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 1-9 of 9 results.

A257750 Quasi-Carmichael numbers.

Original entry on oeis.org

35, 77, 143, 165, 187, 209, 221, 231, 247, 273, 299, 323, 357, 391, 399, 437, 493, 527, 561, 589, 598, 713, 715, 899, 935, 943, 989, 1015, 1073, 1105, 1147, 1189, 1247, 1271, 1295, 1333, 1517, 1537, 1547, 1591, 1595, 1705, 1729, 1739, 1763, 1829, 1885, 1886, 1927
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 07 2015

Keywords

Comments

Quasi-Carmichael numbers are squarefree composites n with the property that for every prime factor p of n, p+b divides n+b positively with b being any integer besides 0.
If b is negative, then it is always larger than 0 minus the square root of the corresponding Quasi-Carmichael number. But if b is positive, how large can it be in relation to its corresponding Quasi-Carmichael number? Conjecture: It is always smaller than the square root of the corresponding Quasi-Carmichael number.
Are 1885 and 1886 the only two consecutive integers such that both numbers are Quasi-Carmichael numbers?
From Robert G. Wilson v, Dec 05 2015: (Start)
The conjecture that b < sqrt(n) is false. Look at n = 87061 = 13*37*181, 87365 = 5*101*173, and 96473 = 13*41*181. Their b values are 299, 331, and 351, while the corresponding sqrt(n) values are 295, 295, and 310, respectively.
For b to result in (n+b)/(p+b) > 0 with n = P_1*p_2*...*p_i and P_1 < p_2 < ... < p_i, -p_1 < b < |(n-p_i^2)/p_i|. (n+b)/(p+b) >= b+1. Solve for b.
Less than 0.5% are even (A262252). Of course they are == 2 (mod 4).
Least k-almost prime quasi-Carmichael number with k>1: 35, 165, 6545, 179998, 7509579, ..., .
(End)

Examples

			a(1) = 35 because this is the first squarefree composite number n such that at least one integer b except 0 exists such that for every prime factor p of n applies that p+b divides n+b (-3): 35 = 5*7 and 2, 4 both divide 32.

Links

Crossrefs

Subsequences: A002997 (Carmichael numbers), A006972 (Lucas-Carmichael numbers), A029553 (-10), A029554 (-9), A029555 (-8), A029556 (-7), A029557 (-6), A029558 (-5), A029559 (-4), A029560 (-3), A029561 (-2), A029562 (+2), A029563 (+3), A029564 (+4), A029565 (+5), A029566 (+6), A029567 (+7), A029568 (+8), A029569 (+9), A029570 (+10), A029590 (Least quasi-Carmichael number of order n), A029591 (Least quasi-Carmichael number of order -n), A257751 (1 base), A257752 (2 bases), A257753 (3 bases), A257754 (4 bases), A257755 (5 bases), A257756 (6 bases), A257757 (7 bases), A258842 (8 bases), A257758 (first occurrences), A259282 (at least one negative base), A259283 (at least one positive base), A257759 (at least one negative base and at least one positive base).

Programs

  • Mathematica
    fQ[n_] := Block[{c = -1, fi = FactorInteger@ n, k, lmt, p}, If[Times @@ (Last@# & /@ fi) == 1 < Plus @@ (Last@# & /@ fi), p = First@# & /@ fi; k = -fi[[1, 1]] + 1; lmt = Abs[(n - fi[[-1, 1]]^2)/fi[[-1, 1]]]; While[k < lmt, If[ Union[ IntegerQ@# & /@ ((n + k)/(p + k))] == {True}, c++; If[c > 0, Goto [fini]]]; k++]]; Label[fini]; c > 0]; Select[ Range@ 2000, fQ] (* Robert G. Wilson v, Dec 05 2015 *)
  • PARI
    for(n=2,1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1),n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k>0, print1(n,", ")))))

Extensions

All terms less than 1000000 checked by Robert G. Wilson v, Dec 13 2015

A257751 Quasi-Carmichael numbers to exactly one base.

Original entry on oeis.org

35, 77, 143, 165, 187, 209, 231, 247, 273, 299, 357, 391, 399, 437, 493, 527, 561, 589, 598, 713, 715, 943, 989, 1015, 1073, 1189, 1247, 1295, 1333, 1537, 1547, 1705, 1729, 1739, 1829, 1886, 1927, 1961, 2015, 2021, 2257, 2279, 2387, 2397, 2419, 2451, 2479, 2501
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 07 2015

Keywords

Comments

See A259238 for the corresponding bases.

Examples

			a(1) = 35 because this is the first squarefree composite number n such that exactly one nonzero integer b exists such that for every prime factor p of n, p+b divides n+b (-3): 35=5*7 and 2, 4 both divide 32.

Links

Crossrefs

Cf. A257750 (every number of bases).
Cf. A257752, A257753, A257754, A257755, A257756, A257757, A258842 (2 to 8 bases).
Cf. A257758 (first occurrences).

Programs

  • PARI
    for(n=2, 1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1), n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==1, print1(n, ", ")))))

A257752 Quasi-Carmichael numbers to exactly two bases.

Original entry on oeis.org

221, 323, 899, 935, 1105, 1147, 1271, 1591, 1595, 1885, 2093, 2465, 2821, 4757, 4807, 4991, 5609, 5963, 6497, 7081, 7843, 9991, 10373, 10403, 10961, 11009, 12319, 13843, 14111, 16031, 17155, 17399, 17653, 17963, 19043, 19721, 20701, 24613, 27331, 28417, 29341
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 07 2015

Keywords

Examples

			a(1) = 221 because this is the first squarefree composite number n such that exactly two integers b except 0 exist such that for every prime factor p of n, p+b divides n+b (-11, -5): 221=13*17 and 2, 6 both divide 210 and 8, 12 both divide 216.

Links

Crossrefs

Cf. A257750 (every number of bases).
Cf. A257751, A257753, A257754, A257755, A257756, A257757, A258842 (1 and 3 to 8 bases).
Cf. A257758 (first occurrences).

Programs

  • PARI
    for(n=2, 1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1), n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==2, print1(n, ", ")))))

A257754 Quasi-Carmichael numbers to exactly four bases.

Original entry on oeis.org

60491, 61937, 65311, 76151, 116843, 127723, 159197, 164009, 168821, 194417, 272483, 284987, 329467, 364087, 369857, 370817, 385241, 389327, 395497, 407837, 423701, 431393, 465043, 509461, 613927, 837209, 853607, 881717, 999919, 1041541, 1117213, 1279903, 1294819
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 12 2015

Keywords

Examples

			a(1) = 60491 because this is the first squarefree composite number n such that exactly four integers b except 0 exist such that for every prime factor p of n, p+b divides n+b (-239, -236, -231, -191): 60491=241*251 and 2, 12 both divide 60252 and 5, 15 both divide 60255 and 10, 20 both divide 60260 and 50, 60 both divide 60300.

Links

Crossrefs

Cf. A257750 (every number of bases).
Cf. A257751, A257752, A257753, A257755, A257756, A257757, A258842 (1, 2, 3, 5, 6, 7 and 8 bases).
Cf. A257758 (first occurrences).

Programs

  • PARI
    for(n=2, 1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1), n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==4, print1(n, ", ")))))

A257757 Quasi-Carmichael numbers to exactly seven bases.

Original entry on oeis.org

777923, 1030189, 1060459, 4903309, 5493247, 5659637, 6431071, 6673087, 6778969, 9790577, 11390429, 11860969, 12053263, 12390319, 12602059, 21215011, 21842629, 22991989, 24005239, 39339667, 39929437, 40080661, 40761169, 42314449, 50979479, 51876007, 54345943
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 12 2015

Keywords

Comments

All known terms have only two prime factors, one slightly larger than the other.
a(435) = 7523021437 = 1597 * 1933 * 2437 is the first term which has more than two prime factors. - Hiroaki Yamanouchi, Sep 28 2015
a(5586) > 10^12. - Hiroaki Yamanouchi, Sep 28 2015

Examples

			a(1) = 777923 because this is the first squarefree composite number n such that exactly seven integers b except 0 exist such that for every prime factor p of n, p+b divides n+b (-879, -878, -875, -872, -867, -863, -839): 777923=881*883 and 2, 4 both divide 777044 and 3, 5 both divide 777045 and 6, 8 both divide 777048 and 9, 11 both divide 777051 and 14, 16 both divide 777056 and 18, 20 both divide 777060 and 42, 44 both divide 777084.

Links

Crossrefs

Cf. A257750 (every number of bases).
Cf. A257751, A257752, A257753, A257754, A257755, A257756, A258842 (1 to 6 and 8 bases).
Cf. A257758 (first occurrences).

Programs

  • PARI
    for(n=2, 1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1), n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==7, print1(n, ", ")))))

Extensions

a(16)-a(27) from Hiroaki Yamanouchi, Sep 26 2015

A257758 Least Quasi-Carmichael number to exactly n bases.

Original entry on oeis.org

35, 221, 1517, 60491, 34933, 189029, 777923, 182293, 11618977, 58076041, 268926877, 1047880741, 342323563, 447110449, 2987821321, 11951066641, 19719180049, 10178985781, 249381206761, 30512751277, 190703385391, 128931982141
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 12 2015

Keywords

Comments

Is this sequence infinite?
10^12 < a(23) <= 4361890724227, a(24) = 805822195351 and a(25) = 560433369241.

Examples

			a(4) = 60491 because this is the first squarefree composite number n such that exactly four integers except 0 exist such that for every prime factor p of n applies that p+b divides n+b (-239, -236, -231, -191): 60491=241*251 and 2, 12 both divide 60252 and 5, 15 both divide 60255 and 10, 20 both divide 60260 and 50, 60 both divide 60300.

Links

Crossrefs

Cf. A257750 (every number of bases).
Cf. A257751, A257752, A257753, A257754, A257755, A257756, A257757, A258842 (1 to 8 bases).

Programs

  • PARI
    for(d=1,9, n=1; until(k==d, n++; if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1),n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==d, print1(n,", "))))))

Extensions

a(10)-a(22) from Hiroaki Yamanouchi, Sep 26 2015

A258842 Quasi-Carmichael numbers to exactly eight bases.

Original entry on oeis.org

182293, 6536953, 13116283, 23337661, 55898473, 56624329, 66112261, 66355291, 66846751, 67239919, 75289033, 76222261, 93331321, 97594157, 110397013, 115175383, 146385797, 147111617, 157333573, 158029141, 159289241, 163825601, 181950817, 187826449, 207820831
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Jun 12 2015

Keywords

Comments

All known terms have only two prime factors whereby the second prime factor is only slightly larger than the first.
a(3384) > 10^12. - Hiroaki Yamanouchi, Sep 26 2015

Examples

			a(1) = 182293 because this is the first squarefree composite number n such that exactly eight integers except 0 exist such that for every prime factor p of n applies that p+b divides n+b (-419, -418, -413, -412, -405, -403, -373, -349): 182293=421*433 and 2, 14 both divide 181874 and 3, 15 both divide 181875 and 8, 20 both divide 181880 and 9, 21 both divide 181881 and 16, 28 both divide 181888 and 18, 30 both divide 181890 and 48, 60 both divide 181920 and 72, 84 both divide 181944.

Links

Crossrefs

Cf. A257750 (every number of bases).
Cf. A257751, A257752, A257753, A257754, A257755, A257756, A257757 (1 to 7 bases).
Cf. A257758 (first occurrences).

Programs

  • PARI
    for(n=2, 1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1), n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==8, print1(n, ", ")))))

Extensions

a(4)-a(25) from Hiroaki Yamanouchi, Sep 26 2015

A257755 Quasi-Carmichael numbers to exactly five bases.

Original entry on oeis.org

34933, 295927, 312157, 346777, 379231, 417091, 444853, 471773, 576077, 582133, 1384753, 1462579, 1687397, 1689991, 1713337, 1861289, 1944869, 3211183, 3654223, 4092493, 4358737, 5134531, 5410051, 5564557, 6863671, 7061321, 7343659, 7531889, 7622221, 7817591
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 12 2015

Keywords

Examples

			a(1) = 34933 because this is the first squarefree composite number n such that exactly five integers except 0 exist such that for every prime factor p of n applies that p+b divides n+b (-178, -175, -173, -157, -133): 34933=181*193 and 3, 15 both divide 34755 and 6, 18 both divide 34758 and 18, 20 both divide 34760 and 24, 36 both divide 34776 and 48, 60 both divide 34800.

Crossrefs

Cf. A257750 (every number of bases).
Cf. A257751, A257752, A257753, A257754, A257756, A257757, A258842 (1, 2, 3, 4, 6, 7 and 8 bases).
Cf. A257758 (first occurrences).

Programs

  • PARI
    for(n=2, 1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1), n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==5, print1(n, ", ")))))

A257756 Quasi-Carmichael numbers to exactly six bases.

Original entry on oeis.org

189029, 404471, 424663, 2595221, 4140901, 4197377, 4347209, 4528159, 4566193, 4631023, 4708819, 4864411, 5175589, 5311729, 6380651, 6400819, 6426029, 7117783, 8173877, 8915971, 10080589, 10460869, 10671173, 11094661, 11538313
Offset: 1

Views

Author

Tim Johannes Ohrtmann, May 12 2015

Keywords

Examples

			a(1) = 189029 because this is the first squarefree composite number n such that exactly six integers b except 0 exist such that for every prime factor p of n, p+b divides n+b (-419, -414, -407, -389, -365, -309): 189029=421*449 and 2, 30 both divide 188610 and 7, 35 both divide 188615 and 14, 42 both divide 188622 and 32, 60 both divide 188640 and 56, 84 both divide 188664 and 112, 140 both divide 188720.

Crossrefs

Cf. A257750 (every number of bases).
Cf. A257751, A257752, A257753, A257754, A257755, A257757, A258842 (1, 2, 3, 4, 5, 7 and 8 bases).
Cf. A257758 (first occurrences).

Programs

  • PARI
    for(n=2, 1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1), n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==6, print1(n, ", ")))))
Showing 1-9 of 9 results.

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