A006450 -id:A006450 - cp's OEIS frontend

A330944 Number of nonprime prime indices of n.

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

Offset: 1

Gus Wiseman, Jan 13 2020

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.

			24 has prime indices {1,1,1,2}, of which {1,1,1} are nonprime, so a(24) = 3.

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 prime index are A076610.
Products of primes of nonprime index are A320628.
Numbers whose prime indices are not all prime are A330945.
Cf. A000040, A000720, A001222, A007097, A018252, A056239, A112798, A302242, A320629, A320633, A330946, A330947.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}/;!PrimeQ[PrimePi[p]]:>k]],{n,30}]

a(n) = my(f=factor(n)); sum(k=1, #f~, if(!isprime(primepi(f[k,1])), f[k,2], 0)); \\ Daniel Suteu, Jan 14 2020

a(n) + A257994(n) = A001222(n).

Additive with a(p^e) = e if primepi(p) is nonprime, and 0 otherwise. - Amiram Eldar, Nov 03 2023

A049081 Primes prime(k) for which A049076(k) = 5.

Original entry on oeis.org

31, 1787, 8527, 19577, 27457, 42043, 72727, 96797, 112129, 137077, 173867, 239489, 250751, 285191, 352007, 401519, 443419, 464939, 490643, 527623, 683873, 718807, 755387, 839483, 864013, 985151, 1021271, 1080923, 1159901, 1278779

Offset: 1

Neil Fernandez

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A049076, A007821, A049078, A049079, A049080, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A049081 := proc(n)
        ithprime(A049080(n)) ;
end proc: # R. J. Mathar, Jul 07 2012

Nest[ Prime, Select[ Range[44], !PrimeQ[ # ] &], 5] (* Robert G. Wilson v, Mar 15 2004 *)

More terms from Robert G. Wilson v, Dec 12 2000

A320629 Products of odd primes of nonprime index.

Original entry on oeis.org

1, 7, 13, 19, 23, 29, 37, 43, 47, 49, 53, 61, 71, 73, 79, 89, 91, 97, 101, 103, 107, 113, 131, 133, 137, 139, 149, 151, 161, 163, 167, 169, 173, 181, 193, 197, 199, 203, 223, 227, 229, 233, 239, 247, 251, 257, 259, 263, 269, 271, 281, 293, 299, 301, 307, 311

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 asymptotic density of this sequence is (1/2) * Product_{p in A006450} (1 - 1/p) = 1/(2*Sum_{n>=1} 1/A076610(n)) < 1/6. - Amiram Eldar, Feb 02 2021

			The sequence of terms begins:
    1 = 1
    7 = prime(4)
   13 = prime(6)
   19 = prime(8)
   23 = prime(9)
   29 = prime(10)
   37 = prime(12)
   43 = prime(14)
   47 = prime(15)
   49 = prime(4)^2
   53 = prime(16)
   61 = prime(18)
   71 = prime(20)
   73 = prime(21)
   79 = prime(22)
   89 = prime(24)
   91 = prime(4)*prime(6)
   97 = prime(25)
  101 = prime(26)
  103 = prime(27)
  107 = prime(28)
  113 = prime(30)
  131 = prime(32)
  133 = prime(4)*prime(8)
  137 = prime(33)
  139 = prime(34)
  149 = prime(35)
  151 = prime(36)
  161 = prime(4)*prime(9)

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

Cf. A000040, A006450, A007821, A018252, A056239, A076610, A112798, A302242, A320533, A320628, A320630, A320631, A320633.

Select[Range[1,100,2],And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]&]

A064988 Multiplicative with a(p^e) = prime(p)^e.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 27, 25, 33, 31, 45, 41, 51, 55, 81, 59, 75, 67, 99, 85, 93, 83, 135, 121, 123, 125, 153, 109, 165, 127, 243, 155, 177, 187, 225, 157, 201, 205, 297, 179, 255, 191, 279, 275, 249, 211, 405, 289, 363, 295, 369, 241, 375, 341, 459, 335, 327

Offset: 1

Vladeta Jovovic, Oct 30 2001

			a(12) = a(2^2*3) = prime(2)^2 * prime(3) = 3^2*5 = 45, where prime(n) = A000040(n).

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization (first 1000 terms from Harry J. Smith)

Cf. A000040, A003961, A003963 (a left inverse), A006450, A048767, A257538, A290641.

Cf. A076610 (terms sorted into ascending order).

a:= n-> mul(ithprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 06 2018

Table[If[n == 1, 1, Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> Prime[p]^e]], {n, 58}] (* Michael De Vlieger, Aug 22 2017 *)

{ for (n=1, 1000, f=factor(n)~; a=1; for (i=1, length(f), a*=prime(f[1, i])^f[2, i]); write("b064988.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 02 2009

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]);); factorback(f);} \\ Michel Marcus, Aug 08 2017

from sympy import factorint, prime
from operator import mul
def a(n): return 1 if n==1 else reduce(mul, [prime(p)**e for p, e in factorint(n).items()])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Aug 08 2017

(define (A064988 n) (if (= 1 n) n (* (A000040 (A020639 n)) (A064988 (A032742 n))))) ;; Antti Karttunen, Aug 08 2017

From Antti Karttunen, Aug 08 & 22 2017: (Start)
For n = p_{i1} * p_{i2} * ... * p_{ik}, where the indices i1, i2, ..., ik of primes p are not necessarily distinct, a(n) = A006450(i1) * A006450(i2) * ... * A006450(ik).
a(n) = A003961(A290641(n)).
A046523(a(n)) = A046523(n). [Preserves the prime signature of n].
A003963(a(n)) = n.
(End)

A330945 Numbers whose prime indices are not all prime numbers.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 84, 86, 87

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:
   2: {{}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
  10: {{},{2}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  20: {{},{},{2}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  28: {{},{},{1,1}}
  29: {{1,3}}

Complement of A076610 (products of primes of prime index).
Numbers n such that A330944(n) > 0.
The restriction to odd terms is A330946.
The restriction to nonprimes is A330948.
The number of prime prime indices is given by A257994.
The number of nonprime prime indices is given by A330944.
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. A000040, A000720, A001222, A018252, A056239, A112798, A302242, A320633, A330943, A330947, A330949.

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

A302590 Squarefree numbers whose prime indices are prime numbers.

Original entry on oeis.org

1, 3, 5, 11, 15, 17, 31, 33, 41, 51, 55, 59, 67, 83, 85, 93, 109, 123, 127, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 241, 249, 255, 277, 283, 295, 327, 331, 335, 341, 353, 367, 381, 401, 415, 431, 451, 461, 465, 471, 509, 527, 537, 545, 547, 561, 563

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.
From David A. Corneth, Feb 05 2021: (Start)
Product_{p in A006450} (p + 1)/p where primepi(p) <= 10^k for k = 3..9 respectively is
3221793975627545730894469494385382768...
3962097386916566795581118542505513350...
4423525010102788492232765893521739629...
4739349879225654126399615785205666552...
4969363158706022367680967716958174889...
5144436325229538304870684054018856517...
5282263225826916578696019016723107071... (End)

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
001: {}
003: {{1}}
005: {{2}}
011: {{3}}
015: {{1},{2}}
017: {{4}}
031: {{5}}
033: {{1},{3}}
041: {{6}}
051: {{1},{4}}
055: {{2},{3}}
059: {{7}}
067: {{8}}
083: {{9}}
085: {{2},{4}}
093: {{1},{5}}
109: {{10}}
123: {{1},{6}}
127: {{11}}
155: {{2},{5}}
157: {{12}}
165: {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A302242, A302243, A302568.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[600],SquareFreeQ[#]&&And@@PrimeQ/@primeMS[#]&]

ok(n)={issquarefree(n) && !#select(p->!isprime(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

Intersection of A005117 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021

A331915 Numbers with exactly one prime prime index, counted with multiplicity.

Original entry on oeis.org

3, 5, 6, 10, 11, 12, 17, 20, 21, 22, 24, 31, 34, 35, 39, 40, 41, 42, 44, 48, 57, 59, 62, 65, 67, 68, 69, 70, 77, 78, 80, 82, 83, 84, 87, 88, 95, 96, 109, 111, 114, 115, 118, 119, 124, 127, 129, 130, 134, 136, 138, 140, 141, 143, 145, 147, 154, 156, 157, 159

Offset: 1

Gus Wiseman, Feb 08 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 begins:
    3: {2}             57: {2,8}            114: {1,2,8}
    5: {3}             59: {17}             115: {3,9}
    6: {1,2}           62: {1,11}           118: {1,17}
   10: {1,3}           65: {3,6}            119: {4,7}
   11: {5}             67: {19}             124: {1,1,11}
   12: {1,1,2}         68: {1,1,7}          127: {31}
   17: {7}             69: {2,9}            129: {2,14}
   20: {1,1,3}         70: {1,3,4}          130: {1,3,6}
   21: {2,4}           77: {4,5}            134: {1,19}
   22: {1,5}           78: {1,2,6}          136: {1,1,1,7}
   24: {1,1,1,2}       80: {1,1,1,1,3}      138: {1,2,9}
   31: {11}            82: {1,13}           140: {1,1,3,4}
   34: {1,7}           83: {23}             141: {2,15}
   35: {3,4}           84: {1,1,2,4}        143: {5,6}
   39: {2,6}           87: {2,10}           145: {3,10}
   40: {1,1,1,3}       88: {1,1,1,5}        147: {2,4,4}
   41: {13}            95: {3,8}            154: {1,4,5}
   42: {1,2,4}         96: {1,1,1,1,1,2}    156: {1,1,2,6}
   44: {1,1,5}        109: {29}             157: {37}
   48: {1,1,1,1,2}    111: {2,12}           159: {2,16}

These are numbers n such that A257994(n) = 1.
Prime-indexed primes are A006450, with products A076610.
The number of distinct prime prime indices is A279952.
Numbers with at least one prime prime index are A331386.
The set S of numbers with exactly one prime index in S are A331785.
The set S of numbers with exactly one distinct prime index in S are A331913.
Numbers with at most one prime prime index are A331914.
Numbers with exactly one distinct prime prime index are A331916.
Numbers with at most one distinct prime prime index are A331995.
Cf. A000040, A000720, A007097, A007821, A018252, A112798, A289509, A320628, A330944, A330945, A331784.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[primeMS[#],_?PrimeQ]==1&]

A049079 Primes prime(k) for which A049076(k) = 3.

Original entry on oeis.org

5, 59, 179, 331, 431, 599, 919, 1153, 1297, 1523, 1847, 2381, 2477, 2749, 3259, 3637, 3943, 4091, 4273, 4549, 5623, 5869, 6113, 6661, 6823, 7607, 7841, 8221, 8719, 9461, 9739, 9859, 11743, 11953, 12097, 12301, 12547, 13469, 13709, 14177, 14723, 14867

Offset: 1

Neil Fernandez

nonn

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

Cf. A049076, A007821, A049078, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046, A006450.

A049079 := proc(n)
        ithprime(A049078(n)) ;
end proc: # R. J. Mathar, Jul 07 2012

Nest[ Prime, Select[ Range[60], !PrimeQ[ # ] &], 3] (* Robert G. Wilson v, Mar 15 2004 *)

Definition edited by Zak Seidov, Sep 15 2013

A302540 Numbers whose prime indices other than 1 are prime numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 20, 22, 24, 25, 27, 30, 31, 32, 33, 34, 36, 40, 41, 44, 45, 48, 50, 51, 54, 55, 59, 60, 62, 64, 66, 67, 68, 72, 75, 80, 81, 82, 83, 85, 88, 90, 93, 96, 99, 100, 102, 108, 109, 110, 118, 120, 121, 123, 124

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A281113, A291686, A302242, A302243, A302534, A302539.

Select[Range[400],#===1||And@@(#===1||PrimeQ[#]&)/@PrimePi/@FactorInteger[#][[All,1]]&]

ok(n)={!#select(p->p>2 && !isprime(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

Sum_{n>=1} 1/a(n) = 2 * Sum_{n>=1} 1/A076610(n) = 2 * Product_{p in A006450} p/(p-1) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021

A049090 Primes for which A049076 >= 4.

Original entry on oeis.org

11, 31, 127, 277, 709, 1063, 1787, 2221, 3001, 4397, 5381, 7193, 8527, 9319, 10631, 12763, 15299, 15823, 19577, 21179, 22093, 24859, 27457, 30133, 33967, 37217, 38833, 40819, 42043, 43651, 52711, 55351, 57943, 60647, 66851, 68639, 72727

Offset: 1

Neil Fernandez

nonn

Union of A049080, A049081, A058322, A058324, etc. - R. J. Mathar, Jul 07 2012

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from Robert G. Wilson v)
R. G. Batchko, A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes, arXiv preprint arXiv:1405.2900 [math.GM], 2014.
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
N. Fernandez, More terms of this and other sequences related to A049076.

Cf. A000040, A006450, A038580, A049076, A049202, A049203, A057847, A057849, A057850, A057851, A058332, A093047.

map(ithprime@@3, select(isprime, [$1..157])); # Peter Luschny, Feb 17 2014

Nest[ Prime, Range[40], 4] (* Robert G. Wilson v, Mar 15 2004 *)

list(lim)=my(v=List(),q,r,s); forprime(p=2,lim,if(isprime(q++)&&isprime(r++)&&isprime(s++),listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Feb 16 2017

a(n) = A006450(A006450(n)). - James G. Merickel, Feb 14 2010
a(n) = A000040(A038580(n)). - R. J. Mathar, Jul 07 2012
a(n) ~ n (log n)^4. - Charles R Greathouse IV, Feb 16 2017

Name corrected by Sean A. Irvine, Jul 18 2021

cp's OEIS Frontend

A330944 Number of nonprime prime indices of n.

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A049081 Primes prime(k) for which A049076(k) = 5.

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A320629 Products of odd primes of nonprime index.

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A064988 Multiplicative with a(p^e) = prime(p)^e.

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A330945 Numbers whose prime indices are not all prime numbers.

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A302590 Squarefree numbers whose prime indices are prime numbers.

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A331915 Numbers with exactly one prime prime index, counted with multiplicity.

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A049079 Primes prime(k) for which A049076(k) = 3.

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