Klaus Brockhaus - cp's OEIS frontend

A192993 Numbers that are in more than one way the concatenation of the decimal representation of two nonzero squares.

164, 1441, 1625, 1961, 2564, 4841, 12116, 14449, 16400, 25625, 46241, 48464, 115625, 116641, 144100, 148841, 160025, 162500, 163844, 169169, 184964, 193636, 196100, 256400, 361225, 368649, 466564, 484100, 493025, 961009, 973441, 1166464

Offset: 1

Klaus Brockhaus and Zak Seidov, Jul 14 2011

Subsequence of A191933.
If k is a term, then k followed by two zeros is also a term.
None of the terms < 40000000 is in more than two ways the concatenation of the decimal representation of two nonzero squares.
A038670 is a subsequence. - Reinhard Zumkeller, Jul 15 2011

			2564 is the concatenation of 256 and 4 as well as of 25 and 64; 256, 4, 25, 64 are squares, so 2564 is a term.

Klaus Brockhaus, Table of n, a(n) for n = 1..100 (terms < 40000000)

Cf. A000290, A191933.

import Data.List (findIndices)
a192993 n = a192993_list !! (n-1)
a192993_list = findIndices (> 1) $ map a193095 [0..]
-- Reinhard Zumkeller, Jul 17 2011

SplitToSquares:=function(n); V:=[]; S:=Intseq(n); for j in [1..#S-1] do A:=[ S[k]: k in [1..j] ]; a:=Seqint(A); B:=[ S[k]: k in [j+1..#S] ]; b:=Seqint(B); if a gt 0 and A[#A] gt 0 and IsSquare(a) and IsSquare(b) then Append(~V, []); end if; end for; return V; end function; [ p: p in [1..1200000] | #P gt 1 where P is SplitToSquares(p) ]; /* to obtain the splittings replace " p: " with " : " */

f@n_ := DeleteDuplicates@
  Select[First@# & /@
    Select[Partition[
      Sort@(FromDigits@Flatten@IntegerDigits@# & /@
         Tuples[Range@Sqrt[10^(n - 1) - 1]^2, {2}]), 2, 1],
     Differences@# == {0} &], # <
10^n &]; f@7 (* Hans Rudolf Widmer, Jun 12 2023 *) (* Numbers with at most n digits that are in more than one way the concatenation of the decimal representation of two nonzero squares. *)

A191933 Numbers that are the concatenation of the decimal representation of two nonzero squares.

Original entry on oeis.org

11, 14, 19, 41, 44, 49, 91, 94, 99, 116, 125, 136, 149, 161, 164, 169, 181, 251, 254, 259, 361, 364, 369, 416, 425, 436, 449, 464, 481, 491, 494, 499, 641, 644, 649, 811, 814, 819, 916, 925, 936, 949, 964, 981, 1001, 1004, 1009, 1100, 1121, 1144, 1169, 1196

Offset: 1

Klaus Brockhaus, Jun 19 2011

Complement of A193096; A193095(a(n)) > 0; A038670, A039686, A167535, A192993, A193097 and A193144 are subsequences. [Reinhard Zumkeller, Jul 17 2011]

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A000290, A192993.

import Data.List (findIndices)
a191933 n = a191933_list !! (n-1)
a191933_list = findIndices (> 0) $ map a193095 [0..]
-- Reinhard Zumkeller, Jul 17 2011

CheckSplits:=function(n); v:=false; S:=Intseq(n); for j in [1..#S-1] do A:=[ S[k]: k in [1..j] ]; a:=Seqint(A); B:=[ S[k]: k in [j+1..#S] ]; b:=Seqint(B); if a gt 0 and A[#A] gt 0 and IsSquare(a) and IsSquare(b) then v:=true; end if; end for; return v; end function; [ p: p in [1..1200] | CheckSplits(p) ];

Take[Union[Flatten[Table[FromDigits[Flatten[{IntegerDigits[m^2], IntegerDigits[n^2]}]], {m, 20}, {n, 20}]]], 50] (* Alonso del Arte, Aug 11 2011 *)
squareQ[n_] := IntegerQ[Sqrt[n]]; okQ[n_] := MatchQ[IntegerDigits[n], {a__ /; squareQ[FromDigits[{a}]], b__ /; First[{b}] > 0 && squareQ[FromDigits[ {b}]]}]; Select[Range[2000], okQ] (* Jean-François Alcover, Dec 13 2016 *)

A192618 Prime powers p^k with even exponents k > 0 such that (1 + p^k)/2 is prime.

Original entry on oeis.org

9, 25, 81, 121, 361, 625, 841, 2401, 3481, 3721, 5041, 6241, 10201, 14641, 17161, 19321, 28561, 32761, 39601, 73441, 83521, 121801, 143641, 167281, 201601, 212521, 271441, 279841, 323761, 326041, 398161, 410881, 436921, 546121, 564001, 674041

Offset: 1

Klaus Brockhaus, Jul 05 2011

nonn

Subsequence of A056798.
From R. J. Mathar, Jul 11 2011: (Start)
For odd k we first have the case k=1, where (1+p)/2 is either classified as A005383 or A176897.
For odd k >= 3, (1+p^k)/2 is not prime. [Sketch of proof: for p=2 it is not integer. Otherwise for odd k, (1+p^k)/(1+p) = Sum_{j=0..k-1} (-p)^j, an integer, so 1+p^k is a multiple of 1+p. For odd p, (1+p^k)/2 is a multiple of (1+p)/2 and therefore composite.] (End)

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A056798.

e:=20; u:=1000; z:=Min(2^e, u^2); S:=[ q: p in PrimesUpTo(u), k in [2..e by 2] | q le z and IsEven(1+q) and IsPrime((1+q) div 2) where q is p^k ]; Sort(~S); S;

Select[Union[Flatten[Table[Prime[n]^k, {n, 142}, {k, 0, 32, 2}]]], PrimeQ[(# + 1)/2] &] (* Alonso del Arte, Jul 05 2011 *)

A191859 The primes created by concatenation of anti-divisors in A191647.

Original entry on oeis.org

2, 3, 23, 347, 349, 311, 391627, 3471331, 384067, 2310175897, 239111323273399, 23167, 3784097136227, 235983149249, 3428116271, 37111677121283, 23293, 3471949133311, 231314228398154359, 378112153101159371, 2379127163381

Offset: 1

Klaus Brockhaus, Jun 18 2011

a(n) is the concatenation of the anti-divisors of A191647(n).

			A191647(6) = 16, the anti-divisors of 16 are 3, 11. Hence a(6) = 311.
A191647(8) = 46, the anti-divisors of 46 are 3, 4, 7, 13, 31. Hence a(8) = 3471331.

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A191647, A066272, A130846, A120712.

Antidivisors:=func< n | [ d: d in [2..n-1] | n mod d ne 0 and ( (IsEven(d) and 2*n mod d eq 0) or (IsOdd(d) and ((2*n-1) mod d eq 0 or (2*n+1) mod d eq 0)) ) ] >; CAD:=function(n); A:=Antidivisors(n); S:=[]; for k in [1..#A] do S:= Intseq(A[k]) cat S; end for; p:=Seqint(S); return p; end function; A191859List:=func< m | [ p: n in [1..m] | IsPrime(p) where p is CAD(n) ] >; A191859List(600);

A190758 Primes p such that x^41 = 2 has a solution mod p, and p is congruent to 1 mod 41.

Original entry on oeis.org

17467, 18287, 31817, 42641, 116359, 139483, 163673, 172283, 176383, 181549, 190979, 225829, 226813, 231323, 259531, 288313, 299137, 307009, 352109, 404507, 421891, 445097, 464777, 484621, 528163, 592861, 604997, 609179, 611393, 629843

Offset: 1

Klaus Brockhaus, May 18 2011

nonn

Klaus Brockhaus, Table of n, a(n) for n = 1..5000

Cf. A049573, A059236, A142199.

forprime(p=2, 700000, if(trap(, 0, sqrtn(Mod(2, p), 41); 1), if(p%41==1, print1(p, ", "))));

A186041 Numbers of the form 3k + 2, 5k + 3, or 7*k + 4.

Original entry on oeis.org

2, 3, 4, 5, 8, 11, 13, 14, 17, 18, 20, 23, 25, 26, 28, 29, 32, 33, 35, 38, 39, 41, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 62, 63, 65, 67, 68, 71, 73, 74, 77, 78, 80, 81, 83, 86, 88, 89, 92, 93, 95, 98, 101, 102, 103, 104, 107, 108, 109, 110, 113, 116, 118, 119, 122

Offset: 1

Klaus Brockhaus, Feb 11 2011, Mar 09 2011

n is in the sequence iff n is in A016789 or in A016885 or in A017029.
First differences are periodic with period length 57. Least common multiple of 3, 5, 7 is 105; number of terms <= 105 is 57.
Sequence is not essentially the same as A053726: a(n) = A053726(n-3) for 3 < n < 33, a(34)=62, A053726(34-3)=61.
Sequence is not essentially the same as A104275: a(n) = A104275(n-2) for 3 < n < 33, a(34)=62, A104275(34-3)=61.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A016789, A016885, A017029, A053726, A104275.

IsA186041:=func< n | exists{ k: k in [0..n div 3] | n in [3*k+2, 5*k+3, 7*k+4] } >; [ n: n in [1..200] | IsA186041(n) ];

Take[With[{no=50},Union[Join[3Range[0,no]+2,5Range[0,no]+3,7Range[0,no]+4]]],70]  (* Harvey P. Dale, Feb 16 2011 *)

a(n) = a(n-57) + 105.
a(n) = a(n-1) + a(n-57) - a(n-58).
G.f.: x*(x^57 + x^56 + x^55 + x^54 + 3*x^53 + 3*x^52 + 2*x^51 + x^50 + 3*x^49 + x^48 + 2*x^47 + 3*x^46 + 2*x^45 + x^44 + 2*x^43 + x^42 + 3*x^41 + x^40 + 2*x^39 + 3*x^38 + x^37 + 2*x^36 + 2*x^35 + x^34 + 2*x^33 + x^32 + x^31 + 2*x^30 + 3*x^29 + 3*x^28 + 2*x^27 + x^26 + x^25 + 2*x^24 + x^23 + 2*x^22 + 2*x^21 + x^20 + 3*x^19 + 2*x^18 + x^17 + 3*x^16 + x^15 + 2*x^14 + x^13 + 2*x^12 + 3*x^11 + 2*x^10 + x^9 + 3*x^8 + x^7 + 2*x^6 + 3*x^5 + 3*x^4 + x^3 + x^2 + x + 2) / ((x - 1)^2*(x^2 + x + 1)*(x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^36 - x^35 + x^33 - x^32 + x^30 - x^29 + x^27 - x^26 + x^24 - x^23 + x^21 - x^20 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)).

A186042 Numbers of the form 2k + 1, 3k + 2, or 5*k + 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 18, 19, 20, 21, 23, 25, 26, 27, 28, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 78, 79, 80, 81, 83, 85, 86, 87, 88, 89, 91, 92, 93, 95, 97

Offset: 1

Klaus Brockhaus, Feb 11 2011, Mar 09 2011

n is in the sequence iff n is in A005408 or in A016789 or in A016885.

First differences are periodic with period length 22. Least common multiple of 2, 3, 5 is 30; number of terms <= 30 is 22.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1).

Cf. A005408, A016789, A016885.

IsA186042:=func< n | exists{ k: k in [0..n div 2] | n in [2*k+1, 3*k+2, 5*k+3] } >; [ n: n in [1..100] | IsA186042(n) ];

LinearRecurrence[{2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1},{1,2,3,5,7,8,9,11,13,14,15,17,18,19,20,21,23,25,26,27,28,29},71] (* Ray Chandler, Jul 12 2015 *)

isok(n) = (n % 2) || ((n % 3)==2) || ((n % 5)==3); \\ Michel Marcus, Jul 26 2017

a(n) = a(n-22) + 30.
a(n) = a(n-1) + a(n-22) - a(n-23).
G.f.: x*(x^21 + x^19 + x^17 + x^16 + x^15 + x^13 + x^11 + x^10 + x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + 1) / ((x - 1)^2*(x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)).

A182603 Number of conjugacy classes in GL(n,8).

Original entry on oeis.org

1, 7, 63, 504, 4088, 32697, 262080, 2096577, 16776648, 134213128, 1073737224, 8589897288, 68719439943, 549755515008, 4398046212672, 35184369697407, 281474974319672, 2251799794521144, 18014398490350584, 144115187922510840, 1152921504453534648

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006951, A006952, A049314, A049315, A049316, A182604 - A182612.

/* The program does not work for n>6: */ [1] cat [NumberOfClasses(GL(n, 8)): n in [1..6]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*8^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*8^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

G.f.: prod((1-x^k)/(1-8*x^k),k=1..infinity).

Extended by D. S. McNeil, Dec 06 2010

MAGMA code edited by Vincenzo Librandi, Jan 23 2013

A182604 Number of conjugacy classes in GL(n,9).

Original entry on oeis.org

1, 8, 80, 720, 6552, 58960, 531360, 4782160, 43045920, 387413208, 3486777120, 31380993360, 282429470960, 2541865231440, 22876791858720, 205891126722080, 1853020183479912, 16677181651254480, 150094635248646000, 1350851717237225040, 12157665458621220720

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182605, A182606, A182607, A182608, A182609, A182610, A182611, A182612.

/* The program does not work for n>6: */ [1] cat [NumberOfClasses(GL(n, 9)): n in [1..6]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*9^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*9^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-9*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-9*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 24 2013

A182612 Number of conjugacy classes in GL(n,27).

Original entry on oeis.org

1, 26, 728, 19656, 531414, 14348152, 387419760, 10460332792, 282429516096, 7625596933890, 205891131543552, 5559060551656248, 150094635282119528, 4052555152616676888, 109418989131110078784, 2954312706539971597184, 79766443076861647780830

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182605, A182606, A182607, A182608, A182609, A182610, A182611.

/* The program does not work for n>4: */ [1] cat [ NumberOfClasses(GL(n, 27)) : n in [1..4] ];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*27^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*27^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-27*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-27*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 24 2013

cp's OEIS Frontend

User: Klaus Brockhaus

A192993 Numbers that are in more than one way the concatenation of the decimal representation of two nonzero squares.

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A191933 Numbers that are the concatenation of the decimal representation of two nonzero squares.

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A192618 Prime powers p^k with even exponents k > 0 such that (1 + p^k)/2 is prime.

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A191859 The primes created by concatenation of anti-divisors in A191647.

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A190758 Primes p such that x^41 = 2 has a solution mod p, and p is congruent to 1 mod 41.

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A186041 Numbers of the form 3*k + 2, 5*k + 3, or 7*k + 4.

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A186042 Numbers of the form 2*k + 1, 3*k + 2, or 5*k + 3.

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A182603 Number of conjugacy classes in GL(n,8).

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A186041 Numbers of the form 3k + 2, 5k + 3, or 7*k + 4.

A186042 Numbers of the form 2k + 1, 3k + 2, or 5*k + 3.