A006450 -id:A006450 - cp's OEIS frontend

A346068 Numbers that are the product of distinct primes with prime subscripts raised to prime powers.

1, 9, 25, 27, 121, 125, 225, 243, 289, 675, 961, 1089, 1125, 1331, 1681, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 6075, 6889, 7225, 7803, 8649, 11881, 11979, 15125, 15129, 16129, 24025, 24649, 25947, 27225, 28125, 29403, 29791, 30375, 31329, 32041, 33275, 34969

Offset: 1

Ilya Gutkovskiy, Jul 30 2021

nonn

			675 = 3^3 * 5^2 = prime(prime(1))^prime(2) * prime(prime(2))^prime(1), therefore 675 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A056166 and A076610.

Cf. A006450, A302590, A302596, A321874.

Join[{1}, Select[Range[35000], AllTrue[Join[PrimePi[(t = Transpose @ FactorInteger[#])[[1]]], t[[2]]], PrimeQ] &]] (* Amiram Eldar, Jul 30 2021 *)

from sympy import factorint, isprime, primepi
def ok(n):
    f = factorint(n)
    if not all(isprime(e) for e in f.values()): return False
    return all(isprime(primepi(p)) for p in f)
print(list(filter(ok, range(35000)))) # Michael S. Branicky, Jul 30 2021

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + Sum_{q prime} 1/p^q) = 1.2271874... - Amiram Eldar, Jul 31 2021

A237284 Number of ordered ways to write 2*n = p + q with p, q and A000720(p) all prime.

Original entry on oeis.org

0, 0, 1, 2, 2, 1, 2, 3, 2, 2, 4, 3, 1, 3, 2, 1, 5, 3, 1, 3, 3, 3, 4, 5, 2, 3, 4, 1, 4, 3, 3, 6, 2, 1, 6, 6, 3, 4, 7, 1, 4, 6, 3, 5, 6, 2, 4, 4, 2, 6, 5, 3, 5, 4, 3, 7, 8, 2, 4, 8, 1, 4, 5, 3, 6, 5, 4, 2, 7, 5, 6, 6, 3, 4, 6, 2, 5, 7, 2, 4

Offset: 1

Zhi-Wei Sun, Feb 06 2014

nonn

Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 6, 13, 16, 19, 28, 34, 40, 61, 166, 278.
This is stronger than Goldbach's conjecture.
The conjecture is true for n <= 5*10^8. - Dmitry Kamenetsky, Mar 13 2020

			a(13) = 1 since 2*13 = 3 + 23 with 3, 23 and A000720(3) = 2 all prime.
a(278) = 1 since 2*278 = 509 + 47 with 509, 47 and A000720(509) = 97 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Carlos Rivera, Conjecture 85. Conjectures stricter that the Goldbach ones, Prime Puzzles
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A000720, A002372, A002375, A006450, A236566.

a[n_]:=Sum[If[PrimeQ[2n-Prime[Prime[k]]],1,0],{k,1,PrimePi[PrimePi[2n-1]]}]
Table[a[n],{n,1,80}]

A330943 Matula-Goebel numbers of singleton-reduced rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 12, 13, 14, 16, 18, 19, 21, 24, 26, 28, 32, 34, 36, 37, 38, 39, 42, 43, 48, 49, 52, 53, 54, 56, 57, 61, 63, 64, 68, 72, 73, 74, 76, 78, 82, 84, 86, 89, 91, 96, 98, 101, 102, 104, 106, 107, 108, 111, 112, 114, 117, 119, 122, 126, 128, 129, 131

Offset: 1

Gus Wiseman, Jan 13 2020

nonn

These trees are counted by A330951.
A rooted tree is singleton-reduced if no non-leaf node has all singleton branches, where a rooted tree is a singleton if its root has degree 1.
The Matula-Goebel number of a rooted tree is the product of primes of the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
A prime index of n is a number m such that prime(m) divides n. A number belongs to this sequence iff it is 1 or its prime indices all belong to this sequence but are not all prime.

			The sequence of all singleton-reduced rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   7: ((oo))
   8: (ooo)
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  19: ((ooo))
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  34: (o((oo)))
  36: (oo(o)(o))
  37: ((oo(o)))

The series-reduced case is A291636.
Unlabeled rooted trees are counted by A000081.
Numbers whose prime indices are not all prime are A330945.
Singleton-reduced rooted trees are counted by A330951.
Singleton-reduced phylogenetic trees are A000311.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000669, A001678, A006450, A007097, A007821, A061775, A196050, A257994, A276625, A277098, A320628, A330944, A330948.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mgsingQ[n_]:=n==1||And@@mgsingQ/@primeMS[n]&&!And@@PrimeQ/@primeMS[n];
Select[Range[100],mgsingQ]

A073124 a(n) = prime(1+prime(n)) - prime(prime(n)).

Original entry on oeis.org

2, 2, 2, 2, 6, 2, 2, 4, 6, 4, 4, 6, 2, 2, 12, 10, 4, 10, 6, 6, 6, 8, 2, 2, 12, 10, 6, 6, 2, 2, 10, 4, 14, 12, 4, 4, 10, 4, 6, 2, 6, 4, 10, 10, 12, 6, 4, 14, 6, 4, 10, 12, 8, 4, 6, 24, 10, 6, 2, 8, 14, 18, 2, 6, 2, 12, 16, 4, 6, 6, 2, 6, 26, 2, 8, 10, 4, 10, 4

Offset: 1

Labos Elemer, Jul 16 2002

nonn

Number of entries in {x,..} such that pi(x) = prime(n).

			For n = 25: prime(25) = 97, pi(x) = 97 holds for 12 numbers x: {509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520} so a(25) = 12. The largest 520 = A073123(25), and the smallest = A006450(25).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A006450, A072677, A073123.

Table[Prime[Prime[n]+1]-Prime[Prime[n]], {n, 1, 256}]
seq[max_] := Module[{p = Prime[Range[max + 1]], m = PrimePi[max], ind}, ind = Prime[Range[m]]; p[[ind + 1]] - p[[ind]]]; seq[400] (* Amiram Eldar, Feb 15 2025 *)

a(n) =  prime(1+prime(n)) - prime(prime(n)); \\ Michel Marcus, Dec 11 2013

a(n) = Card{x; A000720(x) = A000040(n)}.

a(n) = A072677(n) - A006450(n).

More terms from Michel Marcus, Dec 11 2013

A050439 Fifth-order composites.

Original entry on oeis.org

39, 49, 55, 56, 60, 69, 74, 77, 78, 84, 93, 94, 95, 100, 105, 106, 110, 115, 119, 124, 125, 126, 130, 133, 140, 141, 145, 152, 155, 156, 159, 162, 164, 165, 170, 174, 180, 183, 184, 188, 189, 198, 201, 202, 203, 206, 207, 209, 212, 213, 218, 222, 225, 231

Offset: 1

Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

			C(C(C(C(C(8))))) = C(C(C(C(15)))) = C(C(C(25))) = C(C(38)) = C(55) = 77. So 77 is in the sequence.

N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076-A049081, A006450, A050435, A050436, A050438, A050440.

C := remove(isprime,[$4..1000]): seq(C[C[C[C[C[n]]]]],n=1..100);

Let C(n) be the n-th composite number, with C(1)=4. Then these are numbers C(C(C(C(C(n))))).

More terms from Asher Auel Dec 15 2000

A096478 a(n) = A000040(A096477(n)), i.e., prime(a(n)) and prime(a(n)+1) are twin primes.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 41, 43, 83, 89, 109, 113, 173, 277, 307, 313, 353, 373, 463, 563, 577, 601, 613, 643, 673, 719, 743, 1117, 1123, 1171, 1279, 1571, 1621, 1627, 1709, 1741, 1823, 1867, 1907, 1949, 1979, 1987, 1999, 2003, 2063, 2099, 2153, 2287, 2309, 2311

Offset: 1

Labos Elemer, Jun 23 2004

nonn

Gives primes in A029707. - Pierre CAMI, Apr 20 2006

			89 is a term since it is a prime and prime(89 + 1) - prime(89) = 463 - 461 = 2; the prime with subscript 89 (which is prime) and the next prime (i.e., prime(90)) are twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A000040, A006450, A072677, A073124, A096477.

Cf. A029707, A000720.

Prime[Flatten[Position[Table[Prime[Prime[n]+1]-Prime[Prime[n]], {n, 1, 1000}], 2]]]

A141436 Tisdale's sieve: generators for an infinite set of disjoint prime/nonprime sequences (see Comments for precise definition).

Original entry on oeis.org

1, 3, 5, 8, 10, 11, 14, 15, 16, 17, 20, 22, 24, 25, 27, 30, 31, 32, 33, 35, 36, 38, 39, 40, 41, 44, 46, 48, 49, 50, 51, 54, 55, 56, 58, 59, 62, 63, 64, 66, 67, 68, 69, 70, 72, 75, 76, 77, 78, 80, 82, 83, 85, 86, 87, 88, 90, 92, 93, 94, 96, 99, 100, 102, 104, 105, 108, 109, 110, 111

Offset: 1

Daniel Tisdale, Aug 06 2008

The definition: (Start)
Let P = (2,3,5,...) and N = (1,4,6,...) denote the sequences of primes and nonprimes, respectively.
Construct an infinite collection of disjoint sequences S_1, S_2, S_3, ... as follows.
For S_i, the first term S_i(1) is the smallest positive integer not yet used in any S_j with j < i.
Thereafter S_i is extended by the rule S_i(j+1) = either P(S_i(j)) or N(S_i(j)), where we choose P or N so that the primes and nonprimes alternate in S_i.
Then the sequence consists of the initial values S_1(1), S_2(1), S_3(1), ...
We find that S_1 = {1,2,4,7,12,...} (A280028), with smallest missing number 3,
S_2 = {3,6,13,...} (A280029), so now the smallest missing number is 5,
S_3 = {5,9,23,...} (A280030), and so on.
So the sequence begins 1,3,5,...
(End)
Note that the disjointness is a consequence of the definition. For if S_i and S_j have a common term, let S_i(q) = S_j(r) = X (say) be the first time they meet. Then either one or both of q and r are 1, which is impossible by the construction, or q > 1, r > 1. But then S_i(q-1) = S_j(r-1), which is a contradiction. - N. J. A. Sloane, Dec 28 2016
The complementary sequence is 2, 4, 6, 7, 9, 12, 13, 18, 19, etc., see A141437.
The old definition was: "a(k) = k-th prime or nonprime: a(k+1) = a(a(k)); if k is prime, a(k) is nonprime; if k is nonprime, a(k) is prime. {a(1)} = {1,2,4,7,12,...}, {a(3)} = {3,6,13,...}, {a(5)} = {5,9,23,...}. The generators of these mutually disjoint sets are the first elements, {1,3,5,8, etc.}."

B. D. Swan, Table of n, a(n) for n = 1..100000

Cf. A018252, A141437, A102615, A006450, A280028, A280029, A280030.

Conjecture: Equals A102615 UNION A006450. - R. J. Mathar, Aug 14 2008
Proof of Conjecture, from David Applegate, Dec 28 2016: (Start)
First, a simple lemma about simple sieves (doesn't apply to the Sieve of Eratosthenes):
Lemma: Let f be a function mapping positive integers to positive integers, with f(n) > n for all n.
Construct an infinite collection of (not necessarily disjoint) sequences S_1, S_2, S_3, ... as follows.
For S_i, the first term S_i(1) is the smallest positive integer not yet used in any S_j with j < i.
Thereafter S_i is extended by the rule S_i(j+1) = f(S_i(j)) = f^j (S_i(1)).
Then the sequence of initial values S_1(1), S_2(1), S_3(1), ... consists of the positive integers not in the image of f().
Proof: Obviously, if n is not in the image of f(), it can never occur as S_i(j) for j > 1, so it will eventually occur as S_i(1) for some i.
Conversely, if n is in the image of f(), it will occur as S_i(j) for some i and j > 1 such that S_i(1) < n, and hence will not occur as an initial value. We show this by induction n: Let n = f(m) (so m < n). If m is in the image of f(), then by induction it occurs as S_i(j) for some i and j>1, so n=S_i(j+1). Otherwise, if m is not in the image of f(), then it will occur as S_i(1) for some i, so n=S_i(2).
To apply the lemma, just define f(n) = P(n) if n is nonprime, N(n) if n is prime. So, the sequence of initial values will consist of:
primes p such that p does not equal P(n) for n nonprime (A006450) and
nonprimes n such that n does not equal N(p) for p prime (A192615).
(End)

Edited (including a new name) by N. J. A. Sloane, Dec 25 2016

A320630 Products of primes of nonprime squarefree index.

Original entry on oeis.org

2, 4, 8, 13, 16, 26, 29, 32, 43, 47, 52, 58, 64, 73, 79, 86, 94, 101, 104, 113, 116, 128, 137, 139, 146, 149, 158, 163, 167, 169, 172, 181, 188, 199, 202, 208, 226, 232, 233, 256, 257, 269, 271, 274, 278, 292, 293, 298, 313, 316, 317, 326, 334, 338, 344, 347

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

The index of a prime number n is the number m such that n is the m-th prime.

			The sequence of terms begins:
    2 = prime(1)
    4 = prime(1)^2
    8 = prime(1)^3
   13 = prime(6)
   16 = prime(1)^4
   26 = prime(1)*prime(6)
   29 = prime(10)
   32 = prime(1)^5
   43 = prime(14)
   47 = prime(15)
   52 = prime(1)^2*prime(6)
   58 = prime(1)*prime(10)
   64 = prime(1)^6
   73 = prime(21)
   79 = prime(22)
   86 = prime(1)*prime(14)
   94 = prime(1)*prime(15)
  101 = prime(26)
  104 = prime(1)^3*prime(6)
  113 = prime(30)
  116 = prime(1)^2*prime(10)
  128 = prime(1)^7

Cf. A000040, A006450, A007821, A018252, A056239, A076610, A112798, A302242, A302478, A320628, A320629, A320631, A320633.

Select[Range[100],With[{f=PrimePi/@First/@FactorInteger[#]},And[And@@SquareFreeQ/@f,And@@Not/@PrimeQ/@f]]&]

A330948 Nonprime numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106

Offset: 1

Gus Wiseman, Jan 13 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices of prime indices begins:
   4: {{},{}}
   6: {{},{1}}
   8: {{},{},{}}
  10: {{},{2}}
  12: {{},{},{1}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  20: {{},{},{2}}
  21: {{1},{1,1}}
  22: {{},{3}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  28: {{},{},{1,1}}
  30: {{},{1},{2}}
  32: {{},{},{},{},{}}
  34: {{},{4}}
  35: {{2},{1,1}}
  36: {{},{},{1},{1}}
  38: {{},{1,1,1}}

Complement in A330945 of A000040.
Complement in A018252 of A076610.
The restriction to odd terms is A330949.
Nonprime numbers n such that A330944(n) > 0.
Taking odds instead of nonprimes gives A330946.
The number of prime prime indices is given by A257994.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of nonprime index are A320628.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000720, A001222, A056239, A112798, A302242, A320629, A320633, A330943, A330947.

Select[Range[100],!PrimeQ[#]&&!And@@PrimeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]

A372885 Prime numbers whose binary indices (positions of ones in reversed binary expansion) sum to another prime number.

Original entry on oeis.org

2, 3, 11, 23, 29, 41, 43, 61, 71, 79, 89, 101, 103, 113, 131, 137, 149, 151, 163, 181, 191, 197, 211, 239, 269, 271, 281, 293, 307, 331, 349, 353, 373, 383, 401, 433, 457, 491, 503, 509, 523, 541, 547, 593, 641, 683, 701, 709, 743, 751, 761, 773, 827, 863, 887

Offset: 1

Gus Wiseman, May 19 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

The indices of these primes are A372886.

			The binary indices of 89 are {1,4,5,7}, with sum 17, which is prime, so 89 is in the sequence.
The terms together with their binary expansions and binary indices begin:
    2:         10 ~ {2}
    3:         11 ~ {1,2}
   11:       1011 ~ {1,2,4}
   23:      10111 ~ {1,2,3,5}
   29:      11101 ~ {1,3,4,5}
   41:     101001 ~ {1,4,6}
   43:     101011 ~ {1,2,4,6}
   61:     111101 ~ {1,3,4,5,6}
   71:    1000111 ~ {1,2,3,7}
   79:    1001111 ~ {1,2,3,4,7}
   89:    1011001 ~ {1,4,5,7}
  101:    1100101 ~ {1,3,6,7}
  103:    1100111 ~ {1,2,3,6,7}
  113:    1110001 ~ {1,5,6,7}
  131:   10000011 ~ {1,2,8}
  137:   10001001 ~ {1,4,8}
  149:   10010101 ~ {1,3,5,8}
  151:   10010111 ~ {1,2,3,5,8}
  163:   10100011 ~ {1,2,6,8}
  181:   10110101 ~ {1,3,5,6,8}
  191:   10111111 ~ {1,2,3,4,5,6,8}
  197:   11000101 ~ {1,3,7,8}

Robert Israel, Table of n, a(n) for n = 1..10000

For prime instead of binary indices we have A006450, prime case of A316091.
Prime numbers p such that A029931(p) is also prime.
Prime case of A372689.
The indices of these primes are A372886.
A000040 lists the prime numbers, A014499 their binary indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A058698 counts partitions of prime numbers, strict A064688.
A372687 counts strict partitions of prime binary rank, counted by A372851.
A372688 counts partitions of prime binary rank, with Heinz numbers A277319.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000009, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372850, A372887.

filter:= proc(p)
  local L,i,t;
  L:= convert(p,base,2);
  isprime(add(i*L[i],i=1..nops(L)))
end proc:
select(filter, [seq(ithprime(i),i=1..200)]); # Robert Israel, Jun 19 2025

Select[Range[100],PrimeQ[#] && PrimeQ[Total[First/@Position[Reverse[IntegerDigits[#,2]],1]]]&]

cp's OEIS Frontend

A346068 Numbers that are the product of distinct primes with prime subscripts raised to prime powers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A237284 Number of ordered ways to write 2*n = p + q with p, q and A000720(p) all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A330943 Matula-Goebel numbers of singleton-reduced rooted trees.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A073124 a(n) = prime(1+prime(n)) - prime(prime(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A050439 Fifth-order composites.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A096478 a(n) = A000040(A096477(n)), i.e., prime(a(n)) and prime(a(n)+1) are twin primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A141436 Tisdale's sieve: generators for an infinite set of disjoint prime/nonprime sequences (see Comments for precise definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A320630 Products of primes of nonprime squarefree index.

Views

Author

Keywords

Comments

Examples