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 1-7 of 7 results.

A229276 Composite squarefree numbers n such that p-tau(n) divides n+sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

Original entry on oeis.org

6, 10, 15, 66, 145, 231, 435, 1221, 11571, 99093, 105502, 292434, 449854, 585429, 643858, 968014, 1372494, 1787091, 1939434, 4659114, 5524014, 5654334, 6250371, 6974007, 19495374, 19821714, 28488039, 34701369, 46183893, 81133734, 213352233, 230140869
Offset: 1

Views

Author

Paolo P. Lava, Sep 19 2013

Keywords

Comments

Subsequence of A120944.

Examples

			Prime factors of 435 are 3, 5, 29 and sigma(435) = 720, tau(435) = 8.
435 + 720 = 1155 and 1155 / (3 - 8) = -231, 1155 / (5 - 8) = -385, 1155 / (29 - 8) = 55.

Links

Crossrefs

Cf. A000005, A000203, A228299-A228302, A229273-A229275.

Programs

  • Maple
    with (numtheory); P:=proc(q) local a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 or a[i][1]=tau(n) then ok:=0; break;
else if not type((n+sigma(n))/(a[i][1]-tau(n)), integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(2*10^6);

Extensions

a(21)-a(33) from Giovanni Resta, Sep 20 2013
First term deleted by Paolo P. Lava, Sep 23 2013

A229274 Composite squarefree numbers n such that p+tau(n) divides n-sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

Original entry on oeis.org

51, 93, 177, 219, 303, 471, 597, 681, 723, 807, 849, 933, 1059, 1101, 1227, 1437, 1563, 1689, 1731, 1857, 1941, 1983, 2319, 2361, 2487, 2571, 2823, 2949, 2991, 3117, 3327, 3369, 3453, 3579, 3747, 3831, 3873, 3957, 4083, 4377, 4461, 4629, 4713, 4839, 4881
Offset: 1

Views

Author

Paolo P. Lava, Sep 19 2013

Keywords

Comments

Subsequence of A120944.

Examples

			Prime factors of 177 are 3, 59 and sigma(177) = 240 , tau(177) = 4.
177 - 240 = -63 and (-63) / (3 + 4) = -9, (-63) / (59 + 4) = -1.

Links

Crossrefs

Cf. A000005, A000203.
Cf. A228299, A228300, A228301, A228302, A229273, A229275, A229276.

Programs

  • Maple
    with (numtheory); P:=proc(q) local a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 then ok:=0; break;
else if not type((n-sigma(n))/(a[i][1]+tau(n)), integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^6);

Extensions

First term deleted by Paolo P. Lava, Sep 23 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 *)

A274445 a(n) is the smallest composite squarefree number k such that (p+n) | (k-1) for every prime p dividing k.

Original entry on oeis.org

385, 91, 65, 451, 33, 170171, 145, 1261, 161, 78409, 469, 294061, 649, 13051, 1921, 5251, 721, 8453501, 145, 300243, 1121, 47611, 3601, 1915801, 1057, 41311, 545, 5671, 1261, 19723133, 4321, 37759, 6913, 451, 4033, 102821, 1513, 40891, 11521, 1259497, 721, 364781, 145
Offset: 1

Views

Author

Paolo P. Lava, Jun 23 2016

Keywords

Examples

			For n=1, prime factors of 385 are 5, 7 and 11. (385 - 1)/(5 + 1) = 384/6 = 64, (385 - 1)/(7 + 1) = 384/8 = 48 and (385 - 1)/(11 + 1) = 384/12 = 32.
For n=2, prime factors of 91 are 7 and 13. (91 - 1)/(7 + 2) = 90/9 = 10 and (91 - 1)/(13 + 2) = 90/15 = 6.

Links

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274443, A274444, 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 not type((n-1)/(p[d][1]+k),integer) then ok:=0; break; fi; od;
if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
  • Mathematica
    t = Select[Range[10^6], SquareFreeQ@ # && CompositeQ@ # &]; Table[ SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k, Divisible[k - 1, # + n] &]]], {n, 17}] (* Michael De Vlieger, Jun 24 2016, Version 10 *)
  • PARI
    isok(k,n)=if (! issquarefree(k), return (0)); vp = factor(k) [,1]; if (#vp == 1, return (0)); for (i=1, #vp, if ((k-1) % (n+vp[i]), return (0));); 1;
a(n) = my(k=2); while (! isok(k,n), k++); k; \\ Michel Marcus, Jun 28 2016

Extensions

a(18), a(24), a(30) added by Giovanni Resta, Jun 23 2016
More terms from Michel Marcus, Jun 28 2016

A229275 Composite squarefree numbers n such that p+tau(n) divides n+sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

Original entry on oeis.org

10778, 16471, 17353, 439453, 1304443, 3719678, 9234253, 17270678, 20512335, 21179143, 50706307, 77292313, 95506557, 103081993, 104707029, 140419077, 240626953, 287947933, 822767689, 982374757, 1608154233, 1918313911, 2219891947, 2471777007, 2632397677
Offset: 1

Views

Author

Paolo P. Lava, Sep 19 2013

Keywords

Comments

Subsequence of A120944.

Examples

			Prime factors of 10778 are 2, 17, 317 and sigma(10778) = 17172, tau(10778) = 8.
10778 + 17172 = 27950 and 27950 / (2 + 8) = 2795, 27950 / (17 + 8) = 1118, 27950 / (317 + 8) = 86.

Crossrefs

Cf. A000005, A000203, A228299-A228302, A229273, A229274, A229276.

Programs

  • Maple
    with (numtheory); P:=proc(q) local a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 then ok:=0; break;
else if not type((n+sigma(n))/(a[i][1]+tau(n)), integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(2*10^6);

Extensions

a(4) corrected and a(7)-a(26) by Giovanni Resta, Sep 20 2013
First term deleted by Paolo P. Lava, Sep 23 2013

A274446 a(n) is the smallest composite squarefree number k such that (p+n) | (k+1) for all primes dividing k.

Original entry on oeis.org

399, 299, 55, 611, 143, 5549, 39, 155, 493, 615383, 713, 3247, 119, 1304489, 1333, 31415, 2599, 749, 2183, 440153, 155, 75499, 119, 168600949, 4223, 223649, 559, 66299, 6407, 15157, 3431, 85499, 799, 31589, 7313
Offset: 1

Views

Author

Paolo P. Lava, Jun 23 2016

Keywords

Examples

			Prime factors of 399 are 3, 7 and 19. (399 + 1) / (3 + 1) = 400 / 4 = 100, (399 + 1) / (7 + 1) = 400 / 8 = 50 and (399 + 1) / (19 + 1) = 400 / 20 = 20.
Prime factors of 299 are 13 and 23. (399 + 1) / (13 + 2) = 300 / 15 = 20 and (399 + 1) / (23 + 2) = 300 / 25 = 12.

Crossrefs

Cf. A208728, A225702-A225720, A226020, A226111-A226114, A226364, A226448, A228299-A228302, A229273-A229276, A229321-A229324, A274443, A274444, A274445.

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 not type((n+1)/(p[d][1]+k),integer) then ok:=0; break; fi; od;
if ok=1 then print(n); break; fi; fi; od; od; end: P(10^9);
  • Mathematica
    t = Select[Range[2000000], SquareFreeQ@ # && CompositeQ@ # &]; Table[SelectFirst[t, Function[k, AllTrue[First /@ FactorInteger@ k, Divisible[k + 1, # + n] &]]], {n, 23}] (* Michael De Vlieger, Jun 24 2016, Version 10 *)
  • PARI
    isok(k,n) = {if (! issquarefree(k), return (0)); vp = factor(k) [,1]; if (#vp == 1, return (0)); for (i=1, #vp, if ((k+1) % (n+vp[i]), return (0));); 1;}
a(n) = {my(k=2); while (! isok(k,n), k++); k;} \\ Michel Marcus, Jun 28 2016

Extensions

a(24) from Giovanni Resta, Jun 23 2016
Showing 1-7 of 7 results.

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