A093777 -id:A093777 - cp's OEIS frontend

A093778 Primes p used as initial values for Euclid-Mullin sequences (variant A000945) instead of 2, such that all provide {p,2,3,5,7,11,13,q,...} initial segments in which the first six primes occur from 2nd to 7th terms.

99109, 159169, 189199, 399409, 459469, 609619, 669679, 699709, 819829, 1030039, 1090099, 1150159, 1270279, 1300309, 1390399, 1420429, 1810819, 1870879, 1930939, 1960969, 2021029, 2051059, 2141149, 2201209, 2261269, 2321329

Offset: 1

Labos Elemer, May 03 2004

nonn

			Initial segments of Euclid-Mullin sequences provided by
a[33]=3132139, a[34] and a[35] initial values:
{3132139,2,3,5,7,11,13,94058134171}}
{3282289,2,3,5,7,11,13,59}},
{3372379,2,3,5,7,11,13,29}}

Cf. A000945, A051308-A051334, A056756, A093777.

b[x_] :=First[Flatten[FactorInteger[Apply[Times, Table[b[j], {j, 1, x - 1}]] +1]]];b[1] = 1; Do[b[1] = Prime[j], el=8; If[Equal[Table[b[w], {w, 2, 7}], {2, 3, 5, 7, 11, 13}], Print[{j, Table[b[w], {w, 1, el}]}]], {j, 100000, 1000000}]

A093782 a(n) is the smallest initial value (a prime) for the Euclid-Mullin (EM) sequence in which the p=5 prime emerges as n-th term, i.e., arises at the n-th position.

Original entry on oeis.org

5, 0, 17, 19, 127, 61, 2, 31, 97, 13, 23, 269, 53, 239, 181, 449, 541, 11, 953, 1741, 179, 1889, 823, 3209, 13619, 383, 6971, 10331, 45959, 13721

Offset: 1

Labos Elemer, May 04 2004

The sequence is not monotonic and it seems that p=5 may arise at any position > 2. a(2)=0 means that 5 is never the 2nd term in an EM sequence of A000945-type because a(2)=2 or 3.

a(31)>=8581. [Sean A. Irvine, Oct 31 2011]

			The sequence for 17 is 17, 2, 5, ... where the 5 is at the third place, therefore a(3)=17.
For n=15 we have the sequence 181, 2, 3, 1087, 73, 7, 29, 151, 61, 98689, 11, 10929259909, 678859, 97, 5, ...
a(16) = 449 uses the sequence 449, 2, 29, 3, 7, 349, 190861819, 166273, 16091, 11, 3807491, 53, 17, 313, 23, 5, ...
The sequence for 11 is 11, 2, 23, 3, 7, 13, 10805892983887, 73, 6397, 19, 489407, 2753, 87491, 18618443, 5, ... with the 5 at the 18th place, so a(18)=11.

Cf. A000945, A051308-A051334, A056756, A093777-A093781.

Corrected by R. J. Mathar, Oct 06 2006
a(16) = 449 was conjectured by R. J. Mathar and confirmed by Don Reble, Oct 07 2006
a(19)-a(24) from David Wasserman, Apr 20 2007
a(25)-a(30) from Sean A. Irvine, Oct 30 2011

A094152 a(n) is the position of prime 7 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

Original entry on oeis.org

3, 3, 15, 1, 5, 6, 5, 24, 10, 6, 7, 6, 5, 4, 7, 5, 3, 5, 6, 16, 5, 6, 5, 28, 6, 3, 5, 36, 7, 15, 4, 15, 7, 7, 8, 7, 7, 5, 7, 14, 5, 6, 19, 16, 17, 5, 4, 12, 5, 8, 10, 17, 5, 5, 8, 10, 3, 5, 7, 30, 5, 5, 20, 3, 5, 6, 6, 4, 9, 9, 3, 9, 5, 6, 8, 8

Offset: 1

Labos Elemer, May 05 2004

nonn

			n=8: p(8)=19, the corresponding EM sequence is A051312 in which p=7 arises at the 24th position as follows:
{19, 2, 3, 5, 571, 271, 457, 397, 1123, 23, 103, 42572757267735264511, 313, 17, 16013177, 7951, 1259, 41, 1531, 11, 83, 53, 67, 7, 21397}, thus a(8)=24.

Sean A. Irvine, Table of n, a(n) for n = 1..3505

Cf. A000945, A051308-A051334, A056756, A093777-A093784.

More terms from Sean A. Irvine, Sep 20 2012

A094153 a(n) is least prime p such that 7 is the n-th term in the Euclid-Mullin sequence starting at p, or 0 if no such prime p exists.

Original entry on oeis.org

7, 0, 2, 43, 11, 13, 31, 149, 347, 23, 439, 223, 461, 173, 5, 71, 197, 1153, 191, 307, 1657, 971, 9473, 19, 2399, 1607, 6781, 89, 9187, 281, 23623, 15077, 25579, 17203

Offset: 1

Labos Elemer, May 05 2004

The sequence is not monotonic. Compare to A093882.

Next term exceeds 50000. - Sean A. Irvine, Jan 12 2012

			a(5)=11 because p=7 first arises in EM at position 5, which is initiated with 11: {11,2,23,3,7,10627,433}; see A051309.

Cf. A000945, A051308-A051334, A056756, A093777-A093783.

Definition clarified, terms corrected and extended by Sean A. Irvine, Apr 15 2011
More terms from Sean A. Irvine, May 22 2011
25579 and 17203 from Sean A. Irvine, Jan 11 2012

A093779 a(n) is the position of prime 3 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

Original entry on oeis.org

2, 1, 4, 3, 4, 3, 4, 3, 4, 4, 3, 3, 4, 3, 4, 4, 5, 3, 3, 4, 3, 3, 4, 4, 3, 5, 3, 4, 3, 4, 3, 4, 4, 3, 5, 3, 3, 3, 4, 4, 4, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 5, 3, 3, 4, 3, 4, 3, 5, 3, 4, 3, 3, 4, 3, 5, 4, 3, 3, 3, 5, 5, 3, 4, 3, 4, 3, 4, 3, 3, 4, 4, 3, 5, 3, 4, 6, 3, 4, 3, 5, 4, 5, 3, 3, 3, 4, 5, 4, 3

Offset: 1

Labos Elemer, May 03 2004

nonn

			p=3 arises first as n-th term for n=1,2,3,4 as follows: {3,2,7,43,13,53,5}, {2,3,7,43,13,53,5}, {7,2,3,43,13,53,5}, {5,2,11,3,331,19}, ... i.e., started at suitable initial primes;
p=2 arises always as 2nd or once as first term in case of various EM-sequences.

Cf. A000945, A051308-A051334, A056756, A093777, A093778.

A093780 a(n) is the smallest prime used as initial value for Euclid-Mullin (EM) sequence (of variant A000945), such that in the corresponding EM-sequence the p=3 prime arises at the n-th position.

Original entry on oeis.org

3, 2, 7, 5, 59, 479, 821, 1871, 17393, 43019, 284783, 1572149, 2737793, 32938853, 24254639

Offset: 1

Labos Elemer, May 03 2004

			p=3 arises first as n-th term for n=1,2,3,4,...,9th as follows:
{3,2,7,43,13,53,5},{2,3,7,43,13,53,5},{7,2,3,43,13,53,5},
{5,2,11,3,331,19},{269,2,7,3767,3,42559567},{479,2,7,19,5,3},
{821,2,31,109,7,509,3},{1871,2,19,7,37,13,23,3},
{17393,2,43,37,7,4129,13,5,3},

Cf. A000945, A051308-A051334, A056756, A093777-A093779.

More terms from David Wasserman, Apr 12 2007

A093781 a(n) is the position of the prime 5 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

Original entry on oeis.org

7, 7, 1, 7, 18, 10, 3, 4, 11, 7, 8, 8, 10, 7, 3, 13, 8, 6, 7, 8, 6, 4, 7, 8, 9, 4, 6, 3, 4, 11, 5, 8, 3, 4, 4, 8, 8, 13, 3, 10, 21, 15, 6, 8, 3, 4, 13, 5, 3, 4, 8, 14, 6, 10, 3, 6, 12, 6, 10, 6, 6, 13, 8, 4, 6, 3, 11, 5, 3, 4, 13, 6, 10, 8, 4, 26, 8, 7, 11, 4, 7, 10, 7, 5, 4, 7, 16, 8, 7, 9, 3, 5, 5, 6

Offset: 1

Labos Elemer, May 04 2004

nonn

a(38) = 13 because prime(38) = 163 and the corresponding EM sequence is {163, 2, 3, 11, 7, 75307, 3931, 5399, 3041, 409, 179, 92958641873, 5, 2003, ...}, where 5 appears at the 13th position. - David Wasserman, Apr 19 2007

David Wasserman, Apr 19 2007, Table of n, a(n) for n = 1..1000

Cf. A000945, A051308-A051334, A056756, A093777-A093780.

Cf. A093782.

em(i) = local(p, c, n, f, q); p = prime(i); if (p == 5, return(1)); c = 1; n = p; while (1, c++; f = factor(n + 1, 2^31 - 1); q = f[1, 1]; if (!isprime(q), f = factor(n + 1); q = f[1, 1]); if (q == 5, return(c)); n *= q); \\ David Wasserman, Apr 19 2007

More terms from David Wasserman, Apr 19 2007

cp's OEIS Frontend

A093778 Primes p used as initial values for Euclid-Mullin sequences (variant A000945) instead of 2, such that all provide {p,2,3,5,7,11,13,q,...} initial segments in which the first six primes occur from 2nd to 7th terms.

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A093782 a(n) is the smallest initial value (a prime) for the Euclid-Mullin (EM) sequence in which the p=5 prime emerges as n-th term, i.e., arises at the n-th position.

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A094152 a(n) is the position of prime 7 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

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A094153 a(n) is least prime p such that 7 is the n-th term in the Euclid-Mullin sequence starting at p, or 0 if no such prime p exists.

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A093779 a(n) is the position of prime 3 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

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A093780 a(n) is the smallest prime used as initial value for Euclid-Mullin (EM) sequence (of variant A000945), such that in the corresponding EM-sequence the p=3 prime arises at the n-th position.

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A093781 a(n) is the position of the prime 5 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

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