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 1-8 of 8 results.

A279097 Numbers k such that prime(k) divides primorial(j) + 1 for some j.

Original entry on oeis.org

1, 2, 4, 8, 11, 17, 18, 21, 25, 32, 34, 35, 39, 40, 42, 47, 48, 58, 59, 63, 65, 66, 67, 69, 90, 91, 97, 105, 110, 122, 140, 144, 151, 152, 162, 166, 168, 173, 174, 175, 177, 179, 180, 186, 205, 207, 208, 210, 211, 218, 221, 233, 243, 249, 256, 260, 261, 262
Offset: 1

Views

Author

Jon E. Schoenfield, Mar 24 2017

Keywords

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
Primorial(j) + 1 is the j-th Euclid number, A006862(j).

Examples

			1 is in the sequence because primorial(0) + 1 = 1 + 1 = 2 is divisible by prime(1) = 2.
4 is in the sequence because primorial(2) + 1 = 2*3 + 1 = 7 is divisible by prime(4) = 7.
8 is in the sequence because primorial(7) + 1 = 2*3*5*7*11*13*17 + 1 = 510511 is divisible by prime(8) = 19.
59 is in the sequence because primorial(7) + 1 = 510511 is divisible by prime(59) = 277 (and primorial(17) + 1 = 1922760350154212639071 is divisible by prime(59) as well).
5 is not in the sequence because there is no number j such that primorial(j) + 1 is divisible by prime(5) = 11:
    primorial(1) + 1 = 2       + 1 =   3 == 3 (mod 11)
    primorial(2) + 1 = 2*3     + 1 =   7 == 7 (mod 11)
    primorial(3) + 1 = 2*3*5   + 1 =  31 == 9 (mod 11)
    primorial(4) + 1 = 2*3*5*7 + 1 = 211 == 2 (mod 11)
and primorial(j) + 1 = 2*...*11*... + 1  == 1 (mod 11) for all j >= 5.

Links

Crossrefs

Cf. A000040, A002110, A006862, A113165, A279098, A279099, A283928.

Programs

  • Mathematica
    np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[262], np[#] > 0 &] (* Giovanni Resta, Mar 29 2017 *)

A279098 Numbers k such that prime(k) divides primorial(j) + 1 for exactly one integer j.

Original entry on oeis.org

1, 2, 4, 8, 11, 17, 18, 21, 25, 32, 34, 35, 39, 40, 42, 47, 48, 58, 63, 65, 66, 67, 69, 90, 91, 97, 105, 110, 122, 140, 144, 151, 152, 162, 166, 168, 173, 174, 175, 179, 180, 186, 205, 207, 208, 210, 211, 218, 233, 243, 249, 256, 261, 262, 297, 308, 316, 318
Offset: 1

Views

Author

Jon E. Schoenfield, Mar 24 2017

Keywords

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
Primorial(j) + 1 is the j-th Euclid number, A006862(j).

Examples

			59 is not in this sequence because both primorial(7) + 1 = 510511 and primorial(17) + 1 = 1922760350154212639071 are divisible by prime(59) = 277.

Links

Crossrefs

Subsequence of A279097 (which also includes numbers k such that prime(k) divides primorial(j) + 1 for more than one integer j).
Cf. A000040, A002110, A006862, A113165, A279099, A283928.

Programs

  • Mathematica
    np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[262], np[#] == 1 &] (* Giovanni Resta, Mar 29 2017 *)

A279099 Numbers k such that prime(k) divides primorial(j) + 1 for exactly two integers j.

Original entry on oeis.org

59, 177, 221, 260, 285, 431, 476, 489, 625, 653, 686, 860, 957, 1320, 1334, 1734, 1987, 2140, 2215, 2854, 2991, 3051, 3341, 3455, 3535, 3591, 3645, 3695, 3798, 4020, 4032, 4079, 4612, 4614, 4856, 4904, 5019, 5234, 5263, 5842, 6178, 6184, 6491, 6639, 6745, 7151
Offset: 1

Views

Author

Jon E. Schoenfield, Mar 24 2017

Keywords

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
Primorial(j) + 1 is the j-th Euclid number, A006862(j).

Examples

			59 is in this sequence because prime(59) = 277 divides primorial(j) + 1 for exactly two integers j: 7 and 17.
436 is not in this sequence because prime(436) = 3041 divides primorial(j) + 1 for exactly three integers j: 206, 263, and 409.

Links

Crossrefs

Subsequence of A279097 (which includes all numbers k such that prime(k) divides primorial(j) + 1 for one or more integers j); cf. A279098 (exactly one integer j).
Cf. A000040, A002110, A006862, A113165, A279098, A283928.

Programs

  • Mathematica
    np[1]=1; np[n_] := Block[{c=0, p=Prime[n], trg, x=1}, trg = p-1; Do[x = Mod[x Prime[k], p]; If[trg == x, c++], {k, n-1}]; c]; Select[Range[1000], np[#] == 2 &] (* Giovanni Resta, Mar 29 2017 *)

A283928 Numbers k such that prime(k) divides primorial(j) + 1 for exactly three integers j.

Original entry on oeis.org

436, 2753, 13396, 19960, 24293, 26157, 58492, 58723, 61935, 121992, 136592, 145803, 149027, 159752, 179811, 180776, 184575, 194499, 262321, 268645, 280911, 315198, 327876, 339951, 364307, 390394, 413010, 433626, 444744, 492661, 510412, 518156, 541925, 542177
Offset: 1

Views

Author

Jon E. Schoenfield, Mar 24 2017

Keywords

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
Primorial(j) + 1 is the j-th Euclid number, A006862(j).

Examples

			436 is in this sequence because prime(436) = 3041 divides primorial(j) + 1 for exactly three integers j: 206, 263, and 409.
180707 is not in this sequence because prime(180707) = 2464853 divides primorial(j) + 1 for exactly five integers j: 75366, 79914, 139731, 139990, and 175013. - _Jon E. Schoenfield_, Mar 30 2017

Links

Crossrefs

Subsequence of A279097 (which includes all numbers k such that prime(k) divides primorial(j) + 1 for one or more integers j); cf. A279098 (exactly one integer j), A279099 (exactly two).
Cf. A000040, A002110, A006862, A113165.

Programs

  • Magma
    countReqd:=3; kMaxTest:=20000; P:=PrimesInInterval(2,NthPrime(kMaxTest)); itos:=IntegerToString; a:=[]; for k in [1..kMaxTest] do p:=P[k]; pMinus1:=p-1; primorialModp:=1; jSuccess:=[]; if primorialModp eq pMinus1 then jSuccess:=[1]; end if; for j in [1..k-1] do primorialModp:=(primorialModp*P[j]) mod p; if primorialModp eq pMinus1 then jSuccess[#jSuccess+1]:=j; end if; end for; if #jSuccess eq countReqd then a[#a+1]:=k; "a("*itos(#a)*") = " * itos(k) * "; successes at j =", jSuccess; end if; end for; a; // Jon E. Schoenfield, Mar 25 2017

Extensions

a(10)-a(34) from Jon E. Schoenfield, Apr 02 2017

A297891 Numbers that divide exactly two Euclid numbers.

Original entry on oeis.org

277, 1051, 1381, 1657, 1867, 3001, 3373, 3499, 4637, 4877, 5147, 6673, 7547, 10859, 10987, 14797, 17291, 18749, 19531, 25939, 27337, 27953, 31013, 32203, 32983, 33547, 34123, 34591, 35747, 38047, 38197, 38711, 44293, 44357, 47059, 47569, 48809, 51151, 51437
Offset: 1

Views

Author

Jon E. Schoenfield, Jan 07 2018

Keywords

Comments

The k-th Euclid number, A006862(k), is 1 plus the product of the first k primes, i.e., 1 + A002110(k). A113165 lists the numbers (> 1) that divide at least one Euclid number; a(1) = 277 = A113165(19); a(2) = 1051 = A113165(41); a(53) = 92143 = A113165(995).
Up to N = 10^5, roughly 5% of the terms in A113165 are also in this sequence. Does that ratio continue to hold as N increases?
It appears that the vast majority of terms in A113165 are prime, but that sequence contains a number of composite numbers as well, beginning with A113165(59) = 1843 = 19*97, A113165(125) = 5263 = 19*277, A113165(195) = 10147 = 73*139, and A113165(231) = 12629 = 73*173. But do any composites divide more than one Euclid number?

Examples

			a(1) = 277 because 277 is the smallest number that divides exactly two Euclid numbers: 1 + 2*3*5*7*11*13*17 = 510511 and 1 + 2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59 = 1922760350154212639071.

Links

Crossrefs

Cf. A002110 (primorials), A006862 (Euclid numbers), A113165 (numbers > 1 that divide Euclid numbers).

A297893 Numbers that divide exactly three Euclid numbers.

Original entry on oeis.org

3041, 24917, 144671, 224251, 278191, 301927, 726071, 729173, 772691, 1612007, 1822021, 1954343, 2001409, 2157209, 2451919, 2465917, 2522357, 2668231, 3684011, 3779527, 3965447, 4488299, 4683271, 4869083, 5244427, 5650219, 6002519, 6324191, 6499721, 7252669
Offset: 1

Views

Author

Jon E. Schoenfield, Jan 07 2018

Keywords

Comments

A113165 lists numbers those numbers (> 1) that divide at least one Euclid number; A297891 lists those that divide exactly two Euclid numbers.
Is this sequence infinite?
Does this sequence contain any nonprimes?
Are there any numbers > 1 that divide more than three Euclid numbers?
The first numbers that divide 4 and 5 Euclid numbers are 15415223 and 2464853, respectively. - Giovanni Resta, Jun 26 2018

Examples

			a(1) = 3041 because 3041 is the smallest number that divides exactly three Euclid numbers: 1 + A002110(206), 1 + A002110(263), and 1 + A002110(409); these numbers have 532, 712, and 1201 digits, respectively.

Links

Crossrefs

Cf. A002110 (primorials), A006862 (Euclid numbers), A113165 (numbers > 1 that divide Euclid numbers), A297891 (numbers > 1 that divide exactly two Euclid numbers).

Extensions

a(14)-a(30) from Giovanni Resta, Jun 26 2018

A284754 a(n) is the smallest number k such that prime(k) divides primorial(j) + 1 for exactly n integers j.

Original entry on oeis.org

1, 59, 436, 995752, 180707
Offset: 1

Views

Author

Jon E. Schoenfield, Apr 01 2017

Keywords

Comments

As used here, "primorial(j)" refers to the product of the first j primes, i.e., A002110(j).
Primorial(j) + 1 is the j-th Euclid number, A006862(j).
a(n) > 10^7 for n > 5. - Giovanni Resta, Apr 03 2017

Examples

			a(1) = 1 because the first prime, prime(1) = 2, divides primorial(j) + 1 for exactly one integer j, namely, j = 0 (since primorial(0) = 1).
a(2) = 59 because prime(59) = 277 divides primorial(j) + 1 for exactly two integers j (i.e., 7 and 17), and 59 is the smallest integer for which this is the case.
a(3) = 436 because prime(436) = 3041 divides primorial(j) + 1 for exactly three integers j (i.e., 206, 263, and 409), and 436 is the smallest integer for which this is the case.
a(5) = 180707 because prime(180707) = 2464853 divides primorial(j) + 1 for exactly five integers j (i.e., 75366, 79914, 139731, 139990, and 175013), and 180707 is the smallest integer for which this is the case.

Crossrefs

Cf. A000040, A002110, A006862, A113165, A279097, A279098, A279099, A283928.

Extensions

a(4) from Giovanni Resta, Apr 02 2017

A297894 Composite numbers that divide at least one Euclid number.

Original entry on oeis.org

1843, 5263, 10147, 12629, 24047, 26869, 30031, 136109, 189001, 356189, 510511, 648077, 709493, 960359, 1293109, 1459817, 1513817, 1755431, 2263607, 2290129, 2578327, 2825041, 3173707, 3415703, 3440471, 4629071, 5007641, 5497781, 5698237, 6021971, 8614843
Offset: 1

Views

Author

Jon E. Schoenfield, Jan 07 2018

Keywords

Comments

The k-th Euclid number, A006862(k), is 1 plus the product of the first k primes, i.e., 1 + A002110(k). A113165 lists the numbers (> 1) that divide at least one Euclid number. It appears that the vast majority of terms in A113165 are prime; this sequence lists the composite numbers in A113165.
No composite less than 10^8 divides more than one Euclid number.

Examples

			a(1) = 1843 because 1843 = 19*97 is the smallest composite number that divides a Euclid number: 1843 divides 1 + A002110(7) = 1 + 2*3*5*7*11*13*17 = 510511 = 19*97*277. (Thus, 5263 (= 19*277), 26869 (= 97*277), and 19*97*277 = 510511 itself are also composites that divide a Euclid number; 5263 = a(2), 26869 = a(6), and 510511 = a(11).)

Crossrefs

Cf. A002110 (primorials), A006862 (Euclid numbers), A113165 (numbers > 1 that divide Euclid numbers), A297891 (numbers > 1 that divide exactly two Euclid numbers).
Showing 1-8 of 8 results.

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