A317241 -id:A317241 - cp's OEIS frontend

A317390 A(n,k) is the n-th positive integer that has exactly k representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes; square array A(n,k), n>=1, k>=0, read by antidiagonals.

2, 1, 5, 25, 3, 7, 43, 29, 4, 11, 211, 61, 37, 6, 15, 638, 261, 91, 40, 8, 23, 664, 848, 421, 111, 41, 9, 26, 1613, 1956, 921, 426, 121, 49, 10, 27, 2991, 3321, 2058, 969, 441, 124, 51, 12, 28, 7021, 3004, 3336, 2092, 1002, 484, 171, 52, 13, 31, 11306, 7162, 3319, 3368, 2094, 1026, 535, 184, 67, 14, 33

Offset: 1

Alois P. Heinz, Jul 27 2018

			A(6,2) = 49: 1 + 3 * (1 + 5 * (1 + 2)) = 1 + 2 * (1 + 23) = 49.
Square array A(n,k) begins:
   2,  1, 25,  43, 211,  638,  664, 1613, 2991, ...
   5,  3, 29,  61, 261,  848, 1956, 3321, 3004, ...
   7,  4, 37,  91, 421,  921, 2058, 3336, 3319, ...
  11,  6, 40, 111, 426,  969, 2092, 3368, 3554, ...
  15,  8, 41, 121, 441, 1002, 2094, 3741, 3928, ...
  23,  9, 49, 124, 484, 1026, 2283, 3914, 4846, ...
  26, 10, 51, 171, 535, 1106, 2381, 3979, 5552, ...
  27, 12, 52, 184, 540, 1156, 2388, 4082, 5886, ...
  28, 13, 67, 187, 591, 1191, 2432, 4126, 6293, ...

Columns k=0-10 give: A317242, A317391, A317392, A317393, A317394, A317395, A317396, A317397, A317398, A317399, A317400.
Row n=1 gives A317385.
A(n,n) gives A317537.
Cf. A317241.

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() local h, p, q; p, q:= proc() [] end, 0;
      proc(n, k)
        while nops(p(k))

b[n_, s_List] := 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 = q + 1; h = b[q, {}]; p[h] = Append[p[h], q]]; p[k][[n]]];
Table[Table[A[n, d - n], {n, 1, d}], {d, 1, 11}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

A317241(A(n,k)) = k.

A317240 Number of representations of n of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of (not necessarily distinct) primes.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 0, 1, 2, 0, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 4, 1, 3, 2, 0, 1, 2, 1, 3, 2, 1, 1, 3, 0, 2, 3, 2, 1, 3, 1, 3, 3, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 2, 3, 2, 1, 3, 1, 4, 2, 2, 1, 3, 1, 4, 3, 2, 1, 5, 3, 3, 4, 0, 2, 2, 1, 3, 2, 2, 1, 5, 1, 3

Offset: 1

Alois P. Heinz, Jul 24 2018

nonn

			a(13) = 2: 1 + 2 * (1 + 5) = 1 + 3 * (1 + 3) = 13.
a(31) = 3: 1 + 2 * (1 + 2 * (1 + 2 * (1 + 2))) = 1 + 3 * (1 + 3 * (1 + 2)) = 1 + 5 * (1 + 5) = 31.

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A180337, A317241, A317384.

a:= proc(n) option remember; `if`(n=1, 1,
      add(a((n-1)/p), p=numtheory[factorset](n-1)))
    end:
seq(a(n), n=1..200);

pp[n_] := pp[n] = FactorInteger[n][[All, 1]];
q[n_] := q[n] = Switch[n, 1, True, 2, False, _, AnyTrue[pp[n-1], q[(n-1)/#]&]];
a[n_] := a[n] = Which[n == 1, 1, !q[n], 0, True, Sum[a[(n-1)/p], {p, pp[n-1]}]];
Array[a, 105] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

a(n) = Sum_{prime p|(n-1)} a((n-1)/p) for n>1, a(1) = 1.
a(n) = 0 <=> n in { A180337 }.
a(n) >= A317241(n).
G.f. A(x) satisfies: A(x) = x * (1 + A(x^2) + A(x^3) + A(x^5) + ... + A(x^prime(k)) + ...). - Ilya Gutkovskiy, May 09 2019

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

A317390 A(n,k) is the n-th positive integer that has exactly k representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes; square array A(n,k), n>=1, k>=0, read by antidiagonals.

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A317240 Number of representations of n of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of (not necessarily distinct) primes.

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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.

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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.

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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.

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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.

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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.

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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.

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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.