A002035 -id:A002035 - cp's OEIS frontend

A352518 Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves prime numbers.

225, 675, 1089, 1125, 2601, 3025, 3267, 3375, 6075, 7225, 7803, 8649, 11979, 15125, 15129, 24025, 25947, 27225, 28125, 29403, 30375, 31329, 33275, 34969, 35937, 36125, 40401, 42025, 44217, 45387, 54675, 62001, 65025, 70227, 81675, 84375, 87025, 93987

Offset: 1

Gus Wiseman, Mar 24 2022

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 terms together with their prime indices (not factors) begin:
     225: {2,2,3,3}
     675: {2,2,2,3,3}
    1089: {2,2,5,5}
    1125: {2,2,3,3,3}
    2601: {2,2,7,7}
    3025: {3,3,5,5}
    3267: {2,2,2,5,5}
    3375: {2,2,2,3,3,3}
    6075: {2,2,2,2,2,3,3}
    7225: {3,3,7,7}
    7803: {2,2,2,7,7}
    8649: {2,2,11,11}
   11979: {2,2,5,5,5}
   15125: {3,3,3,5,5}
   15129: {2,2,13,13}
   24025: {3,3,11,11}
   25947: {2,2,2,11,11}
   27225: {2,2,3,3,5,5}
   28125: {2,2,3,3,3,3,3}
For example, 7803 = prime(1)^3 prime(4)^2.

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

These partitions are counted by A352493.
This is the restriction of A346068 to numbers that are not a prime power.
The prime-power version is A352519, counted by A230595.
A000040 lists the primes.
A000961 lists prime powers.
A001694 lists powerful numbers, counted by A007690.
A038499 counts partitions of prime length.
A053810 lists all numbers p^q for p and q prime, counted by A001221.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A001597, A002035, A007821, A007916, A101436, A117958, A164336, A268335, A330945.

Select[Range[10000],!PrimePowerQ[#]&& And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&& And@@PrimeQ/@Last/@FactorInteger[#]&]

Sum_{n>=1} 1/a(n) = (Product_{p prime-indexed prime} (1 + Sum_{q prime} 1/p^q)) - (Sum_{p prime-indexed prime} Sum_{q prime} 1/p^q) - 1 = 0.0106862606... . - Amiram Eldar, Aug 04 2024

A352519 Numbers of the form prime(p)^q where p and q are primes. Prime powers whose prime index and exponent are both prime.

Original entry on oeis.org

9, 25, 27, 121, 125, 243, 289, 961, 1331, 1681, 2187, 3125, 3481, 4489, 4913, 6889, 11881, 16129, 24649, 29791, 32041, 36481, 44521, 58081, 68921, 76729, 78125, 80089, 109561, 124609, 134689, 160801, 161051, 177147, 185761, 205379, 212521, 259081, 299209

Offset: 1

Gus Wiseman, Mar 26 2022

nonn

Alternatively, numbers of the form prime(prime(i))^prime(j) for some positive integers i, j.

			The terms together with their prime indices begin:
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    125: {3,3,3}
    243: {2,2,2,2,2}
    289: {7,7}
    961: {11,11}
   1331: {5,5,5}
   1681: {13,13}
   2187: {2,2,2,2,2,2,2}
   3125: {3,3,3,3,3}
   3481: {17,17}
   4489: {19,19}
   4913: {7,7,7}
   6889: {23,23}
  11881: {29,29}
  16129: {31,31}
  24649: {37,37}
  29791: {11,11,11}

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

Numbers of the form p^q for p and q prime are A053810, counted by A001221.
These partitions are counted by A230595.
This is the prime power case of A346068.
For numbers that are not a prime power we have A352518, counted by A352493.
A000040 lists the primes.
A000961 lists prime powers.
A001597 lists perfect powers.
A001694 lists powerful numbers, counted by A007690.
A056166 = prime exponents are all prime, counted by A055923.
A076610 = prime indices are all prime, counted by A000607, powerful A339218.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A164336 lists all possible power-towers of prime numbers.
A257994 counts prime indices that are themselves prime, nonprime A330944.
A325131 = disjoint indices from exponents, counted by A114639.
Cf. A000720, A002035, A005117, A007821, A038499, A101436, A181819, A181821.

N:= 10^7: # for terms <= N
M:=numtheory:-pi(numtheory:-pi(isqrt(N))):
PP:= {seq(ithprime(ithprime(i)),i=1..M)}:
R:= NULL:
for p in PP do
  q:= 1:
  do
    q:= nextprime(q);
    t:= p^q;
    if t > N then break fi;
    R:= R, t;
  od;
od:
sort([R]); # Robert Israel, Dec 08 2022

Select[Range[10000],PrimePowerQ[#]&&MatchQ[FactorInteger[#],{{?(PrimeQ[PrimePi[#]]&),k?PrimeQ}}]&]

from sympy import primepi, integer_nthroot, primerange
def A352519(n):
    def f(x): return int(n+x-sum(primepi(primepi(integer_nthroot(x,p)[0])) for p in primerange(x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

A340901 Additive with a(p^e) = (-p)^e.

Original entry on oeis.org

0, -2, -3, 4, -5, -5, -7, -8, 9, -7, -11, 1, -13, -9, -8, 16, -17, 7, -19, -1, -10, -13, -23, -11, 25, -15, -27, -3, -29, -10, -31, -32, -14, -19, -12, 13, -37, -21, -16, -13, -41, -12, -43, -7, 4, -25, -47, 13, 49, 23, -20, -9, -53, -29, -16, -15, -22, -31

Offset: 1

Sebastian Karlsson, Jan 26 2021

sign

The sequence contains every integer infinitely many times.
Proof (outline):
1. Every integer m > 9 is the sum of distinct odd primes [R. E. Dressler].
2. Any integer k (positive as negative) can be written as k = 4^e - m, for sufficiently large and infinitely many e > 0 and m > 9.
3. Pick an arbitrary integer k and write it like k = 4^e - m. Let p_1, p_2, ..., p_i be distinct odd primes such that p_1 + p_2 + ... + p_i = m. Then a(p_1*p_2*...*p_i*4^e) = 4^e - m = k. Since there are infinitely many representations of any k of the form 4^e - m, this means that there are infinitely many n such that a(n) = k.
Q.E.D.

			a(20) = a(2^2*5) = (-2)^2 + (-5) = -1.

Sebastian Karlsson, Table of n, a(n) for n = 1..10000
R. E. Dressler, A stronger Bertrand's postulate with an application to partitions, Proc. Amer. Math. Soc., 33 (1972), 226-228.

Cf. A001414, A002035, A008472, A008475, A008476, A316523.

a[n_] := Total@ (((-First[#])^Last[#]) & /@ FactorInteger[n]); a[1] = 0; Array[a, 100] (* Amiram Eldar, May 15 2023 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, (-f[k,1])^f[k,2]); \\ Michel Marcus, Jan 26 2021
(APL, Dyalog dialect) A340901 ← {1=⍵:0 ⋄ +/{(-⍺)*≢⍵}⌸factors(⍵)} ⍝ Needs also factors function from https://dfns.dyalog.com/c_factors.htm - Antti Karttunen, Feb 16 2024

from sympy import primefactors as pf, multiplicity as mult
def a(n):
    return sum([(-p)**mult(p, n) for p in pf(n)])
for n in range(1, 59):
    print(a(n), end=', ')

a(A002035(n)) = - A008475(A002035(n)).

a(n^2) = A008475(n^2).

A072586 Number of numbers <= n having prime factors with odd exponents only.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 16, 17, 17, 18, 19, 19, 20, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 32, 33, 33, 33, 34, 35, 35, 35, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 43, 44, 45, 45, 45, 46, 47, 48, 48, 49, 50, 51, 51, 52

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

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

Cf. A000196, A002035, A065463, A295316.

Accumulate @ Table[If[n > 1 && AllTrue[FactorInteger[n][[;; , 2]], OddQ], 1, 0], {n, 1, 100}] (* Amiram Eldar, Sep 18 2022 *)

a(n) ~ c * n , where c = A065463. - Amiram Eldar, Sep 18 2022

A072588 Numbers having at least one prime factor with an odd and one with an even exponent.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 60, 63, 68, 72, 75, 76, 80, 84, 90, 92, 98, 99, 108, 112, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 171, 172, 175, 176, 180, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228, 234, 236, 240, 242, 244

Offset: 1

Reinhard Zumkeller, Jun 23 2002

nonn

= Complement(Union(A002035, A000290)) = Intersection(A000037, A072587);
a(k)=A070011(k) for 0A070011(26)=120 is not a term, as 120=5*3*2^3 having only odd exponents (A002035(85)=120), and a(54)=240 is not a term of A070011, as from 240=5*3*2^4 follows that A001222(240)/A001221(240)=6/3=2 is an integer.
The asymptotic density of this sequence is 1 - A065463 = 0.2955577990... - Amiram Eldar, Sep 18 2022
Numbers k such that A007913(k) properly divides A007947(k). (Same as A072587 without square terms). A number k is in this sequence iff 1 < A007913(k) < A007947(k) < k, and A007913(k)|A007947(k), equivalently iff k is not in A000037 and A336643(k) is squarefree. - David James Sycamore, Sep 20 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000037, A000290, A001221, A001222, A002035, A065463, A070011, A072587, A124010.

Cf. A007913, A007947, A336643.

a072588 n = a072588_list !! (n-1)
a072588_list = filter f [1..] where
   f x = any odd es && any even es  where es = a124010_row x
-- Reinhard Zumkeller, Nov 15 2012

oeeQ[n_]:=Module[{fi=Transpose[FactorInteger[n]][[2]]},Count[fi,?OddQ]>0  && Count[fi,?EvenQ]>0]; Select[Range[250],oeeQ] (* Harvey P. Dale, Jun 21 2015 *)

is(n)=#Set(factor(n)[,2]%2)==2 \\ Charles R Greathouse IV, Oct 16 2015

A167759 Numbers k such that d(k) is an isolated number (A167706).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92

Offset: 1

Juri-Stepan Gerasimov, Nov 11 2009

nonn

Isolated numbers (A167706) are 2, 4, 6, 12, 18, 23, 30, 37, .... Sequence lists numbers k such that the number of divisors of k is isolated number. Also, the positions of isolated numbers in A000005.

			A000005(a(1)=2)=2; A000005(a(2)=3)=2; A000005(a(3)=5)=2; A000005(a(4)=6)=4.

Cf. A000005, A002035, A167706.

isA007510 := proc(n) if isprime(n) then not isprime(n+2) and not isprime(n-2) ; else false; end if; end proc: isA014574 := proc(n) isprime(n+1) and isprime(n-1) ; end proc: isA167706 := proc(n) isA007510(n) or isA014574(n) ; end proc: isA167759 := proc(n) isA167706(numtheory[tau](n)) ; end proc: for n from 1 to 100 do if isA167759(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Nov 16 2009

A000005(a(n)) is in A167706.

Edited by Jon E. Schoenfield, May 10 2019

cp's OEIS Frontend

A352518 Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves prime numbers.

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A352519 Numbers of the form prime(p)^q where p and q are primes. Prime powers whose prime index and exponent are both prime.

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A340901 Additive with a(p^e) = (-p)^e.

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A072586 Number of numbers <= n having prime factors with odd exponents only.

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A072588 Numbers having at least one prime factor with an odd and one with an even exponent.

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A167759 Numbers k such that d(k) is an isolated number (A167706).

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