Phil Carmody - cp's OEIS frontend

A255724 List of numbers such that no number, nor its reverse, is in the concatenation of all previous terms.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 29, 30, 33, 35, 36, 37, 38, 40, 44, 46, 47, 49, 50, 55, 57, 58, 60, 66, 69, 70, 77, 80, 88, 90, 99, 100, 102, 103, 104, 105, 106, 107, 108, 109, 112, 114, 115, 116, 117, 118, 122

Offset: 1

Phil Carmody, Mar 19 2015

			After listing 1..11, 12 is not listed as "12" is found in the concatenated earlier terms. After continuing with 13..18, 19 is not listed as "91" is likewise found.

Cf. A048991.

lista(nn) = my(d, v=[]); for(n=1, nn, for(i=0, #v-#d=digits(n), (v[i+1..i+#d]==d || v[i+1..i+#d]==Vecrev(d)) && next(2)); print1(n, ", "); v=concat(v, d)) \\ Jinyuan Wang, Aug 23 2021

$s="";$=0;do{$++;if(index($s,$)<0 && index($s,reverse)<0){print("$ ");$s.=$_}}while(1);

More terms from Jinyuan Wang, Aug 23 2021

A222714 Smallest i such that prime(n) divides gcd(sigma(i), phi(i)) (cf. A009223).

Original entry on oeis.org

3, 14, 88, 116, 989, 477, 6901, 7067, 6439, 10207, 4976, 10877, 13529, 44461, 79523, 22577, 250277, 62023, 107869, 161027, 75008, 49769, 55277, 183296, 75077, 612463, 381923, 412163, 712423, 153679, 32576, 137549, 450181, 154289, 1776377, 1642577, 491723, 637981, 3903791, 239777, 642251, 1572889, 1608983, 1192739, 2791489, 316409, 888731, 4773091, 4942243, 1256293

Offset: 1

Phil Carmody, Mar 01 2013

nonn

			Given A009223 = 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 4, 2, 6, 8, 1, 2, 3, ...
prime(1)=2 first divides A009223(3); prime(2)=3 first divides A009223(14)=6; prime(3)=5 first divides both sigma(88)=180 and phi(88)=40, so A222714(3)=88.

Cf. A009223. Subsequence of A222713.

A009223_hunt(x)=local(n=0,g);while(n++,g=A009223(n);if(g%x,,return(n)));
for(x=1,50,print1(A009223_hunt(prime(x))", "))

A222713 Least number k such that n divides gcd(sigma(k), phi(k)) (A009223).

Original entry on oeis.org

1, 3, 14, 12, 88, 14, 116, 15, 190, 88, 989, 35, 477, 116, 209, 105, 6901, 190, 7067, 88, 196, 989, 6439, 35, 15049, 477, 2754, 172, 10207, 209, 4976, 336, 989, 6901, 1189, 190, 10877, 7067, 477, 248, 13529, 377, 44461, 989, 418, 6439, 79523, 105, 10244, 15049

Offset: 1

Phil Carmody, Mar 01 2013

nonn

For each n there are infinitely many numbers k for which n divides sigma(k) and phi(k). - Marius A. Burtea, Mar 28 2019

			Given A009223 = 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 4, 2, 6, 8, 1, 2, 3, ...
1 first divides A009223(1); 2 first divides A009223(3); 3 first divides A009223(14)=6.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (terms 1-388 from Marius A. Burtea, 389-2808 from David A. Corneth)

Cf. A009223, A222714.

[Min([n: n in [1..300000] | IsIntegral(SumOfDivisors(n)/m) and IsIntegral(EulerPhi(n)/m) ]): m in [1..70]]; // Marius A. Burtea, Mar 28 2019

v:=[];
for n in [1..60] do
m:=1;
        while  not EulerPhi(m) mod n  eq 0 or not SumOfDivisors(m) mod n  eq 0 do
           v[n]:=0;
           m:=m+1;
        end while;
     v[n]:=m;
end for;
v; // Marius A. Burtea, Mar 30 2019

Array[Block[{i = 1}, While[Mod[GCD[DivisorSigma[1, i], EulerPhi@ i], #] != 0, i++]; i] &, 50] (* Michael De Vlieger, Mar 28 2019 *)

a(n)={my(k=1); while(gcd(sigma(k), eulerphi(k))%n!=0, k++); k}

A222712 Record values of gcd(sigma(n), phi(n)) (A009223).

Original entry on oeis.org

1, 2, 4, 6, 8, 24, 48, 72, 120, 192, 240, 336, 720, 960, 1440, 1872, 2016, 2880, 3360, 4032, 5760, 6720, 6912, 7392, 8640, 10080, 17280, 20160, 34560, 40320, 44352, 60480, 69120, 74880, 95040, 96768, 100800, 120960, 134784, 201600, 241920, 322560, 354816, 411840, 483840

Offset: 1

Phil Carmody, Mar 01 2013

nonn

RECORDS transform of A009223.

Donovan Johnson, Table of n, a(n) for n = 1..500
N. J. A. Sloane, Transforms (The RECORDS transform returns both the high-water marks and the places where they occur).

Cf. A009223, A222711 (indexes where these records are attained).

mg=0;for(x=1,1000000,g=A009223(x);if(g>mg,print1(g", ");mg=g))

A222711 Numbers k such that gcd(sigma(k), phi(k)) (A009223) attains record values.

Original entry on oeis.org

1, 3, 12, 14, 15, 35, 105, 190, 248, 357, 616, 812, 1045, 3080, 3135, 3339, 4064, 5049, 8323, 8636, 10659, 12441, 16065, 19780, 20026, 23374, 24871, 29029, 50065, 58435, 64285, 87685, 124355, 132957, 137885, 140335, 248501, 263055, 317205, 353133, 423657, 596037, 655707, 734517, 894387

Offset: 1

Phil Carmody, Mar 01 2013

RECORDS transform of A009223.

Donovan Johnson, Table of n, a(n) for n = 1..500
N. J. A. Sloane, Transforms (The RECORDS transform returns both the high-water marks and the places where they occur).

Cf. A009223, A222712.

a[1] = 1; record = 1; a[n_] := a[n] = For[k = a[n-1] + 1, True, k++, g = GCD[DivisorSigma[1, k], EulerPhi[k]]; If[g > record, record = g; Return[k]]]; Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Oct 07 2013 *)
DeleteDuplicates[Table[{k,GCD[DivisorSigma[1,k],EulerPhi[k]]},{k,900000}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jun 22 2024 *)

mg=0;for(x=1,1000000,g=A009223(x);if(g>mg,print1(x", ");mg=g))

A222007 a(n+1) is the smallest prime factor of any (Product_{k=1..j} a(k)) + (Product_{k=j+1..n} a(k)) for j=0..n.

Original entry on oeis.org

2, 3, 5, 11, 41, 7, 23, 17, 19, 13, 37, 53, 73, 151, 29, 43, 31, 59, 71, 47, 79, 61, 107, 83, 103, 163, 109, 89, 101, 113, 67, 97, 137, 131, 139, 127, 229, 149, 173, 227, 179, 239, 181, 191, 193, 167, 197, 241, 277, 157, 233, 211, 397, 257, 271, 283, 251, 281, 313, 269, 347, 349, 317, 263, 379, 223, 367, 199, 353, 401, 421, 463, 293, 337, 383, 389, 331, 431, 359, 443

Offset: 1

Phil Carmody, Feb 23 2013

			For n=4, a = <2,3,5>, yielding sums <1+2*3*5, 2+3*5, 2*3+5, 2*3*5+1> = <31,17,11,31>, with least prime factor a(4)=11.

A modification of A000945, the Euclid-Mullin sequence, which looks only at factors from the j=n term.

prodsum(ls) = local(left=1, right=prod(x=1,#ls,ls[x]), o=vector(#ls)); for(x=1,#ls,left*=ls[x];right/=ls[x];o[x]=left+right);o
newlpf(v) = local(l=0, fs); for(x=1,#v,fs=factor(v[x],if(l>500000,0,l));if(!l||fs[1,1]

A219178 a(n) = first unlucky number removed at the n-th stage of Lucky sieve.

Original entry on oeis.org

2, 5, 19, 27, 45, 55, 85, 109, 139, 157, 175, 213, 255, 265, 337, 363, 387, 411, 423, 457, 513, 547, 597, 637, 675, 715, 789, 807, 843, 871, 907, 987, 1033, 1083, 1113, 1125, 1267, 1297, 1315, 1371, 1407, 1465, 1515, 1555, 1609, 1651, 1671, 1707, 1851, 1873, 1927, 1969

Offset: 1

Phil Carmody, Nov 15 2012

First numbers removed by each lucky number in the lucky number sieve. - This is the original definition of the sequence, still valid from a(2) onward.

a(1) = 2, because at the first stage of Lucky sieve, all even numbers are removed, of which 2 is the first one. - Antti Karttunen, Feb 26 2015

			1 and 2 are a special case in the lucky number sieve, (1 is the lucky number, but every 2nd element is removed) so are ignored [in the original version of the sequence, which started from a(2). Now we have a(1) = 2. - _Antti Karttunen_, Feb 26 2015]. The 2nd lucky number, 3, removes { 5, 11, ... } from the list, so a(2) = 5. The 3rd lucky number, 7, removes { 19, 39, ... } from the list, so a(3)=19.

Antti Karttunen, Table of n, a(n) for n = 1..333
Wikipedia, Lucky Number

Column 1 of A255543, Column 2 of A255545 (And apart from the first term, also column 2 of A255551).

Cf. A000959, A001248, A254100, A255549, A255550, A255553.

rows = 52; cols = 1; L = 2 Range[0, 10^4] + 1; A = Join[{2 Range[cols]}, Reap[For[n = 2, n <= rows, r = L[[n++]]; L0 = L; L = ReplacePart[L, Table[r i -> Nothing, {i, 1, Length[L]/r}]]; Sow[Complement[L0, L][[1 ;; cols]]]]][[2, 1]]]; Table[A[[n, 1]], {n, 1, rows}] (* Jean-François Alcover, Mar 15 2016 *)

(define (A219178 n) (A255543bi n 1)) ;; Code for A255543bi given in A255543.

From Antti Karttunen, Feb 26 2015: (Start)
a(n) = A255543(n,1).
Other identities.
For all n >= 2, a(n) = A255553(A001248(n)).
(End)

Term a(1) = 2 prepended, without changing the rest of sequence. Name changed, with the original, more restrictive definition moved to the Comments section. - Antti Karttunen, Feb 26 2015

A179113 Odd primes which can never divide 2^a+2^b+1.

Original entry on oeis.org

31, 89, 127, 223, 233, 431, 601, 881, 911, 1103, 1801, 2089, 2351, 3191, 3391, 4513, 5209, 6361, 8191, 9623, 9719, 11447, 11471, 13367, 14951, 15193, 15809, 18041, 18121, 18199, 18287, 20231, 23279, 23671, 39551, 43441, 50023, 53993, 54217, 55441, 55871, 59233

Offset: 1

Phil Carmody, Jan 04 2011

nonn

Contains the Mersenne primes M_p for p>3 as a subsequence, as 2^a+2^b cannot exceed 2^(p-1)+2^(p-2) which is less than 2^p-2 is p>3.

Mariusz Skałba conjectures that this sequence has density zero among all primes but contains infinitely many primes based on the following observations. For any prime p in this sequence, the multiplicative order of 2 modulo p is Tomohiro Yamada, Aug 08 2019

			31 is on the list as you can't sum any two of {1, 2, 4, 8, 16} to make 30 (mod 31).

Robert Israel, Table of n, a(n) for n = 1..129
Mariusz Skałba, Two conjectures on primes dividing 2^a+2^b+1, Elemente der Mathematik 59 (2004), issue 4, pp. 171-173.

N:= 10000; # to test the first N primes for membership
A179113:= proc(p)
          local x, R;
x:= 1; R:= {};
do
  R:= R union {p-1-x};
  if member(x,R) then return(false) end if;
  x:= 2*x mod p;
  if x = 1 then return(true) end if;
end do;
end proc;
select(A179113,[seq(ithprime(i),i=2..N)]);
# Robert Israel, May 19 2013

n = 10000; (* to test the first n primes for membership *) A179113[p_] := Module[{x = 1, r = {}}, While[True, r = r ~Union~ {p-1-x}; If[MemberQ[r, x], Return[False]]; x = Mod[2*x, p]; If[x == 1, Return[True]]]];Reap[Do[If[A179113[p], Print[p]; Sow[p]], {p, Prime /@ Range[2, n]}]][[2, 1]] (* Jean-François Alcover, Dec 02 2013, translated from Robert Israel's Maple program *)

forprime(p=3,1000,pol=x+O(x^p);t=2;while(t-1,pol+=x^t;t=t*2%p);pol2=pol*pol;if(!polcoeff(pol2,p-1),print1(p", ")))

More terms from Robert Israel, May 19 2013

A171245 Reciprocals of the deviation of continued fraction convergents from Pi.

Original entry on oeis.org

7, -790, 12016, -3748629, 1730431227, -3015428685, 8172837003, -34313104062, 114738541304, -620832653094, 2474837590227, -45157370341559, 1726856191927439, -6094457427533473, 12791098836150787, -51642743340781227, 325724575136738920, -1813676066719263909

Offset: 1

Phil Carmody, Dec 06 2009

sign

Clearly the terms alternate in sign, but the initial choice of sign was arbitrary.

			The first convergent to Pi, 3, is just over 1/7 too low, so 7 is the first term.
The second convergent, 22/7, is 0.001264+ too high, which is just over 1/790, so -790 is the second term.

Cf. A001203.

for(x=1,10,t=contfracpnqn(contfrac(Pi,x));print(truncate(1/(Pi-t[1,1]/t[2,1]))))

More terms from Jinyuan Wang, Aug 23 2021

A174106 Smallest primes such that when primes up to and including the n-th term in this sequence are summed, the result will be divisible by 10^n.

Original entry on oeis.org

5, 23, 35677, 106853, 632501, 31190879, 58369153, 707712517, 26219976521, 87424229843, 1642257355619, 2962734127453

Offset: 1

Phil Carmody, Mar 07 2010

In the long term, the prime sums' residues modulo 10^n are uniformly distributed, so one would expect the n-th term to be about the same size as the n-th prime.

			2+3+5 = 10, so 5 is the 1st term.
2+3+5+7+11+13+17+19+23 = 100, so 23 is the 2nd term.

cp's OEIS Frontend

User: Phil Carmody

A255724 List of numbers such that no number, nor its reverse, is in the concatenation of all previous terms.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Perl

Extensions

A222714 Smallest i such that prime(n) divides gcd(sigma(i), phi(i)) (cf. A009223).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A222713 Least number k such that n divides gcd(sigma(k), phi(k)) (A009223).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

A222712 Record values of gcd(sigma(n), phi(n)) (A009223).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A222711 Numbers k such that gcd(sigma(k), phi(k)) (A009223) attains record values.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A222007 a(n+1) is the smallest prime factor of any (Product_{k=1..j} a(k)) + (Product_{k=j+1..n} a(k)) for j=0..n.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A219178 a(n) = first unlucky number removed at the n-th stage of Lucky sieve.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

Extensions

A179113 Odd primes which can never divide 2^a+2^b+1.

Views

Author

Keywords

Comments

Examples

Links