A317240 -id:A317240 - cp's OEIS frontend

A317241 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 distinct primes.

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 0, 1, 0, 0, 2, 1, 1, 2, 2, 1, 3, 1, 1, 1, 0, 1, 2, 0, 2, 2, 1, 1, 1, 0, 0, 1, 1, 1, 3, 1, 0, 1, 1, 0, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 0, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Alois P. Heinz, Jul 24 2018

nonn

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

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

Cf. A317240, A317242, A317385.

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:= n-> b(n, {}):
seq(a(n), n=1..200);

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_] := b[n, {}];
Array[a, 200] (* Jean-François Alcover, May 26 2019, after Alois P. Heinz *)

a(n) = 0 <=> n in { A317242 }.

a(n) <= A317240(n).

A317384 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 (not necessarily distinct) primes.

Original entry on oeis.org

2, 1, 13, 31, 43, 91, 111, 231, 175, 274, 351, 471, 703, 526, 463, 931, 823, 1723, 1579, 1279, 1903, 2083, 1791, 2143, 2227, 2443, 2671, 2781, 2335, 3807, 3163, 3631, 3199, 4243, 5314, 5482, 5107, 4671, 6231, 6681, 8863, 7483, 6111, 6331, 7879, 8031, 8023, 9351

Offset: 0

Alois P. Heinz, Jul 26 2018

nonn

			a(1) = 1: 1.
a(2) = 13: 1 + 2 * (1 + 5) = 1 + 3 * (1 + 3) = 13.
a(3) = 31: 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 = 0..1000

Cf. A317240, A317385.

b:= proc(n) option remember; `if`(n=1, 1,
      add(b((n-1)/p), p=numtheory[factorset](n-1)))
    end:
a:= proc(n) option remember; local k;
      for k while n<>b(k) do od; k
    end:
seq(a(n), n=0..50);

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)/#]&]];
b[n_] := b[n] = Which[n == 1, 1, ! q[n], 0, True, Sum[b[(n-1)/p], {p, pp[n-1]}]];
a[n_] := Module[{k}, For[k = 1, True, k++, If[n == b[k], Return[k]]]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 07 2023, after Alois P. Heinz *)

a(n) = min { j > 0 : A317240(j) = n }.

A180337 Numbers which cannot be expressed as a sum 1 + p1 + p1p2 + p1p2*p3 + ... for some collection of primes {p1, p2, p3, ...}.

Original entry on oeis.org

2, 5, 11, 23, 26, 47, 56, 95, 116, 122, 236, 254, 518, 530, 1082, 2210

Offset: 1

Jack W Grahl, Aug 28 2010

nonn

I conjecture, but have not been able to prove, that this sequence is finite with only the terms given above. In that case it can be constructed by taking a1=2, and adjoining all numbers aj*ak + 1, where aj and ak are two prime members of the sequence.

Any number which can be expressed as p*q + 1, where p is prime and q does not belong to the sequence, does not belong to the sequence either.

			8 is not a member of the sequence since it is equal to 1 + 7.
9 is not a member of the sequence since it can be written 1 + 2 + 2*3.
10 is not a member of the sequence since it is equal to 1 + 3 + 3*2.
11 is a member of the sequence. If 11 could be written in this form, then p1 must divide 10. We would have 11 = 1 + p1(1 + p2 + ...), which would imply that 5 is not a member of the sequence if p1 = 2, or vice versa. Since both 2 nor 5 are members, so is 11.

All terms given above belong to A009293.

Cf. A317240, A317242.

q:= proc(n) option remember; is(n=1 or ormap(p->
      q((n-1)/p), numtheory[factorset](n-1)))
    end:
remove(q, [$1..3000])[];  # Alois P. Heinz, Jul 24 2018

q[1] = True; q[2] = False;
q[n_] := q[n] = AnyTrue[FactorInteger[n-1][[All, 1]], q[(n-1)/#]&];
Select[Range[3000], !q[#]&] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)

#!/usr/bin/perl $max = 10; if (defined($ARGV[0])) { $max = $ARGV[0]; } $primes{1} = 0; $list{1} = 1; $list{2} = 0; print "2, "; foreach $k (2..$max){ $p = 1; $l = 0; foreach $j (1..$k) { if ($primes{$j}){ if (($k % $j) == 0){ $p = 0; if ($list{$k / $j}){ $l = 1; } } } } $primes{$k} = $p; $list{$k + 1} = $l || $p; if (!$list{$k + 1}){ $t = $k + 1; print "$t, " } }

A317240(a(n)) = 0. - Alois P. Heinz, Jul 24 2018

A380635 a(1) = 1; a(n+1) = Sum_{d^2|n} a(n/d^2).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 7, 7, 8, 8, 10, 10, 10, 10, 12, 13, 13, 14, 16, 16, 16, 16, 19, 19, 19, 19, 24, 24, 24, 24, 28, 28, 28, 28, 32, 34, 34, 34, 39, 40, 41, 41, 46, 46, 48, 48, 53, 53, 53, 53, 58, 58, 58, 60, 67, 67, 67, 67, 74, 74, 74, 74, 84, 84, 84, 85, 93, 93, 93, 93

Offset: 1

Ilya Gutkovskiy, Jan 28 2025

nonn

Cf. A003238, A076752, A167865, A307779, A317240.

a:= proc(n) option remember; uses numtheory; `if`(n=1, 1,
      add(`if`(issqr(d), a((n-1)/d), 0), d=divisors(n-1)))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 28 2025

a[1] = 1; a[n_] := a[n] = DivisorSum[n - 1, a[(n - 1)/#] &, IntegerQ[Sqrt[#]] &]; Table[a[n], {n, 1, 80}]
nmax = 80; A[] = 0; Do[A[x] = x (1 + Sum[A[x^(k^2)], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * (1 + A(x) + A(x^4) + A(x^9) + ... + A(x^(k^2)) + ...).

cp's OEIS Frontend

A317241 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 distinct primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A317384 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 (not necessarily distinct) primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A180337 Numbers which cannot be expressed as a sum 1 + p1 + p1*p2 + p1*p2*p3 + ... for some collection of primes {p1, p2, p3, ...}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Perl

Formula

A380635 a(1) = 1; a(n+1) = Sum_{d^2|n} a(n/d^2).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A180337 Numbers which cannot be expressed as a sum 1 + p1 + p1p2 + p1p2*p3 + ... for some collection of primes {p1, p2, p3, ...}.