cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A246702 The number of positive k < (2n-1)^2 such that (2^k - 1)/(2n - 1)^2 is an integer.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 10, 2, 1, 1, 1, 6, 3, 2, 1, 9, 2, 3, 3, 2, 2, 6, 1, 13, 9, 1, 1, 10, 5, 1, 3, 2, 8, 3, 2, 2, 1, 1, 10, 3, 8, 7, 9, 2, 2, 3, 1, 2, 26, 1, 3, 9, 4, 2, 9, 4, 1, 6, 1, 18, 9, 1, 7, 3, 2, 1, 3, 2, 5, 10, 1, 10, 6, 38, 3, 3, 4, 1, 41, 2
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Nov 15 2014

Keywords

Comments

a(n) is the number of integers k in range 1 .. A016754(n-1)-1 such that A000225(k) is an integral multiple of A016754(n-1). - Antti Karttunen, Nov 15 2014
Conjecture: the positions of 1's, a(k)=1, are exactly given by the 2k-1 which are elements of A167791. - Antti Karttunen, Nov 15 2014
From Charlie Neder, Oct 18 2018: (Start)
It would appear that, if 2k-1 is in A167791, then so is (2k-1)^2, and so a(k) = 1 would follow by definition.
Conjecture: Let B be the first value such that (2k-1)^2 divides 2^B - 1. Then either 2k-1 divides B, or 2k-1 is a Wieferich prime (A001220). (End)

Examples

			a(2) = 1 because (2^6 - 1)/(2*2 - 1)^2 = 7 is an integer and 6 < 9.
a(3) = 1 because (2^20 - 1)/(2*3 - 1)^2 = 41943 is an integer and 20 < 25.
a(3) = 2 because (2^21 - 1)/(2*4 - 1)^2 = 42799 is an integer and 21 < 49; and also (2^42 - 1)/(2*4 - 1)^2 = 89756051247 is an integer and 42 < 49.

Links

Crossrefs

A246703 gives the positions of records.
Cf. A000225, A002326, A016754, A049094, A237043.

Programs

  • Maple
    A246702 := proc(n)
    local a,klim,k ;
    a := 0 ;
    klim := (2*n-1)^2 ;
    for k from 1 to klim-1 do
        if modp(2^k-1,klim) = 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A246702(n),n=1..80) ; # R. J. Mathar, Nov 15 2014
  • Mathematica
    A246702[n_] := Module[{a, klim, k}, a = 0; klim = (2*n-1)^2; For[k = 1, k <= klim-1, k++, If[Mod[2^k-1, klim] == 0, a = a+1]]; a];
Table[A246702[n], {n, 1, 84}] (* Jean-François Alcover, Oct 04 2017, translated from R. J. Mathar's Maple code *)
  • PARI
    a(n)=my(t=(2*n-1)^2,m=Mod(1,t)); sum(k=1,t-1,m*=2;m==1) \\ Charles R Greathouse IV, Nov 16 2014
  • PARI
    a246702(n) = my(m=(2*n-1)^2); (m-1)\znorder(Mod(2,m)); \\ Max Alekseyev, Oct 11 2023
  • Scheme
    (define (A246702 n) (let ((u (A016754 (- n 1)))) (let loop ((k (- u 1)) (s 0)) (cond ((zero? k) s) ((zero? (modulo (A000225 k) u)) (loop (- k 1) (+ s 1))) (else (loop (- k 1) s)))))) ;; Antti Karttunen, Nov 15 2014

Formula

a(n) = floor( 4*n*(n-1) / A002326(2*n*(n-1)) ). - Max Alekseyev, Oct 11 2023

Extensions

Corrected by R. J. Mathar, Nov 15 2014

A172418 Numbers k that have measure of smoothness J larger than 3, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

Original entry on oeis.org

16, 32, 64, 81, 128, 243, 256, 288, 324, 384, 432, 486, 512, 576, 625, 648, 729, 768, 864, 972, 1024, 1152, 1250, 1280, 1296, 1458, 1536, 1600, 1728, 1944, 2000, 2048, 2187, 2304, 2401, 2500, 2560, 2592, 2916, 3072, 3125, 3136, 3200, 3456, 3584, 3645
Offset: 1

Views

Author

Artur Jasinski, Feb 02 2010

Keywords

Comments

Subsequence of A049094.

Links

Crossrefs

Cf. A007947, A049094, A059172, A172419, A172420, A172421, A172422.

Programs

  • Mathematica
    aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 3, AppendTo[aa, c]], {c, 2, 10000}]; aa

A172419 Numbers k that have measure of smoothness J larger than 4, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

Original entry on oeis.org

32, 64, 128, 243, 256, 512, 729, 1024, 1458, 1536, 1728, 1944, 2048, 2187, 2304, 2592, 2916, 3072, 3125, 3456, 3888, 4096, 4374, 4608, 5184, 5832, 6144, 6561, 6912, 7776, 8192, 8748, 9216, 10240, 10368, 11664, 12288, 12500, 12800, 13122, 13824, 15552
Offset: 1

Views

Author

Artur Jasinski, Feb 02 2010

Keywords

Comments

Subsequence of A049094 and A172418.

Links

Crossrefs

Cf. A007947, A049094, A172418, A172420, A172421, A172422.

Programs

  • Mathematica
    aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 4, AppendTo[aa, c]], {c, 2, 10000}]; aa

A172420 Numbers k that have measure of smoothness J larger than 5, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

Original entry on oeis.org

64, 128, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 8748, 9216, 10368, 11664, 12288, 13122, 13824, 15552, 15625, 16384, 17496, 18432, 19683, 20736, 23328, 24576, 26244, 27648, 31104, 32768, 34992, 36864, 39366, 41472, 46656, 49152
Offset: 1

Views

Author

Artur Jasinski, Feb 02 2010

Keywords

Comments

This sequence is a subsequence of A049094, A172418, and A172419.

Links

Crossrefs

Cf. A007947, A049094, A059172, A172418, A172419, A172421, A172422.

Programs

  • Mathematica
    aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 5, AppendTo[aa, c]], {c, 2, 10000}]; aa
Select[Range[2,50000],Log[Times@@FactorInteger[#][[All,1]],#]>5&] (* Harvey P. Dale, Apr 30 2018 *)

A172421 Numbers k that have measure of smoothness J larger than 6, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

Original entry on oeis.org

128, 256, 512, 1024, 2048, 2187, 4096, 6561, 8192, 16384, 19683, 32768, 49152, 52488, 55296, 59049, 62208, 65536, 69984, 73728, 78125, 78732, 82944, 93312, 98304, 104976, 110592, 118098, 124416, 131072, 139968, 147456, 157464, 165888, 177147, 186624, 196608
Offset: 1

Views

Author

Artur Jasinski, Feb 02 2010

Keywords

Comments

This sequence is a subsequence of A049094, A172418, A172419, and A172420.

Links

Crossrefs

Cf. A007947, A049094, A059172, A172418, A172419, A172420, A172422.

Programs

  • Mathematica
    aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 6, AppendTo[aa, c]], {c, 2, 10000}]; aa

Extensions

More terms from Amiram Eldar, Mar 10 2020

A172422 Numbers k that have measure of smoothness J larger than 7, where J = log(k)/log(rad(k)) and rad(k) is the product of the distinct prime divisors of k (A007947).

Original entry on oeis.org

256, 512, 1024, 2048, 4096, 6561, 8192, 16384, 19683, 32768, 59049, 65536, 131072, 177147, 262144, 294912, 314928, 331776, 354294, 373248, 390625, 393216, 419904, 442368, 472392, 497664, 524288, 531441, 559872, 589824, 629856, 663552, 708588, 746496, 786432
Offset: 1

Views

Author

Artur Jasinski, Feb 02 2010

Keywords

Comments

This sequence is a subsequence of A049094, A172418, A172419, A172420, and A172421.

Links

Crossrefs

Cf. A007947, A049094, A059172, A172418, A172419, A172420, A172421.

Programs

  • Mathematica
    aa = {}; Do[kk = FactorInteger[c]; nn = 1; Do[nn = nn*kk[[n]][[1]], {n, 1, Length[kk]}]; If[Log[c]/Log[nn] > 7, AppendTo[aa, c]], {c, 2, 10000}]; aa

Extensions

More terms from Amiram Eldar, Mar 10 2020

A280296 Squarefree numbers k such that 2^k - 1 is divisible by a square > 1.

Original entry on oeis.org

6, 21, 30, 42, 66, 78, 102, 105, 110, 114, 138, 155, 174, 186, 210, 222, 231, 246, 253, 258, 273, 282, 310, 318, 330, 354, 357, 366, 390, 399, 402, 426, 438, 462, 465, 474, 483, 498, 506, 510, 534, 546, 570, 582, 602, 606, 609, 618, 642, 651, 654, 678, 690, 714, 759, 762, 770, 777, 786, 798, 822
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 31 2016

Keywords

Comments

Intersection of A049094 and A005117. - Michel Marcus, Dec 31 2016

Examples

			6 is in this sequence because 2^6 - 1 = 63 is divisible by 9 = 3^2.

Links

Crossrefs

Cf. A005117, A049094, A237043, A280203, A280208, A280209.

Programs

  • Magma
    [n: n in [1..200] | IsSquarefree(n) and not IsSquarefree(2^n-1)];

Extensions

a(38)=498 inserted by Amiram Eldar, Oct 23 2019

A375710 Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.

Original entry on oeis.org

5, 6, 9, 19, 20, 21, 33, 34, 36, 49, 57, 58, 62, 63, 66, 76, 77, 88, 89, 91, 96, 97, 103, 104, 113, 114, 118, 119, 130, 131, 132, 136, 142, 149, 150, 161, 162, 174, 175, 187, 188, 189, 190, 201, 202, 206, 215, 217, 218, 225, 226, 231, 232, 245, 246, 249, 253
Offset: 1

Views

Author

Gus Wiseman, Sep 09 2024

Keywords

Comments

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

Examples

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 2 after the fifth and sixth terms.

Crossrefs

Positions of 2's in A078147.
For prime numbers we have A029707.
For nonprime numbers we appear to have A014689.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A373573, A375927.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[1000], !SquareFreeQ[#]&]],2]

Formula

Complement of A375709 U A375711 U A375712.

A375711 Numbers k such that A013929(k+1) - A013929(k) = 3. In other words, the k-th nonsquarefree number is 3 less than the next nonsquarefree number.

Original entry on oeis.org

3, 16, 23, 27, 31, 44, 46, 51, 55, 60, 68, 74, 79, 86, 95, 101, 105, 107, 112, 116, 121, 126, 129, 146, 147, 152, 159, 164, 167, 172, 177, 182, 185, 191, 195, 199, 204, 209, 220, 223, 229, 234, 237, 242, 244, 257, 262, 270, 275, 285, 286, 291, 299, 305, 312
Offset: 1

Views

Author

Gus Wiseman, Sep 09 2024

Keywords

Comments

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

Examples

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 3 after the third term.

Crossrefs

Positions of 3's in A078147.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A375927.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[1000],!SquareFreeQ[#]&]],3]

Formula

Complement of A375709 U A375710 U A375712.

A375712 Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.

Original entry on oeis.org

1, 4, 7, 11, 12, 13, 14, 22, 25, 26, 29, 32, 35, 39, 40, 41, 42, 50, 53, 54, 61, 64, 70, 71, 72, 75, 78, 81, 82, 83, 84, 87, 90, 98, 99, 102, 109, 110, 117, 120, 123, 124, 127, 135, 139, 140, 144, 151, 154, 155, 156, 157, 160, 163, 168, 169, 170, 173, 176, 179
Offset: 1

Views

Author

Gus Wiseman, Sep 09 2024

Keywords

Comments

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
- A375709 (difference 1)
- A375710 (difference 2)
- A375711 (difference 3)
- A375712 (difference 4)

Examples

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 4 after the first, fourth, and seventh terms.

Crossrefs

For prime numbers we have A029709.
Positions of 4's in A078147.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A029707, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A375927.

Programs

  • Mathematica
    Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&]],4]

Formula

Complement of A375709 U A375710 U A375711.
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