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.

Showing 1-10 of 46 results. Next

A225702 Composite squarefree numbers n such that p-2 divides n+2 for each prime p dividing n.

Original entry on oeis.org

273, 54943, 67303, 199393, 831283, 1097305, 1363723, 1569103, 1590433, 3199579, 3282433, 3503773, 5645563, 5659333, 9260053, 9733843, 9984115, 10738033, 16645363, 19229533, 32168743, 37759363, 38645233, 50806585, 53825497, 56451373, 58327423, 62207173
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 1097305 are 5, 11, 71 and 281. We have that (1097305+2)/(5-2)= 365769, (1097305+2)/(11-2) = 121923, (1097305+2)/(71-2)= 15903 and (1097305+2)/(281-2) = 3933.

Crossrefs

Cf. A208728, A225703-A225720.

Programs

  • Maple
    with(numtheory); A225702:=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: A225702(10^9,2);
  • Mathematica
    t = {}; n = 0; len = -2; While[len <= 262, n++; {p, e} = Transpose[FactorInteger[n]]; If[Length[p] > 1 && Union[e] == {1} && Mod[n, 2] > 0 && Union[Mod[n + 2, p - 2]] == {0}, AppendTo[t, n]; len = len + Length[IntegerDigits[n]] + 2]]; t
  • PARI
    is(n,f=factor(n))=if(#f[,2]<3 || vecmax(f[,2])>1 || f[1,1]==2, return(0)); for(i=1,#f~, if((n+2)%(f[i,1]-2), return(0))); 1 \\ Charles R Greathouse IV, Nov 05 2017

Extensions

Extended by T. D. Noe, May 17 2013

A225720 Composite squarefree numbers n such that p+10 divides n-10 for each prime p dividing n.

Original entry on oeis.org

10, 79222, 206965, 784090, 1673122, 2227123, 4798090, 5202571, 9196330, 13146715, 15015430, 18213595, 19342333, 21735010, 27907435, 28234018, 28240090, 37394146, 38710990, 53990695, 54772453, 70646509, 79671826, 89678830, 107251990, 114572545, 115005187, 137245690
Offset: 1

Views

Author

Paolo P. Lava, May 13 2013

Keywords

Examples

			Prime factors of 2227123 are 19, 251 and 467. We have that (2227123-10)/(19+10) = 76797, (2227123-10)/(251+10) = 8533 and (2227123-10)/(467+10) = 4669.

Crossrefs

Cf. A208728, A225702-A225719.

Programs

  • Maple
    with(numtheory); A225720:=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: A225720(10^9,-10);
  • PARI
    is(n,f=factor(n))=if(#f[,2]<2 || vecmax(f[,2])>1, return(0)); for(i=1,#f~, if((n-10)%(f[i,1]+10), return(0))); 1 \\ Charles R Greathouse IV, Nov 05 2017

Extensions

a(20)-a(27) from Donovan Johnson, Nov 15 2013
a(28) from Charles R Greathouse IV, Nov 05 2017

A226111 Composite squarefree numbers n such that the ratio (n - 1/2)/(p(i) + 1/2) is an integer, where p(i) are the prime factors of n.

Original entry on oeis.org

260813, 960323, 4572113, 5991098, 18912713, 37481945, 68688458, 214337813, 1418459963, 1488523838, 1905782603, 1906387718, 2416383938, 3866147051, 6153859058, 6927221438, 10696723538, 12000312419, 24529142138, 43004079563, 43648495313, 54750300413
Offset: 1

Views

Author

Paolo P. Lava, May 27 2013

Keywords

Comments

Also composite squarefree numbers n such that (2*p(i)+1) | (2*n-1).

Examples

			The prime factors of 5991098 are 2, 103, 127 and 229. We see that (5991098 - 1/2)/(2 + 1/2) = 2396439, (5991098 - 1/2)/(103 + 1/2) = 57885, (5991098 - 1/2)/(127 + 1/2) = 46989 and (5991098 - 1/2)/(229 + 1/2) = 26105. Hence 5991098 is in the sequence.
The prime factors of 1123342 are 2, 11 and 51061. We see that(1123342 - 1/2)/(2 + 1/2) = 748895, (1123342 - 1/2)/(11 + 1/2) = 106985 but (1123342 - 1/2)/(51061 + 1/2) = 2246685/102121. Hence 1123342 is not in the sequence.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226112-A226114.

Programs

  • Maple
    with(numtheory); A226111:=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: A226111(10^9,1/2);

Extensions

a(8)-a(22) from Giovanni Resta, Jun 02 2013

A226114 Composite squarefree numbers n such that the ratio (n + 1/3)/(p(i) - 1/3) is an integer, where p(i) are the prime factors of n.

Original entry on oeis.org

1045, 1639605, 7343133, 7938133, 25615893, 282388773, 296251293, 346148733, 895445173, 1217200533, 1584568533, 2578055893, 3604398933, 4078150853, 5181367893, 5621460973, 7591692693, 8199401613, 9393224533, 9489314501, 12671984033, 12723857813, 14057815893
Offset: 1

Views

Author

Paolo P. Lava, May 27 2013

Keywords

Comments

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

Examples

			The prime factors of 1045 are 5, 11 and 19. We see that (1045 + 1/3)/(5 - 1/3) = 224, (1045 + 1/3)/(11 - 1/3) = 98 and (1045 + 1/3)/(19 - 1/3) = 56. Hence 1045 is in the sequence.
The prime factors of 1639605 are 3, 5, 11, 19 and 523. We see that (1639605 + 1/3)/(3 - 1/3) = 614852, (1639605 + 1/3)/(5 - 1/3) = 351344, (1639605 + 1/3)/(11 - 1/3) = 153713, (1639605 + 1/3)/(19 - 1/3) = 87836 and (1639605 + 1/3)/(523 - 1/3) = 3137. Hence 1639605 is in the sequence.
The prime factors of 1117965 are 3, 5 and 74531. We see that (1117965 + 1/3)/(3 - 1/3) = 419237, (1117965 + 1/3)/(5 - 1/3) = 239564 but (1117965 + 1/3)/(74531 - 1/3) = 419237/27949. Hence 1117965 is not in the sequence.

Links

Crossrefs

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

Programs

  • Maple
    with(numtheory); A226114:=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: A226114(10^9,1/3);

Extensions

a(6)-a(23) from Giovanni Resta, Jun 02 2013

A226020 Composite squarefree numbers n such that the ratio (n + 1/2)/(p(i) + 1/2) is an integer, where p(i) are the prime factors of n.

Original entry on oeis.org

13702, 42997, 1004062, 1684462, 38447662, 40243549, 70801087, 107728582, 409055062, 594021862, 760767262, 1045475437, 1104435202, 1471700587, 1686747562, 1920806662, 3136180162, 3469071937, 5291041297, 7239716347, 7903353667, 12738885862, 22711489762
Offset: 1

Views

Author

Paolo P. Lava, May 23 2013

Keywords

Comments

Also composite squarefree numbers n such that (2*p(i)+1) | (2*n+1).

Examples

			The prime factors of 13702 are 2, 13, 17 and 31. We see that (13702 + 1)/(2 + 1/2) = 5481, (13702 + 1/2)/(13 + 1/2) = 1015, (13702 + 1)/(17 + 1/2) = 783 and ( 13702 + 1/2)/(31 + 1/2) = 435. Hence 13702 is in the sequence.
The prime factors of 1123545 are 3, 5 and 74903. We see that
(1123545 + 1/2)/(3 + 1/2) = 321013, (1123545 + 1/2)/(5 + 1/2) = 204281 but (1123545 + 1/2)/(74903+ 1/2) = 321013/21401. Hence 1123545 is not in the sequence.

Links

Crossrefs

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

Programs

  • Maple
    with(numtheory); A226020:=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: A226020(10^9,1/2);

Extensions

a(9)-a(23) from Giovanni Resta, Jun 02 2013

A225506 -2-Knödel numbers.

Original entry on oeis.org

4, 6, 8, 10, 12, 24, 28, 30, 70, 88, 130, 238, 510, 754, 868, 910, 1330, 2068, 2590, 2728, 3304, 4002, 5110, 5406, 8554, 8710, 12958, 15748, 18430, 20878, 21238, 23902, 24178, 32422, 39928, 46870, 49210, 53590, 55678, 57358, 62248, 67858, 70414, 79378, 88198, 95038, 95758, 95788, 102238, 114478
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 -2. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+2) = 1 mod n.
All terms are even numbers.

Links

Crossrefs

Cf. A208728.
Cf. A225507, A225508, A225509, A225510, A225511, A225512, A225513, A225514.

Programs

  • Maple
    with(numtheory); ListA225506:=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: ListA225506(10^6,-2);
  • Mathematica
    Select[Range[10000], CompositeQ[#] && Divisible[# + 2, CarmichaelLambda[#]] &] (* Amiram Eldar, Mar 28 2019 *)
  • PARI
    is(n) = forprime(p=3, n, if (n%p != 0 && Mod(p,n)^(n+2) != 1, return(0))); 1;
seq(N) = {
  my(a=vector(N), k=0, n=4);
  while(k < N, if(is(n), a[k++] = n); n += 2);
  a;
};
seq(50) \\ Gheorghe Coserea, Dec 23 2018

Extensions

More terms from Gheorghe Coserea, Dec 23 2018

A225507 -3-Knödel numbers.

Original entry on oeis.org

9, 21, 45, 63, 105, 117, 273, 285, 585, 627, 765, 1365, 1449, 1677, 3705, 3885, 4221, 4485, 4797, 7137, 7565, 8109, 10197, 10545, 11445, 13065, 14637, 16965, 19437, 20805, 26061, 27645, 30573, 31317, 33705, 35853, 38805, 39897, 40887, 41181, 48633, 50505, 57057
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 -3. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+3) = 1 mod n.

Links

Crossrefs

Cf. A208728.
Cf. A225506, A225508, A225509, A225510, A225511, A225512, A225513, A225514.

Programs

  • Maple
    with(numtheory); ListA225507:=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: ListA225507(10^6,-3);
  • Mathematica
    Select[Range[10000], CompositeQ[#] && Divisible[# + 3, CarmichaelLambda[#]] &] (* Amiram Eldar, Mar 28 2019 *)

Extensions

Incorrect comment deleted by Joseph DeVincentis, Dec 04 2015
More terms from Amiram Eldar, Mar 28 2019

A225508 -4-Knödel numbers.

Original entry on oeis.org

4, 6, 8, 12, 14, 16, 20, 24, 40, 48, 56, 60, 66, 80, 104, 120, 140, 176, 204, 240, 260, 266, 476, 560, 696, 728, 920, 1020, 1040, 1292, 1508, 1634, 1736, 1820, 1976, 2320, 2544, 2660, 3416, 3440, 3848, 4136, 4686, 4756, 5180, 5456, 6188, 6608, 7280, 8004, 8816
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 -4. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+4) = 1 mod n.
All terms are even numbers.

Links

Crossrefs

Cf. A208728.
Cf. A225506, A225507, A225509, A225510, A225511, A225512, A225513, A225514.

Programs

  • Maple
    with(numtheory); ListA225508:=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: ListA225508(10^6,-4);
  • Mathematica
    Select[Range[10000], CompositeQ[#] && Divisible[# + 4, CarmichaelLambda[#]] &] (* Amiram Eldar, Mar 28 2019 *)

Extensions

More terms from Amiram Eldar, Mar 28 2019

A225509 -5-Knödel numbers.

Original entry on oeis.org

15, 55, 75, 91, 175, 247, 275, 715, 775, 1275, 1435, 2275, 2635, 3075, 3355, 4615, 6355, 6475, 7975, 8827, 9139, 10075, 10675, 11275, 11935, 13515, 14555, 21775, 26455, 28975, 30415, 31675, 32395, 43615, 46075, 47275, 52195, 59755, 64255, 77275, 78403, 81055
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 -5. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+5) = 1 mod n.

Links

Crossrefs

Cf. A208728.
Cf. A225506, A225507, A225508, A225510, A225511, A225512, A225513, A225514,

Programs

  • Maple
    with(numtheory); ListA225509:=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: ListA225509(10^6,-5);
  • Mathematica
    Select[Range[10000], CompositeQ[#] && Divisible[# + 5, CarmichaelLambda[#]] &] (* Amiram Eldar, Mar 28 2019 *)

Extensions

More terms from Amiram Eldar, Mar 28 2019

A225510 -6-Knödel numbers.

Original entry on oeis.org

4, 6, 8, 10, 12, 18, 24, 30, 36, 42, 44, 72, 78, 84, 90, 126, 168, 170, 210, 228, 234, 252, 264, 390, 504, 546, 570, 630, 714, 744, 924, 1110, 1170, 1254, 1530, 1548, 1596, 1638, 2262, 2574, 2604, 2730, 2898, 3354, 3978, 3990, 4674, 5544, 5688, 6204, 7254, 7410
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 -6. Composite numbers n > 0 such that if 1 < a < n and gcd(n,a) = 1 then a^(n+6) = 1 mod n.

Links

Crossrefs

Cf. A208728.
Cf. A225506, A225507, A225508, A225509, A225511, A225512, A225513, A225514.

Programs

  • Maple
    with(numtheory); ListA225510:=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: ListA225510(10^6,-6);
  • Mathematica
    Select[Range[10000], CompositeQ[#] && Divisible[# + 6, CarmichaelLambda[#]] &] (* Amiram Eldar, Mar 28 2019 *)
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