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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A225511 -7-Knödel numbers.

Original entry on oeis.org

33, 65, 245, 345, 1353, 1421, 1505, 2405, 3185, 4433, 4745, 6293, 6923, 7733, 8729, 9065, 9443, 9785, 15113, 16113, 18473, 19565, 21593, 30485, 30705, 32513, 35705, 42833, 45353, 50141, 55685, 57017, 64505, 66521, 67065, 73073, 79553, 80093, 83657, 91553, 96473
Offset: 1

Views

Author

Paolo P. Lava, May 09 2013

Keywords

Comments

Extension of k-Knodel numbers to k negative, in this case equal to -7. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+7) = 1 mod n.

Links

Crossrefs

Cf. A208728.
Cf. A225506, A225507, A225508, A225509, A225510, A225512, A225513, A225514.

Programs

  • Maple
    with(numtheory); ListA225511:=proc(q,k) local a,n,ok;
for n from 2 to q do if not isprime(n) then ok:=1; for a from 1 to n do
if gcd(a,n)=1 then if (a^(n-k)-1) mod n<>0 then ok:=0; break; fi; fi;
od; if ok=1 then print(n); fi; fi; od; end: ListA225511(10^6,-7);
  • Mathematica
    Select[Range[10000], CompositeQ[#] && Divisible[# + 7, CarmichaelLambda[#]] &] (* Amiram Eldar, Mar 28 2019 *)

Extensions

More terms from Amiram Eldar, Mar 28 2019

A225512 -8-Knödel numbers.

Original entry on oeis.org

4, 6, 8, 12, 16, 20, 22, 24, 28, 32, 40, 48, 52, 60, 80, 96, 112, 120, 132, 136, 160, 208, 240, 280, 322, 352, 364, 408, 480, 520, 532, 580, 680, 682, 952, 1036, 1120, 1312, 1392, 1456, 1612, 1768, 1840, 2040, 2080, 2332, 2584, 3016, 3172, 3268, 3472, 3640
Offset: 1

Views

Author

Paolo P. Lava, May 09 2013

Keywords

Comments

Extension of k-Knodel numbers to k negative, in this case equal to -8. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+8) = 1 mod n.

Links

Crossrefs

Cf. A208728.
Cf. A225506, A225507, A225508, A225509, A225510, A225511, A225513, A225514.

Programs

  • Maple
    with(numtheory); ListA225512:=proc(q,k) local a,n,ok;
for n from 2 to q do if not isprime(n) then ok:=1; for a from 1 to n do
if gcd(a,n)=1 then if (a^(n-k)-1) mod n<>0 then ok:=0; break; fi; fi;
od; if ok=1 then print(n); fi; fi; od; end: ListA225512(10^6,-8);
  • Mathematica
    Select[Range[10000], CompositeQ[#] && Divisible[# + 8, CarmichaelLambda[#]] &] (* Amiram Eldar, Mar 28 2019 *)
  • PARI
    is(n) = if (bigomega(n)>1, for (a=2, n-1, if (gcd(n,a)==1 && Mod(a,n)^(n+8)!=1, return (0))); return (1), return (0)) \\ Rémy Sigrist, Mar 03 2019

Extensions

More terms from Rémy Sigrist, Mar 03 2019

A225513 -9-Knödel numbers.

Original entry on oeis.org

9, 15, 21, 27, 39, 63, 135, 171, 189, 195, 231, 315, 351, 513, 651, 663, 819, 855, 999, 1197, 1755, 1881, 2223, 2295, 2331, 3111, 3591, 4095, 4347, 4599, 4995, 5031, 5301, 6327, 7161, 9471, 9855, 10431, 10791, 11115, 11655, 12663, 12987, 13455, 14091, 14391
Offset: 1

Views

Author

Paolo P. Lava, May 09 2013

Keywords

Comments

Extension of k-Knodel numbers to k negative, in this case equal to -9. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+9) = 1 mod n.

Links

Crossrefs

Cf. A208728.
Cf. A225506, A225507, A225508, A225509, A225510, A225511, A225512, A225514.

Programs

  • Maple
    with(numtheory); ListA225513:=proc(q,k) local a,n,ok;
for n from 2 to q do if not isprime(n) then ok:=1; for a from 1 to n do
if gcd(a,n)=1 then if (a^(n-k)-1) mod n<>0 then ok:=0; break; fi; fi;
od; if ok=1 then print(n); fi; fi; od; end: ListA225513(10^6,-9);
  • Mathematica
    Select[Range[10000], CompositeQ[#] && Divisible[# + 9, CarmichaelLambda[#]] &] (* Amiram Eldar, Mar 28 2019 *)

Extensions

More terms from Amiram Eldar, Mar 28 2019

A225514 -10-Knödel numbers.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 24, 26, 30, 50, 56, 102, 110, 150, 152, 182, 276, 330, 350, 494, 550, 650, 770, 962, 1190, 1230, 1430, 1550, 1650, 2550, 2870, 3050, 3410, 3752, 3770, 4510, 4550, 5270, 6150, 6650, 6710, 9230, 9350, 10010, 10850, 11526, 12710, 12950, 15950
Offset: 1

Views

Author

Paolo P. Lava, May 09 2013

Keywords

Comments

Extension of k-Knödel numbers to k negative, in this case equal to -10. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+10) = 1 mod n.

Links

Crossrefs

Cf. A208728.
Cf. A225506, A225507, A225508, A225509, A225510, A225511, A225512, A225513.

Programs

  • Maple
    with(numtheory); ListA225514:=proc(q,k) local a,n,ok;
for n from 2 to q do if not isprime(n) then ok:=1; for a from 1 to n do
if gcd(a,n)=1 then if (a^(n-k)-1) mod n<>0 then ok:=0; break; fi; fi;
od; if ok=1 then print(n); fi; fi; od; end: ListA225514(10^6,-10);
  • Mathematica
    Select[Range[10000], CompositeQ[#] && Divisible[# + 10, CarmichaelLambda[#]] &] (* Amiram Eldar, Mar 28 2019 *)

Extensions

More terms from Amiram Eldar, Mar 28 2019

A225711 Composite squarefree numbers n such that p(i)+1 divides n-1, where p(i) are the prime factors of n.

Original entry on oeis.org

385, 2737, 6061, 6721, 17641, 24769, 25201, 31521, 34561, 49105, 66385, 76609, 79401, 113221, 136081, 180481, 194833, 199801, 254881, 268801, 311905, 321937, 328321, 362881, 436645, 469201, 506521, 545905, 547561, 558145, 628705, 642505, 649153, 778261, 884305
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 24769 are 17, 31 and 47. We have that (24769-1)/(17+1) = 1376, (24769-1)/(31+1) = 774 and (24769-1)/(47+1) = 516.

Links

Crossrefs

Cf. A208728, A225702-A225710, A225712-A225720.

Programs

  • Maple
    with(numtheory); A225711:=proc(i,j) local c, d, n, ok, p, t;
for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][2]>1 or p[d][1]=j then ok:=0; break; fi;
if  not type((n+j)/(p[d][1]-j),integer) then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: A225711(10^9,-1);
  • Mathematica
    t = {}; n = 0; len = -2; While[len <= 262, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Union[Mod[n - 1, p + 1]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)

A226364 Composite squarefree numbers n such that the ratios (n - 1/3)/(p - 1/3) are integers for each prime p dividing n.

Original entry on oeis.org

308267, 1420467, 1445995, 46874667, 153810067, 324218667, 355724747, 393253747, 471957547, 618729307, 886489707, 901990059, 1062803467, 1525582667, 1735517355, 4306362667, 4815895467, 6528285867, 6634856107, 11460166667, 12364885867, 13330858667, 20628538667
Offset: 1

Views

Author

Paolo P. Lava, Jun 05 2013

Keywords

Comments

Also composite squarefree numbers n such that (3p - 1) | (3n - 1).

Examples

			The prime factors of 1445995 are 5, 19, 31 and 491. We see that (1445995 - 1/3)/(5 - 1/3) = 309856, (1445995 - 1/3)/(19 - 1/3) = 77464, (1445995 - 1/3)/(31 - 1/3) = 47152 and (1445995 - 1/3)/(491 - 1/3) = 2947. Hence 1445995 is in the sequence.
The prime factors of 1112307 are 3, 7 and 52967. We see that (1112307 - 1/3)/(3 - 1/3) = 417115, (1112307 - 1/3)/(7 - 1/3) = 166846 but (1112307 - 1/3)/(52967 - 1/3) = 166846/7945. Hence 1112307 is not in the sequence.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226448.

Programs

  • Maple
    with(numtheory); ListA226364:=proc(i, j) local c, d, n, ok, p;
for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][2]>1 or not type((n-j)/(p[d][1]-j),integer) then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: ListA226364(10^9,1/3);
  • PARI
    is(n,P)=n=3*n-1; for(i=1,#P,if(n%(3*P[i]-1),return(0))); 1
list(lim,P=[],n=1,mx=lim\2)=my(v=[],t);if(#P>1&&is(n,P), v=[n]); P=concat(P,0); forprime(p=2,min(lim,mx),P[#P]=p;t=list(lim\p,P,n*p,p-1);if(#t,v=concat(v,t))); v \\ Charles R Greathouse IV, Jun 07 2013

Extensions

a(5)-a(23) from Giovanni Resta, Jun 07 2013

A226448 Composite squarefree numbers k such that the ratios (k - 1/2)/(p - 1/2) are integers for each prime p dividing k.

Original entry on oeis.org

260054438, 597892523, 1200695738, 3287998643, 3423456563, 10524308498, 13292859563, 15646705718, 19441707170, 33309521438, 38848586123, 43312628678, 61899936935, 72422400713, 75439031063, 85338414662, 112419230963, 132624705038, 136084511063, 141236121758
Offset: 1

Views

Author

Paolo P. Lava and Giovanni Resta, Jun 07 2013

Keywords

Comments

Also composite squarefree numbers k such that (2p - 1) | (2k - 1).

Examples

			3287998643 is a term since it is equal to 743*787*5623 and 3287998643-1/2 divided by 743-1/2, 787-1/2 and 5623-1/2 gives 3 integers, namely 4428281, 4180545 and 584793.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364.

Programs

  • Maple
    with(numtheory); ListA226448:=proc(i, j) local c, d, n, ok, p;
for n from 2 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][2]>1 or not type((n-j)/(p[d][1]-j), integer) then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: ListA226448(10^9, 1/2); # Paolo P. Lava, Oct 06 2013
  • PARI
    is(n, P)=n=2*n-1; for(i=1, #P, if(n%(2*P[i]-1), return(0))); 1
list(lim, P=[], n=1, mx=lim\2)=my(v=[], t); if(#P>1&&is(n, P), v=[n]); P=concat(P, 0); forprime(p=2, min(lim, mx), P[#P]=p; t=list(lim\p, P, n*p, p-1); if(#t, v=concat(v, t))); v \\ Charles R Greathouse IV, Jun 07 2013

A225710 Composite squarefree numbers n such that p(i)-10 divides n+10, where p(i) are the prime factors of n.

Original entry on oeis.org

14, 22, 35, 55, 65, 77, 102, 110, 143, 165, 182, 221, 455, 494, 665, 935, 1001, 1173, 1430, 2717, 2795, 4505, 4526, 4862, 5957, 6479, 11526, 27521, 30485, 34661, 35126, 45917, 49715, 52910, 53846, 81686, 90574, 106865, 113477, 118745, 139073, 140822, 147095
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 34661 are 11, 23 and 137. We have that (34661+10)/(11-10) = 34671, (34661+10)/(23-10) = 2667 and (34661+10)/(137-10) = 273.

Crossrefs

Cf. A208728, A225702-A225709, A225711-A225720.

Programs

  • Maple
    with(numtheory); A225710:=proc(i,j) local c, d, n, ok, p, t;
for n from 1 to i do if not isprime(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][2]>1 or p[d][1]=j then ok:=0; break; fi;
if  not type((n+j)/(p[d][1]-j),integer) then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: A225710(10^9,10);
  • Mathematica
    t = {}; n = 0; len = -2; While[len <= 262, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Union[Mod[n + 10, p - 10]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t (* T. D. Noe, May 17 2013 *)

Extensions

Extended by T. D. Noe, May 17 2013

A274443 Least composite squarefree number k such that (p-n) | (k-1) for all primes p dividing n.

Original entry on oeis.org

561, 21, 85, 15, 21, 35, 33, 21, 65, 91, 57, 91, 133, 55, 161, 91, 57, 133, 33, 253, 65, 91, 145, 115, 217, 451, 161, 703, 253, 551, 561, 253, 481, 217, 129, 451, 301, 1081, 161, 1189, 145, 989, 217, 235, 481, 703, 649, 329, 265, 1081, 1121, 1219, 145, 1037, 721
Offset: 1

Views

Author

Paolo P. Lava, Jun 23 2016

Keywords

Examples

			Prime factors of 561 are 3, 11 and 17: (561 - 1) / (3 - 1) = 560 / 2 = 280, (561 - 1) / (11 - 1) = 560 / 10 = 56 and (561 - 1) / (17 - 1) = 560 / 16 = 35.
Prime factors of 21 are 3 and 7: (21 - 1) / (3 - 2) = 20 / 1 = 20, (21 - 1) / (7 - 2) = 20 / 5 = 4.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274444, A274445, A274446.

Programs

  • Maple
    with(numtheory); P:=proc(q) local d,k,n,ok,p;
for k from 1 to q do for n from 2 to q do
if not isprime(n) and issqrfree(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][1]=k then ok:=0; break; else
if not type((n-1)/(p[d][1]-k),integer) then ok:=0; break; fi; fi; od;
if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
  • Mathematica
    t = Select[Range@2000, SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k, If[# == 0, False, Divisible[k - 1, #]] &[# - n] &]]], {n, 55}] (* Michael De Vlieger, Jun 24 2016, Version 10 *)

A274444 a(n) = smallest composite squarefree number k such that (p-n) | (k+1) for all primes dividing k.

Original entry on oeis.org

15, 65, 35, 15, 21, 35, 15, 35, 35, 77, 35, 55, 55, 143, 119, 51, 95, 155, 55, 323, 95, 119, 39, 391, 87, 209, 119, 299, 143, 341, 319, 629, 259, 899, 407, 185, 119, 299, 287, 1517, 203, 799, 159, 155, 407, 1189, 119, 517, 341, 1763, 1363, 629, 335, 2491, 493, 3599
Offset: 1

Views

Author

Paolo P. Lava, Jun 23 2016

Keywords

Examples

			a(1) = 15: Prime factors of 15 are 3 and 5: (15 + 1) / (3 - 1) = 16 / 2 = 8 and (15 + 1) / (5 - 1) = 16 / 4 = 4.
a(2) = 6: Prime factors of 65 are 5 and 13: (65 + 1) / (5 - 2) = 66 / 3 = 22 and (65 + 1) / (13 - 2) = 66 / 11 = 6.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274443, A274445, A274446.

Programs

  • Maple
    with(numtheory); P:=proc(q) local d,k,n,ok,p;
for k from 1 to q do for n from 2 to q do
if not isprime(n) and issqrfree(n) then p:=ifactors(n)[2]; ok:=1;
for d from 1 to nops(p) do if p[d][1]=k then ok:=0; break; else
if not type((n+1)/(p[d][1]-k),integer) then ok:=0; break; fi; fi; od;
if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
  • Mathematica
    t = Select[Range[10^4], SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k,
If[# == 0, False, Divisible[k + 1, #]] &[# - n] &]]], {n, 56}] (* Michael De Vlieger, Jun 24 2016, Version 10 *)
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