351359032, 478326193010497869700, 14349345894391097803752

Offset: 0

Vincenzo Librandi, Apr 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (72,-1942,25992,-190081,779760,-1747800,1944000,-810000).

Cf. A018255 (divisors of 30).

Cf. similar sequence listed in A241029.

Magma
```
[DivisorSigma(n, 30): n in [0..20]];
```
Mathematica
```
Total[#^Range[0, 20] & /@ Divisors[30]]
```

G.f.: (8 - 504*x + 11652*x^2 - 129960*x^3 + 760324*x^4 - 2339280* x^5 + 3495600*x^6 - 1944000*x^7) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 5*x)*(1 - 6*x)*(1 - 10*x)*(1 - 15*x)*(1 - 30*x)).

a(n) = (1 + 2^n)*(1 + 3^n)*(1 + 5^n).

A355713 Numbers k such that k and k+1 have the same sum of 5-smooth divisors.

Original entry on oeis.org

175, 2224, 2575, 4975, 7024, 9424, 9775, 11824, 12175, 14224, 14575, 16975, 19024, 21424, 21775, 23824, 24175, 26224, 26575, 28975, 31024, 33424, 33775, 35824, 36175, 38224, 38575, 40975, 43024, 45424, 45775, 47824, 48175, 50224, 50575, 52975, 55024, 57424, 57775

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A355584(k) = A355584(k+1).
Equivalently, numbers k such that the largest 5-smooth divisors of k and k+1, A355582(k) and A355582(k+1), have the same sum of divisors (A000203).
For all the terms k, both k and k+1 are not squarefree: each of the two largest 5-smooth divisors, of k and k+1, cannot be squarefree, since the squarefree 5-smooth numbers are the divisors of 30 = 2*3*5 (A018255) whose values of sigma (A000203), {1, 3, 4, 6, 12, 18, 24, 72}, are not shared with sigma of any other 5-smooth number.
Apparently, all the terms are of only two types: numbers k such that A355582(k) = 16 and A355582(k+1) = 25, or numbers k such that A355582(k) = 25 and A355582(k+1) = 16. Both types are infinite sequences: The first type is the sequence of numbers of the form 2224 + 2400*m, where m is not congruent to 1 (mod 5), and the second type is the sequence of numbers of the form 175 + 2400*m, where m is not congruent to 3 (mod 5). If there are no other terms, then this sequence is a linear recurrence with a signature (1,0,0,0,0,0,0,1,-1). The question of the existence of other types is equivalent to the question of the existence of two coprime 5-smooth numbers other than 16 and 25 whose sums of divisors are equal.
Are there runs of 3 consecutive numbers with the same sum of 5-smooth divisors? There are no such runs below 5*10^10.

			175 is a term since A355584(175) = A355584(176) = 31.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A002961, A018255, A355584.
Subsequence of A013929 and A068781.
Similar sequences: A002961, A064115, A064125, A293183, A306985, A333949.

f[p_, e_] := If[p > 5, 1, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] == s[# + 1] &]

s(n) = (2^(valuation(n, 2) + 1) - 1) * (3^(valuation(n, 3) + 1) - 1) * (5^(valuation(n, 5) + 1) - 1) / 8;
s1 = s(1); for(k = 2, 6e4, s2 = s(k); if(s1 == s2, print1(k-1,", ")); s1 = s2);

cp's OEIS Frontend

A161715 a(n) = (50*n^7 - 1197*n^6 + 11333*n^5 - 53655*n^4 + 132125*n^3 - 156828*n^2 + 73212*n + 5040)/5040.

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A087005 Divisors of 2310.

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A087006 Divisors of 30030.

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A087007 Divisors of 510510.

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A087008 Divisors of 9699690.

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A241032 Sum of n-th powers of divisors of 30.

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Formula

A355713 Numbers k such that k and k+1 have the same sum of 5-smooth divisors.

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Mathematica

PARI

A161715 a(n) = (50n^7 - 1197n^6 + 11333n^5 - 53655n^4 + 132125n^3 - 156828n^2 + 73212*n + 5040)/5040.