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Least prime divisor of T(n) which does not divide any T(k) with k < n, or 1 if such a primitive prime divisor of T(n) does not exist, where T(n) is the n-th central trinomial coefficient given by A002426.
+10
11
1, 3, 7, 19, 17, 47, 131, 41, 43, 1279, 503, 113, 2917, 569, 198623, 14083, 26693, 201611, 42998951, 41931041, 52635749, 1296973, 169097, 1451, 1304394227, 107, 233, 173, 2062225210273, 719, 191, 31551555041, 6301, 563, 3769, 967, 9539, 5073466546857451, 4542977, 9739
COMMENTS
Conjecture: (i) a(n) > 1 for all n > 1.
(ii) For any integer n > 3, the n-th Motzkin number M(n) given by A001006 has a prime divisor which does not divide any M(k) with k < n.
EXAMPLE
a(11) = 503 since T(11) = 3*17*503 with the prime divisor 503 dividing none of T(1),...,T(10), but 3 divides T(2) = 3 and 17 divides T(5) = 51.
MATHEMATICA
T[n_]:=Sum[Binomial[n, 2k]*Binomial[2k, k], {k, 0, n/2}]
f[n_]:=FactorInteger[T[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[T[n]<2, Goto[cc]]; Do[Do[If[Mod[T[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}];
Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}];
Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 40}]
Least prime divisor of B(n) which does not divide any B(k) with k < n, or 1 if such a primitive prime divisor of B(n) does not exist, where B(n) is the n-th Bell number given by A000110.
+10
11
1, 2, 5, 3, 13, 7, 877, 23, 19, 4639, 22619, 37, 27644437, 1800937, 251, 241, 255755771, 19463, 271, 61, 24709, 17, 89, 123419, 367, 101, 157, 67, 75979, 107, 11, 179167, 5694673, 111509, 980424262253, 193, 44101, 5399, 6353, 3221
COMMENTS
Conjecture: (i) a(n) > 1 for all n > 1.
Conjecture: (ii) For any integer n > 2, the derangement number D(n) given by A000166 has a prime divisor dividing none of those D(k) with 1 < k < n.
MAPLE
a(4) = 3 since B(4) = 3*5 with 3 dividing none of B(1) = 1, B(2) = 2 and B(3) = 5.
MATHEMATICA
b[n_]:=BellB[n]
f[n_]:=FactorInteger[b[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[b[n]<2, Goto[cc]]; Do[Do[If[Mod[b[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 40}]
PROG
(Python)
# Python 3.2 or higher required.
from itertools import accumulate
from sympy import primefactors
A242171_list, bell, blist, b = [1], [1, 1], [1], 1
for _ in range(20):
blist = list(accumulate([b]+blist))
b = blist[-1]
fs = primefactors(b)
for p in fs:
if all([n % p for n in bell]):
break
else:
bell.append(b) # Chai Wah Wu, Sep 19 2014
Least prime divisor of the n-th central Delannoy number D(n) which does not divide any D(k) with k < n, or 1 if such a primitive prime divisor of D(n) does not exist.
+10
11
3, 13, 7, 107, 11, 89, 31, 265729, 19, 9887, 23, 113, 79, 373, 53, 3089, 151, 127, 719, 193, 43, 482673878761, 47, 61403, 109, 37889, 1223, 3251609, 59, 181, 22504880485262968151, 3598831, 67, 69593, 179, 13828116559, 4247285503, 1579, 19095283759, 619
COMMENTS
Conjecture:
(i) a(n) > 1 for all n > 0.
(ii) For any integer n > 0, the n-th Apéry number A(n) = Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2 has a prime divisor which does not divide any A(k) with k < n.
EXAMPLE
a(3) = 7 since D(3) = 3^2*7 with 7 dividing none of D(1) = 3 and D(2) = 13.
MATHEMATICA
d[n_]:=Sum[Binomial[n+k, k]*Binomial[n, k], {k, 0, n}]
f[n_]:=FactorInteger[d[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[d[n]<2, Goto[cc]]; Do[Do[If[Mod[d[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 40}]
Least prime p such that B_{2*n} == 0 (mod p) but there is no k < n with B_{2k} == 0 (mod p), or 1 if such a prime p does not exist, where B_m denotes the m-th Bernoulli number.
+10
10
1, 1, 1, 1, 5, 691, 7, 3617, 43867, 283, 11, 103, 13, 9349, 1721, 37, 17, 26315271553053477373, 19, 137616929, 1520097643918070802691, 59, 23, 653, 417202699, 577, 39409, 113161, 29, 2003, 31, 1226592271, 839, 101, 688531
COMMENTS
Conjecture: a(n) is prime for any n > 4.
It is known that (-1)^(n-1)*B_{2*n} > 0 for all n > 0.
See also A242194 for a similar conjecture involving Euler numbers.
EXAMPLE
a(14) = 9349 since the numerator of |B_{28}| is 7*9349*362903 with B_2*B_4*B_6*...*B_{26} not congruent to 0 modulo 9349, but B_{14} == 0 (mod 7).
MATHEMATICA
b[n_]:=Numerator[Abs[BernoulliB[2n]]]
f[n_]:=FactorInteger[b[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[b[n]<2, Goto[cc]]; Do[Do[If[Mod[b[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 35}]
(* Second program: *)
LPDtransform[n_, fun_] := Module[{d}, d[p_] := AllTrue[Range[n - 1], !Divisible[fun[#], p]&]; SelectFirst[FactorInteger[fun[n]][[All, 1]], d] /. Missing[_] -> 1];
A242193list[sup_] := Table[LPDtransform[n, Function[k, Abs[BernoulliB[2k]] // Numerator]], {n, 1, sup}]
PROG
(Sage)
def LPDtransform(n, fun):
d = lambda p: all(not p.divides(fun(k)) for k in (1..n-1))
return next((p for p in prime_divisors(fun(n)) if d(p)), 1)
A242193list = lambda sup: [LPDtransform(n, lambda k: abs(bernoulli(2*k)).numerator()) for n in (1..sup)]
Least prime divisor of E_{2*n} which does not divide any E_{2*k} with k < n, or 1 if such a primitive prime divisor of E_{2*n} does not exist, where E_m denotes the m-th Euler number given by A122045.
+10
8
1, 5, 61, 277, 19, 13, 47, 17, 79, 41737, 31, 2137, 67, 29, 15669721, 930157, 4153, 37, 23489580527043108252017828576198947741, 41, 137, 587, 285528427091, 5516994249383296071214195242422482492286460673697, 5639, 53, 2749, 5303, 1459879476771247347961031445001033, 6821509
COMMENTS
Conjecture: a(n) is prime for any n > 1.
It is known that (-1)^n*E_{2*n} > 0 for all n = 0, 1, ....
See also A242193 for a similar conjecture involving Bernoulli numbers.
EXAMPLE
a(4) = 277 since E_8 = 5*277 with 277 not dividing E_2*E_4*E_6, but 5 divides E_4 = 5.
MATHEMATICA
e[n_]:=Abs[EulerE[2n]]
f[n_]:=FactorInteger[e[n]]
p[n_]:=p[n]=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[e[n]<2, Goto[cc]]; Do[Do[If[Mod[e[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 30}]
(* Second program: *)
LPDtransform[n_, fun_] := Module[{}, d[p_, m_] := d[p, m] = AllTrue[ Range[m-1], ! Divisible[fun[#], p]&]; f[m_] := f[m] = FactorInteger[ fun[m]][[All, 1]]; SelectFirst[f[n], d[#, n]&] /. Missing[_] -> 1];
a[n_] := a[n] = LPDtransform[n, Function[k, Abs[EulerE[2k]]]];
PROG
(Sage) # uses[LPDtransform from A242193] A242194list = lambda sup: [LPDtransform(n, lambda k: euler_number(2*k)) for n in (1..sup)]
Least prime divisor of the n-th tangent number T_n which does not divide any T_k with k < n, or 1 if such a primitive prime divisor of T_n does not exist.
+10
7
1, 2, 1, 17, 31, 691, 43, 257, 73, 41, 89, 103, 2731, 113, 151, 37, 43691, 109, 174763, 61681, 337, 59, 178481, 97, 251, 157, 39409, 113161, 67, 1321, 266689, 641, 839, 101, 281, 433, 223, 229, 121369, 631
COMMENTS
Conjecture: (i) a(n) is prime for any n > 3.
(ii) For the n-th Springer number S_n given by A001586, if n is greater than one and not equal to 5, then S_n has a prime divisor which does not divide any S_k with k < n. See also A242193 and A242194 for similar conjectures involving Bernoulli numbers and Euler numbers.
EXAMPLE
a(4) = 17 since T_4 = 2^4*17 with 17 dividing none of T_1 = 1, T_2 = 2 and T_3 = 2^4.
MATHEMATICA
t[n_]:=(-1)^(n-1)*2^(2n)(2^(2n)-1)BernoulliB[2n]/(2n)
f[n_]:=FactorInteger[t[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[t[n]<2, Goto[cc]]; Do[Do[If[Mod[t[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 40}]
PROG
(Sage) # uses[LPDtransform from A242193] def Tnum(n): return (-1)^(n-1)*2^(2*n)*(2^(2*n)-1)*bernoulli(2*n)/(2*n)
A242195list = lambda sup: [LPDtransform(n, Tnum) for n in (1..sup)]
Least prime divisor of A005260(n) which does not divide any previous term A005260(k) with k < n, or 1 if such a primitive prime divisor of A005260(n) does not exist.
+10
4
2, 3, 41, 5, 7, 349, 61, 75617, 31, 13, 499, 643897693, 17, 19, 1729774061, 101, 2859112064587, 138407, 83, 167, 59, 29, 653, 257, 997540809461453561581, 347, 13679, 37, 160449179727717672892660463, 211, 151, 43, 97, 73, 47
COMMENTS
Conjecture: a(n) is prime for any n > 0. In general, for any r > 2, if n is large enough then f_r(n) = sum_{k=0..n}C(n,k)^r has a prime divisor which does not divide any previous terms f_r(k) with k < n.
MATHEMATICA
u[n_]:=Sum[Binomial[n, k]^4, {k, 0, n}]
f[n_]:=FactorInteger[u[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[u[n]<2, Goto[cc]]; Do[Do[If[Mod[u[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 35}]
Least prime divisor of the n-th Domb number D(n) which does not divide any D(k) with k < n, or 1 if such a primitive prime divisor of D(n) does not exist.
+10
4
2, 7, 1, 97, 11, 23, 19, 643, 659, 1753, 4922329, 613, 341447, 1193, 2213, 2040452101603, 491, 82461839, 733, 113, 1108394340978316050481, 1034497328556150923437, 593, 73, 17117, 804943, 422291, 1559, 858631, 337655751557
COMMENTS
Conjecture: a(n) is prime except for n = 3.
EXAMPLE
a(4) = 97 since D(4) = 2^2*7*97 with 97 dividing none of D(1) = 2^2, D(2) = 2^2*7 and D(3) = 2^8.
MATHEMATICA
d[n_]:=Sum[Binomial[n, k]^2*Binomial[2k, k]*Binomial[2(n-k), n-k], {k, 0, n}]
f[n_]:=FactorInteger[d[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[d[n]<2, Goto[cc]]; Do[Do[If[Mod[d[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 30}]
Least prime p such that H(n) == 0 (mod p) but H(k) == 0 (mod p) for no 0 < k < n, or 1 if such a prime p does not exist, where H(n) denotes the n-th harmonic number sum_{k=1..n}1/k.
+10
3
1, 3, 11, 5, 137, 7, 1, 761, 7129, 61, 97, 13, 29, 1049, 41233, 17, 37, 19, 7440427, 11167027, 18858053, 23, 583859, 577, 109, 34395742267, 521, 375035183, 4990290163, 31, 2667653736673, 2917, 269, 3583, 397, 1297, 10839223, 199, 737281, 41
COMMENTS
Conjecture: a(n) is prime except for n = 1, 7.
EXAMPLE
a(4) = 5 since H(4) = 25/12 == 0 (mod 5), but none of H(1) = 1, H(2) = 3/2 and H(3) = 11/6 is congruent to 0 modulo 5.
MATHEMATICA
h[n_]:=Numerator[HarmonicNumber[n]]
f[n_]:=FactorInteger[h[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[h[n]<2, Goto[cc]]; Do[Do[If[Mod[h[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 40}]
CROSSREFS
Cf. A000040, A001008, A002805, A242169, A242170, A242171, A242173, A242174, A242193, A242194, A242195, A242207, A242222.
Least prime divisor of 2^n - n which does not divide any 2^k - k with 0 < k < n, or 1 if such a primitive prime divisor of 2^n - n does not exist.
+10
2
1, 2, 5, 3, 1, 29, 11, 31, 503, 13, 7, 1021, 8179, 1637, 4679, 1, 8737, 131063, 524269, 262139, 2097131, 349, 131, 773, 271, 197, 457, 1493, 317, 17, 6733, 73, 41, 157109, 83, 53, 1741, 3329, 49977801259, 997, 149, 2199023255531, 61, 4398046511093, 3769453
COMMENTS
Conjecture: a(n) = 1 only for n = 1, 5, 16.
In constrast, a classical theorem of Bang asserts that if n > 1 is different from 6 then 2^n - 1 has a prime divisor which does not divide any 2^k - 1 with 0 < k < n.
REFERENCES
A. S. Bang, Taltheoretiske Undersgelser, Tidsskrift fur Mat. 4(1886), no. 5, 70--80, 130--137.
EXAMPLE
a(4) = 3 since 2^4 - 4 = 2^2*3 with 3 dividing none of 2^1 - 1 = 1, 2^2 - 2 = 2 and 2^3 - 3 = 5.
MATHEMATICA
u[n_]:=2^n-n
f[n_]:=FactorInteger[u[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[u[n]<2, Goto[cc]]; Do[Do[If[Mod[u[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 45}]
CROSSREFS
Cf. A000040, A000325, A112927, A242169, A242170, A242171, A242173, A242174, A242193, A242194, A242195, A242207.
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