A317390 -id:A317390 - cp's OEIS frontend

A317242 Positive integers having no representation of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

2, 5, 7, 11, 15, 23, 26, 27, 28, 31, 33, 35, 36, 47, 50, 56, 57, 63, 66, 78, 81, 82, 95, 96, 106, 116, 119, 120, 122, 129, 136, 156, 162, 166, 167, 190, 193, 215, 218, 219, 227, 236, 244, 254, 263, 286, 289, 330, 335, 342, 352, 359, 387, 393, 395, 396, 414

Offset: 1

Alois P. Heinz, Jul 24 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=0 of A317390.

Cf. A180337 (subsequence), A317241.

q:= proc(n, s) option remember; is (n=1 or ormap(p->
      q((n-1)/p, s union {p}), numtheory[factorset](n-1) minus s))
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 2, 1+a(n-1)) while q(k, {}) do od; k
    end:
seq(a(n), n=1..100);

b[n_, s_] := b[n, s] = If[n == 1, 1, Sum[If[p == 1, 0, b[(n - 1)/p, s ~Union~ {p}]], {p, FactorInteger[n - 1][[All, 1]] ~Complement~ s}]];
Position[Array[b[#, {}]&, 10^5], 0] // Flatten (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz in A317241 *)

A317241(a(n)) = 0.

A317385 Smallest positive integer that has exactly n representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

2, 1, 25, 43, 211, 638, 664, 1613, 2991, 7021, 11306, 9439, 17361, 23230, 40886, 38341, 49063, 36583, 99111, 111229, 110631, 171718, 233451, 255531, 309141, 327643, 369519, 521266, 489406, 738544, 682690, 812826, 1048594, 1015096, 2003002, 2118439, 1602360, 2204907, 2850772, 2702743, 2794198

Offset: 0

Alois P. Heinz, Jul 26 2018

nonn

			a(1) = 1: 1.
a(2) = 25: 1 + 2 * (1 + 11) = 1 + 3 * (1 + 7) = 25.
a(3) = 43: 1 + 2 * (1 + 5 * (1 + 3)) = 1 + 3 * (1 + 13) = 1 + 7 * (1 + 5) = 43.

Row n=1 of A317390.

Cf. A317241, A317384.

b:= proc(n, s) option remember; `if`(n=1, 1, add(b((n-1)/p
      , s union {p}), p=numtheory[factorset](n-1) minus s))
    end:
a:= proc(n) option remember; local k;
      for k while n<>b(k, {}) do od; k
    end:
seq(a(n), n=0..15);

b[n_, s_] := b[n, s] = If[n == 1, 1, Sum[If[p == 1, 0, b[(n - 1)/p, s~Union~{p}]], {p, FactorInteger[n - 1][[All, 1]]~Complement~s}]];
A[n_, k_] := Module[{h, p, q = 0}, p[_] = {}; While[Length[p[k]] < n, q++; h = b[q, {}]; p[h] = Append[p[h], q]]; p[k][[n]]];
a[k_] := If[k == 0, 2, A[1, k]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz in A317390 *)

a(n) = min { j > 0 : A317241(j) = n }.

A317391 Positive integers that have a unique representation of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 30, 32, 34, 38, 39, 42, 44, 45, 46, 48, 53, 54, 55, 58, 59, 60, 62, 64, 65, 68, 69, 70, 72, 73, 74, 75, 76, 79, 80, 83, 84, 86, 90, 92, 93, 94, 98, 99, 100, 101, 102, 104, 105, 107, 108, 109

Offset: 1

Alois P. Heinz, Jul 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=1 of A317390.

Cf. A317241.

b:= proc(n, s) option remember; local p, r; if n=1 then 1 else r:=0;
      for p in numtheory[factorset](n-1) minus s while r<2
        do r:= r+b((n-1)/p, s union {p}) od; `if`(r<2, r, 2)
      fi
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 1, 1+a(n-1)) while b(k, {})<>1 do od; k
    end:
seq(a(n), n=1..100);

A317241(a(n)) = 1.

A317392 Positive integers that have exactly two representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

25, 29, 37, 40, 41, 49, 51, 52, 67, 71, 77, 85, 87, 88, 89, 97, 103, 112, 115, 123, 125, 126, 127, 130, 137, 139, 145, 146, 148, 149, 155, 157, 161, 169, 175, 181, 183, 186, 191, 199, 202, 209, 214, 217, 222, 223, 229, 232, 235, 238, 239, 241, 243, 248, 249

Offset: 1

Alois P. Heinz, Jul 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=2 of A317390.

Cf. A317241.

b:= proc(n, s) option remember; local p, r; if n=1 then 1 else r:=0;
      for p in numtheory[factorset](n-1) minus s while r<3
        do r:= r+b((n-1)/p, s union {p}) od; `if`(r<3, r, 3)
      fi
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 1, 1+a(n-1)) while b(k, {})<>2 do od; k
    end:
seq(a(n), n=1..100);

A317241(a(n)) = 2.

A317393 Positive integers that have exactly three representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

43, 61, 91, 111, 121, 124, 171, 184, 187, 205, 221, 231, 256, 265, 267, 268, 274, 277, 281, 283, 291, 311, 323, 326, 331, 337, 371, 373, 375, 379, 386, 411, 412, 427, 428, 435, 443, 451, 456, 457, 471, 474, 475, 482, 491, 494, 505, 507, 508, 511, 519, 521, 523

Offset: 1

Alois P. Heinz, Jul 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=3 of A317390.

Cf. A317241.

b:= proc(n, s) option remember; local p, r; if n=1 then 1 else r:=0;
      for p in numtheory[factorset](n-1) minus s while r<4
        do r:= r+b((n-1)/p, s union {p}) od; `if`(r<4, r, 4)
      fi
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 1, 1+a(n-1)) while b(k, {})<>3 do od; k
    end:
seq(a(n), n=1..100);

A317241(a(n)) = 3.

A317394 Positive integers that have exactly four representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

211, 261, 421, 426, 441, 484, 535, 540, 591, 621, 634, 667, 683, 691, 715, 726, 732, 761, 771, 776, 778, 794, 818, 853, 862, 871, 925, 970, 979, 987, 989, 1011, 1021, 1023, 1038, 1074, 1086, 1105, 1114, 1141, 1171, 1176, 1184, 1190, 1197, 1222, 1261, 1266

Offset: 1

Alois P. Heinz, Jul 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=4 of A317390.

Cf. A317241.

b:= proc(n, s) option remember; local p, r; if n=1 then 1 else r:=0;
      for p in numtheory[factorset](n-1) minus s while r<5
        do r:= r+b((n-1)/p, s union {p}) od; `if`(r<5, r, 5)
      fi
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 1, 1+a(n-1)) while b(k, {})<>4 do od; k
    end:
seq(a(n), n=1..100);

A317241(a(n)) = 4.

A317395 Positive integers that have exactly five representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

638, 848, 921, 969, 1002, 1026, 1106, 1156, 1191, 1248, 1276, 1310, 1326, 1341, 1431, 1444, 1480, 1499, 1548, 1592, 1641, 1730, 1764, 1772, 1786, 1856, 1888, 1911, 1996, 2005, 2025, 2038, 2050, 2053, 2061, 2121, 2129, 2131, 2133, 2146, 2171, 2224, 2256, 2258

Offset: 1

Alois P. Heinz, Jul 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=5 of A317390.

Cf. A317241.

b:= proc(n, s) option remember; local p, r; if n=1 then 1 else r:=0;
      for p in numtheory[factorset](n-1) minus s while r<6
        do r:= r+b((n-1)/p, s union {p}) od; `if`(r<6, r, 6)
      fi
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 1, 1+a(n-1)) while b(k, {})<>5 do od; k
    end:
seq(a(n), n=1..100);

A317241(a(n)) = 5.

A317396 Positive integers that have exactly six representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

664, 1956, 2058, 2092, 2094, 2283, 2381, 2388, 2432, 2466, 2533, 2624, 2701, 2775, 2822, 2853, 2976, 3070, 3193, 3220, 3316, 3326, 3436, 3442, 3461, 3485, 3529, 3568, 3571, 3576, 3620, 3746, 3784, 3785, 3797, 3826, 3839, 3913, 4005, 4026, 4031, 4213, 4234

Offset: 1

Alois P. Heinz, Jul 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=6 of A317390.

Cf. A317241.

b:= proc(n, s) option remember; local p, r; if n=1 then 1 else r:=0;
      for p in numtheory[factorset](n-1) minus s while r<7
        do r:= r+b((n-1)/p, s union {p}) od; `if`(r<7, r, 7)
      fi
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 1, 1+a(n-1)) while b(k, {})<>6 do od; k
    end:
seq(a(n), n=1..100);

A317241(a(n)) = 6.

A317397 Positive integers that have exactly seven representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

1613, 3321, 3336, 3368, 3741, 3914, 3979, 4082, 4126, 4219, 4561, 4777, 4798, 4824, 4929, 4936, 4948, 5083, 5314, 5371, 5559, 5656, 5825, 5877, 5946, 5986, 6096, 6109, 6111, 6291, 6303, 6376, 6644, 6651, 6673, 6700, 6711, 6786, 6883, 6886, 6917, 6920, 7036

Offset: 1

Alois P. Heinz, Jul 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=7 of A317390.

Cf. A317241.

b:= proc(n, s) option remember; local p, r; if n=1 then 1 else r:=0;
      for p in numtheory[factorset](n-1) minus s while r<8
        do r:= r+b((n-1)/p, s union {p}) od; `if`(r<8, r, 8)
      fi
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 1, 1+a(n-1)) while b(k, {})<>7 do od; k
    end:
seq(a(n), n=1..100);

A317241(a(n)) = 7.

A317398 Positive integers that have exactly eight representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

2991, 3004, 3319, 3554, 3928, 4846, 5552, 5886, 6293, 6784, 7183, 7286, 7396, 7668, 7741, 7743, 7829, 7996, 8095, 8121, 8212, 8477, 8586, 8614, 8856, 8861, 9096, 9307, 9374, 9591, 9626, 9636, 9637, 9721, 9738, 9845, 9891, 9912, 9934, 10011, 10024, 10048, 10251

Offset: 1

Alois P. Heinz, Jul 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=8 of A317390.

Cf. A317241.

b:= proc(n, s) option remember; local p, r; if n=1 then 1 else r:=0;
      for p in numtheory[factorset](n-1) minus s while r<9
        do r:= r+b((n-1)/p, s union {p}) od; `if`(r<9, r, 9)
      fi
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 1, 1+a(n-1)) while b(k, {})<>8 do od; k
    end:
seq(a(n), n=1..100);

A317241(a(n)) = 8.

cp's OEIS Frontend

A317242 Positive integers having no representation of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A317385 Smallest positive integer that has exactly n representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A317391 Positive integers that have a unique representation of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A317392 Positive integers that have exactly two representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A317393 Positive integers that have exactly three representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A317394 Positive integers that have exactly four representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A317395 Positive integers that have exactly five representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A317396 Positive integers that have exactly six representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A317397 Positive integers that have exactly seven representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Views

Author

Keywords

Links