Muniru A Asiru - cp's OEIS frontend

A323874 Irregular triangle of 13^k mod prime(n).

1, 1, 1, 3, 4, 2, 1, 6, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 0, 1, 13, 16, 4, 1, 13, 17, 12, 4, 14, 11, 10, 16, 18, 6, 2, 7, 15, 5, 8, 9, 3, 1, 13, 8, 12, 18, 4, 6, 9, 2, 3, 16, 1, 13, 24, 22, 25, 6, 20, 28, 16, 5, 7, 4, 23, 9, 1, 13, 14, 27, 10, 6, 16, 22, 7, 29, 5

Offset: 1

Muniru A Asiru, Feb 04 2019

Length of the n-th row (n != 6) is the order of 13 modulo the n-th prime.

Except for the sixth row, the first term of each row is 1.

			The first 10 rows are:
  1
  1
  1, 3, 4, 2
  1, 6
  1, 2, 4, 8, 5, 10, 9, 7, 3, 6
  0
  1, 13, 16, 4
  1, 13, 17, 12, 4, 14, 11, 10, 16, 18, 6, 2, 7, 15, 5, 8, 9, 3
  1, 13, 8, 12, 18, 4, 6, 9, 2, 3, 16
  1, 13, 24, 22, 25, 6, 20, 28, 16, 5, 7, 4, 23, 9

Cf. A000040.

Cf. A201908 (2^k), A201909 (3^k), A201910 (5^k), A201911 (7^k), A323873 (11^k), this sequence (13^k).

A000040:=Filtered([1..350],IsPrime);; p:=6;;
R:=List([1..Length(A000040)],n->OrderMod(A000040[p],A000040[n]));;
a1:=List([1..p-1],n->List([0..R[n]-1],k->PowerMod(A000040[p],k,A000040[n])));;
a:=Flat(Concatenation(a1,[0],List([p+1..2*p],n->List([0..R[n]-1],k->PowerMod(A000040[p],k,A000040[n])))));; Print(a);

T:= n-> (p-> `if`(p=13, 0, seq(13&^k mod p,
         k=0..numtheory[order](13, p)-1)))(ithprime(n)):
seq(T(n), n=1..15);  # Alois P. Heinz, Feb 06 2019

With[{q = 13}, Table[If[p == q, {0}, Array[PowerMod[q, #, p] &, MultiplicativeOrder[q, p], 0]], {p, Prime@ Range@ 11}]] // Flatten (* Michael De Vlieger, Feb 25 2019 *)

A323873 Irregular triangle of 11^k mod prime(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 2, 0, 1, 11, 4, 5, 3, 7, 12, 2, 9, 8, 10, 6, 1, 11, 2, 5, 4, 10, 8, 3, 16, 6, 15, 12, 13, 7, 9, 14, 1, 11, 7, 1, 11, 6, 20, 13, 5, 9, 7, 8, 19, 2, 22, 12, 17, 3, 10, 18, 14, 16, 15, 4, 21, 1, 11, 5, 26, 25, 14, 9, 12, 16, 2, 22, 10, 23, 21, 28

Offset: 1

Muniru A Asiru, Feb 04 2019

Length of the n-th row (n != 5) is the order of 11 modulo the n-th prime.

Except for the fifth row, the first term of each row is 1.

			The first 9 rows are:
  1;
  1,  2;
  1;
  1,  4, 2;
  0;
  1, 11, 4,  5,  3,  7, 12, 2,  9,  8, 10,  6;
  1, 11, 2,  5,  4, 10,  8, 3, 16,  6, 15, 12, 13,  7, 9, 14;
  1, 11, 7;
  1, 11, 6, 20, 13,  5,  9, 7,  8, 19,  2, 22, 12, 17, 3, 10, 18, 14, 16, 15, 4, 21;
  ...

Alois P. Heinz, Rows n = 1..120, flattened

Cf. A201908 (2^k), A201909 (3^k), A201910 (5^k), A201911 (7^k), this sequence (11^k), A323874 (13^k).

Cf. A000040.

A000040:=Filtered([1..350],IsPrime);; p:=5;;
R:=List([1..Length(A000040)],n->OrderMod(A000040[p],A000040[n]));;
a1:=List([1..p-1],n->List([0..R[n]-1],k->PowerMod(A000040[p],k,A000040[n])));;
a:=Flat(Concatenation(a1,[0],List([p+1..2*p],n->List([0..R[n]-1],k->PowerMod(A000040[p],k,A000040[n])))));; Print(a);

T:= n-> (p-> `if`(p=11, 0, seq(11&^k mod p,
         k=0..numtheory[order](11, p)-1)))(ithprime(n)):
seq(T(n), n=1..15);  # Alois P. Heinz, Feb 06 2019

Table[If[p == 11, {0}, Array[PowerMod[11, #, p] &, MultiplicativeOrder[11, p], 0]], {p, Prime@ Range@ 10}] (* Michael De Vlieger, Feb 25 2019 *)

A322749 Primes in A032420.

Original entry on oeis.org

3, 5, 7, 31, 37, 41, 61, 103, 137, 2053, 125887, 968467

Offset: 1

Muniru A Asiru, Dec 25 2018

Equivalently, primes p such that 141*2^p+1 is prime.

Subsequence of A032420.

Filtered([1..400],p->IsPrime(p) and IsPrime(141*2^p+1));

select(p->isprime(p) and isprime(141*2^p+1),[$1..400]);

Select[Prime[Range[140]],PrimeQ[141*2^#+1]&] (* Harvey P. Dale, Sep 04 2023 *)

A320865 Powers of 2 with initial digit 9.

Original entry on oeis.org

9007199254740992, 9223372036854775808, 9444732965739290427392, 9671406556917033397649408, 9903520314283042199192993792, 91343852333181432387730302044767688728495783936, 93536104789177786765035829293842113257979682750464

Offset: 1

Muniru A Asiru, Nov 21 2018

Muniru A Asiru, Table of n, a(n) for n = 1..150
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n), A217402 (numbers starting with 9).

Powers of 2 with initial digit k, (k = 1..9): A067488, A067480, A320859, A320860, A320861, A320862, A320863, A320864, this sequence.

Filtered(List([0..200],n->2^n),i->ListOfDigits(i)[1]=9);

select(x->"9"=""||x[1],[2^n$n=0..200])[];

Select[2^Range[200], IntegerDigits[#][[1]] == 9 &] (* Amiram Eldar, Nov 21 2018 *)

select(x->(digits(x)[1]==9), vector(200, n, 2^n)) \\ Michel Marcus, Nov 21 2018

A320864 Powers of 2 with initial digit 8.

Original entry on oeis.org

8, 8192, 8388608, 8589934592, 8796093022208, 81129638414606681695789005144064, 83076749736557242056487941267521536, 85070591730234615865843651857942052864, 87112285931760246646623899502532662132736, 89202980794122492566142873090593446023921664

Offset: 1

Muniru A Asiru, Nov 21 2018

Muniru A Asiru, Table of n, a(n) for n = 1..170
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n), A217401 (numbers starting with 8).

Powers of 2 with initial digit k, (k = 1..8): A067488, A067480, A320859, A320860, A320861, A320862, A320863, this sequence.

Filtered(List([0..200],n->2^n),i->ListOfDigits(i)[1]=8);

select(x->"8"=""||x[1],[2^n$n=0..200])[];

Select[2^Range[200], IntegerDigits[#][[1]] == 8 &] (* Amiram Eldar, Nov 21 2018 *)

select(x->(digits(x)[1]==8), vector(200, n, 2^n)) \\ Michel Marcus, Nov 21 2018

A320859 Powers of 2 with initial digit 3.

Original entry on oeis.org

32, 32768, 33554432, 34359738368, 35184372088832, 36028797018963968, 36893488147419103232, 37778931862957161709568, 302231454903657293676544, 38685626227668133590597632, 309485009821345068724781056, 39614081257132168796771975168, 316912650057057350374175801344

Offset: 1

Muniru A Asiru, Oct 22 2018

Muniru A Asiru, Table of n, a(n) for n = 1..410
Index to divisibility sequences

Cf. A000079 (Powers of 2), A008952 (leading digit of 2^n).
Powers of 2 with initial digit k, (k = 1..4): A067488, A067480, this sequence, A320860.
Cf. A172404.

Filtered(List([0..120],n->2^n),i->ListOfDigits(i)[1]=3);

[2^n: n in [1..100] | Intseq(2^n)[#Intseq(2^n)] eq 3]; // G. C. Greubel, Oct 24 2018

select(x->"3"=""||x[1],[2^n$n=0..120])[];

Select[2^Range[0, 100], First[IntegerDigits[#]] == 3 &] (* G. C. Greubel, Oct 24 2018 *)

lista(nn) = {for(n=1, nn, x = 2^n; if (digits(x=2^n)[1] == 3, print1(x, ", ")););} \\ Michel Marcus, Oct 25 2018

a(n) = 2^A172404(n).

A320860 Powers of 2 with initial digit 4.

Original entry on oeis.org

4, 4096, 4194304, 4294967296, 4398046511104, 4503599627370496, 4611686018427387904, 4722366482869645213696, 4835703278458516698824704, 4951760157141521099596496896, 40564819207303340847894502572032, 41538374868278621028243970633760768

Offset: 1

Muniru A Asiru, Oct 22 2018

Differs from A067482 first at n = 11.

Muniru A Asiru, Table of n, a(n) for n = 1..320
Index to divisibility sequences

Cf. A000079 (Powers of 2), A008952 (leading digit of 2^n), A217397 (numbers starting with 4).

Powers of 2 with initial digit k, (k = 1..4): A067488, A067480, A320859, this sequence.

Filtered(List([0..150],n->2^n),i->ListOfDigits(i)[1]=4);

[2^n: n in [1..160] | Intseq(2^n)[#Intseq(2^n)] eq 4]; // G. C. Greubel, Oct 27 2018

select(x->"4"=""||x[1],[2^n$n=0..150])[];

Select[2^Range[160], First[IntegerDigits[#]] == 4 &] (* G. C. Greubel, Oct 27 2018 *)

select(x->(digits(x)[1]==4), vector(200, n, 2^n)) \\ Michel Marcus, Oct 26 2018

A320862 Powers of 2 with initial digit 6.

Original entry on oeis.org

64, 65536, 67108864, 68719476736, 604462909807314587353088, 618970019642690137449562112, 633825300114114700748351602688, 649037107316853453566312041152512, 664613997892457936451903530140172288, 680564733841876926926749214863536422912

Offset: 1

Muniru A Asiru, Oct 23 2018

Muniru A Asiru, Table of n, a(n) for n = 1..220
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n), A217399 (numbers starting with 6).

Powers of 2 with initial digit k, (k = 1..6): A067488, A067480, A320859, A320860, A320861, this sequence.

Filtered(List([0..180],n->2^n),i->ListOfDigits(i)[1]=6);

[2^n: n in [1..160] | Intseq(2^n)[#Intseq(2^n)] eq 6]; // G. C. Greubel, Oct 27 2018

select(x->"6"=""||x[1],[2^n$n=0..180])[];

Select[2^Range[160], First[IntegerDigits[#]] == 6 &] (* G. C. Greubel, Oct 27 2018 *)

select(x->(digits(x)[1]==6), vector(200, n, 2^n)) \\ Michel Marcus, Oct 26 2018

A320863 Powers of 2 with initial digit 7.

Original entry on oeis.org

70368744177664, 72057594037927936, 73786976294838206464, 75557863725914323419136, 77371252455336267181195264, 79228162514264337593543950336, 713623846352979940529142984724747568191373312, 730750818665451459101842416358141509827966271488

Offset: 1

Muniru A Asiru, Oct 26 2018

Muniru A Asiru, Table of n, a(n) for n = 1..170
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n), A217400 (numbers starting with 7).

Powers of 2 with initial digit k, (k = 1..7): A067488, A067480, A320859, A320860, A320861, A320862, this sequence.

Filtered(List([0..180],n->2^n),i->ListOfDigits(i)[1]=7);

[2^n: n in [1..160] | Intseq(2^n)[#Intseq(2^n)] eq 7]; // G. C. Greubel, Oct 27 2018

select(x->"7"=""||x[1],[2^n$n=0..180])[];

Select[2^Range[160], First[IntegerDigits[#]] == 7 &] (* G. C. Greubel, Oct 27 2018 *)

select(x->(digits(x)[1]==7), vector(200, n, 2^n)) \\ Michel Marcus, Oct 27 2018

A320861 Powers of 2 with initial digit 5.

Original entry on oeis.org

512, 524288, 536870912, 549755813888, 562949953421312, 576460752303423488, 590295810358705651712, 5070602400912917605986812821504, 5192296858534827628530496329220096, 5316911983139663491615228241121378304, 5444517870735015415413993718908291383296

Offset: 1

Muniru A Asiru, Oct 23 2018

Robert Israel, Table of n, a(n) for n = 1..263
Index to divisibility sequences

Cf. A000079 (powers of 2), A008952 (leading digit of 2^n).

Powers of 2 with initial digit k, (k = 1..5): A067488, A067480, A320859, A320860, this sequence.

Filtered(List([0..160],n->2^n),i->ListOfDigits(i)[1]=5);

[2^n: n in [1..200] | Intseq(2^n)[#Intseq(2^n)] eq 5]; // Vincenzo Librandi, Oct 25 2018

select(x->"5"=""||x[1],[2^n$n=0..160])[];
# Alternative:
Res:= NULL: count:= 0:
for k from 1 to 49 do
   n:= ilog2(6*10^k);
   if n > ilog2(5*10^k) then count:= count+1;
     Res:= Res, 2^n;
   fi
od:
Res; # Robert Israel, Oct 26 2018

Select[2^Range[200], First[IntegerDigits[#]]==5 &] (* Vincenzo Librandi, Oct 25 2018 *)

lista(nn) = {for(n=1, nn, x = 2^n; if (digits(x=2^n)[1] == 5, print1(x, ", ")););} \\ Michel Marcus, Oct 25 2018

cp's OEIS Frontend

User: Muniru A Asiru

A323874 Irregular triangle of 13^k mod prime(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

GAP

Maple

Mathematica

A323873 Irregular triangle of 11^k mod prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

A322749 Primes in A032420.

Views

Author

Keywords

Comments

Crossrefs

Programs

GAP

Maple

Mathematica

A320865 Powers of 2 with initial digit 9.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

A320864 Powers of 2 with initial digit 8.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

A320859 Powers of 2 with initial digit 3.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A320860 Powers of 2 with initial digit 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma