A069782 -id:A069782 - cp's OEIS frontend

A069780 a(n) = gcd(d(n^3), d(n)).

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 8, 1, 4, 2, 2, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 8, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 8, 4, 8, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4, 8, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 2, 1, 2, 8, 2, 8, 8

Offset: 1

Labos Elemer, Apr 08 2002

nonn

Terms are usually powers of 2. Smallest number m such that A069780(m)=2^n is A037992(n). The first n such that a(n) is not a power of 2 equals 432: a(432) = gcd(d(80621568), d(432)) = gcd(130,20) = 10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A037992, A069781, A069782.

Table[GCD[DivisorSigma[0, n^3], DivisorSigma[0, n]], {n, 1, 500}]

a(n)=my(f=factor(n)[, 2]); gcd(prod(i=1, #f, 3*f[i]+1), prod(i=1, #f, f[i]+1)) \\ Charles R Greathouse IV, Oct 16 2015

a(n) = gcd(A000005(n^3), A000005(n)).

A069781 Numbers k such that gcd(d(k^3), d(k)) is not a power of 2.

Original entry on oeis.org

432, 576, 648, 1600, 2000, 2160, 2880, 2916, 3024, 3136, 3240, 4032, 4536, 4752, 4800, 5000, 5488, 5616, 6000, 6336, 7128, 7344, 7488, 7744, 8208, 8424, 9408, 9792, 9936, 10125, 10800, 10816, 10944, 11016, 11200, 12312, 12528, 13248, 13392

Offset: 1

Labos Elemer, Apr 08 2002

nonn

The complement of this sequence in the positive integers A000027 is A069782. - M. F. Hasler, Jan 18 2015

The numbers of the form 4*3^(7*m - 1), m >= 1, are terms. - Marius A. Burtea, Oct 18 2019

			For n<100000, gcd[d(n^3),d[n]] = {5,7,10,14,20,28,40,80} which is obtained for n={20736,576,432,2880,54000,20160,2160,15120} respectively.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A061701, A037992, A069780, A069782.

f:=func; [k:k in [1..14000]| not IsIntegral(Log(2,f(k)))]; // Marius A. Burtea, Oct 18 2019

f[x_] := GCD[DivisorSigma[0, x^3], DivisorSigma[0, x]] Do[s=f[n]; If[ !IntegerQ[Log[2, s]], Print[n]], {n, 1, 100000}]
Select[Range[14000],!IntegerQ[Log[2,GCD[DivisorSigma[0,#^3], DivisorSigma[ 0,#]]]]&] (* Harvey P. Dale, Mar 20 2018 *)

is(n)=my(f=factor(n)[,2], g=gcd(prod(i=1,#f,3*f[i]+1), prod(i=1,#f,f[i]+1))); g!=1<Charles R Greathouse IV, Oct 16 2015

log_2(gcd(A000005(n^3), A000005(n))) is nonintegral.

A069783 a(n) = gcd(d(n!^3), d(n!)), where d() is the number of divisors function.

Original entry on oeis.org

1, 2, 4, 8, 16, 2, 4, 32, 32, 10, 20, 8, 16, 32, 448, 448, 1792, 32, 64, 80, 3200, 1280, 2560, 320, 448, 1792, 25088, 101920, 203840, 128, 256, 4096, 81920, 112640, 2048, 8960, 17920, 1024, 2048, 5120, 10240, 5734400, 11468800, 1003520, 250880, 8960, 17920

Offset: 1

Labos Elemer, Apr 08 2002

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A000142, A069780, A069781, A069782.

Table[GCD[DivisorSigma[0, (n!)^3], DivisorSigma[0, n! ]], {n, 1, 100}]

a(n) = my(e = factor(n!)[, 2]); gcd(vecprod(apply(x -> x + 1, e)), vecprod(apply(x -> 3*x + 1, e))) \\ Amiram Eldar, Mar 07 2025

a(n) = A069780(n!).

A069784 Numbers m such that gcd(d((m!)^3), d(m!)) = 2^k, i.e., is a power of 2; d = A000005.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 18, 19, 30, 31, 32, 35, 38, 39

Offset: 1

Labos Elemer, Apr 08 2002

From David A. Corneth, Jul 31 2017: (Start)
Theorem: There are no further terms.
Proof:
Let e_n(n, p) be the exponent of p in n!. The prime p has exponent e_(n, p) = n/p for sqrt(n) < p < n in n!. n/4 <= p < n/3, e_(n, p) = 3 so e_(n, p) * 3 = 9. and for n/5 <= p < n/4, e = 4. The gap g_n between prime(n) and prime(n+1) is about sqrt(n) * log(n). There is a gap of n/4 - n/5 = n/20 between n/5 and n/4. primepi(1000) = 168, so for n > 5*1000, the gap between n/5 and the next prime is about sqrt(168) * log(168) ~= 66. This is much less than n/20. No 40 <= m <= 15000 is in the sequence, which completes the proof. (End)

Cf. A000005, A000142, A069780, A069781, A069782, A069783.

Do[s=GCD[DivisorSigma[0, (n!)^3], DivisorSigma[0, n! ]]; If[IntegerQ[n/100], Print[{n}]]; If[IntegerQ[Log[2, s]], Print[n]], {n, 1, 10000}]

val(n, p) = my(r=0); while(n, r+=n\=p);r
is(n) = {my(p1 = p2 = 1); forprime(p=2, n, v = val(n, p); p1 *= (v + 1); p2 *= (3*v + 1)); g = gcd(p1, p2); g==2^(valuation(g, 2))} \\ David A. Corneth, Jul 31 2017

Keywords fini and full added by David A. Corneth, Jul 31 2017

A069785 a(n) = A061680(n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 15, 15, 3, 5, 135, 135, 99, 99, 9, 63, 21, 21, 459, 459, 135, 19, 15, 15, 15, 21, 189, 189, 585, 585, 18225, 18225, 675, 15, 135, 891, 8505, 25515, 81, 81, 7695, 7695, 1575, 1575, 4725, 6615, 40635, 40635, 945, 1215, 3645, 3645, 151875, 151875

Offset: 1

Labos Elemer, Apr 09 2002

			Observe cases when consecutive terms are equal: n={1,2,3,4,6,10,...,78,80,82,88,96}.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A000142, A061680, A061701, A048691, A069780, A069781, A069782, A069783, A069784.

a[n_] := Module[{e = FactorInteger[n!][[;;, 2]]}, GCD[Times @@ (2*e+1), Times @@ (e+1)]]; Array[a, 100] (* Amiram Eldar, Dec 02 2023 *)

a(n) = {my(e = factor(n!)[,2]); gcd(vecprod(apply(x -> 2*x+1, e)), vecprod(apply(x -> x+1, e)));} \\ Amiram Eldar, Dec 02 2023

a(n) = A061680(A000142(n)). - Amiram Eldar, Dec 02 2023

A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

1, 5, 15, 45, 105, 264, 555, 1221, 2445, 4935, 9324, 17941, 32522, 59400, 104808, 184569, 315711, 540540, 902335, 1504800, 2462724, 4014513, 6444425, 10316250, 16283707, 25610886, 39841865, 61720659, 94687230, 144731706, 219282679, 330996105, 495901413, 740046425

Offset: 1

Olivier Gérard, May 31 2024

nonn

The length of each row is given by A000041.
As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.
A069782, A088480, A090107, A090108, A090109, A115510, A115511, A130446, A131717, A132086, A153671, A167904, A296879, A296882, A296885, A296888, A296891, A296894, A296897, A296900, A296903, A296906, A303502, A317945, A335280, A361374.

			Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
.
    1 |  1
    5 |  2  3
   15 |  4  5  6
   45 |  7  8  9 10 11
  105 | 12 13 14 15 16 17 18
  264 | 19 20 21 22 23 24 25 26 27 28 29
  555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
.
The sequence gives the row sums of this triangle.

Cf. A000027, seen as a triangle with shape A000041.
Cf. A373301, the same principle, but starting from integer zero instead of 1.
Cf. A006003, row sums of the integers but for the linear triangle.

Module[{s = 0},
 Table[s +=
   PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -
   s*(s - 1)/2, {n, 1, 30}]]

cp's OEIS Frontend

A069780 a(n) = gcd(d(n^3), d(n)).

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A069781 Numbers k such that gcd(d(k^3), d(k)) is not a power of 2.

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A069783 a(n) = gcd(d(n!^3), d(n!)), where d() is the number of divisors function.

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A069784 Numbers m such that gcd(d((m!)^3), d(m!)) = 2^k, i.e., is a power of 2; d = A000005.

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A069785 a(n) = A061680(n!).

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A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

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Mathematica