A049084 -id:A049084 - cp's OEIS frontend

A095874 a(n) = k if n = A000961(k) (powers of primes), a(n) = 0 if n is not in A000961.

1, 2, 3, 4, 5, 0, 6, 7, 8, 0, 9, 0, 10, 0, 0, 11, 12, 0, 13, 0, 0, 0, 14, 0, 15, 0, 16, 0, 17, 0, 18, 19, 0, 0, 0, 0, 20, 0, 0, 0, 21, 0, 22, 0, 0, 0, 23, 0, 24, 0, 0, 0, 25, 0, 0, 0, 0, 0, 26, 0, 27, 0, 0, 28, 0, 0, 29, 0, 0, 0, 30, 0, 31, 0, 0, 0, 0, 0, 32, 0, 33, 0, 34, 0, 0, 0, 0, 0, 35, 0

Offset: 1

Reinhard Zumkeller, Jun 10 2004

nonn

The name has been edited to clarify that the indices k refer to A000961 ("powers of primes" = {1} U A246655) and not to the list A246655 of proper prime powers. - M. F. Hasler, Jun 16 2021

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

Cf. A000961 (right inverse), A049084, A097621.

a095874 n | y == n    = length xs + 1
          | otherwise = 0
          where (xs, y:ys) = span (< n) a000961_list
-- Reinhard Zumkeller, Feb 16 2012, Jun 26 2011

Join[{1},Module[{k=2},Table[If[PrimePowerQ[n],k;k++,0],{n,2,100}]]] (* Harvey P. Dale, Aug 15 2020 *)

a(n)=if(isprimepower(n), sum(i=1,logint(n,2), primepi(sqrtnint(n,i)))+1, n==1) \\ Charles R Greathouse IV, Apr 29 2015

{M95874=Map(); A095874(n,k)=if(mapisdefined(M95874,n,&k),k, isprimepower(n), mapput(M95874,n, k=sum(i=1,exponent(n), primepi(sqrtnint(n,i)))+1); k,n==1)} \\ Variant with memoization, possibly useful to compute A097621, A344826 and related. One may omit "isprimepower(n)," (possibly requiring factorization) and ",n==1" if n is known to be a power of a prime, i.e., to get a left inverse for A000961. - M. F. Hasler, Jun 15 2021

from sympy import primepi, integer_nthroot, primefactors
def A095874(n): return 1+int(primepi(n)+sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length()))) if n==1 or len(primefactors(n))==1 else 0 # Chai Wah Wu, Jan 19 2025

a(n) = Sum_{1 <= k <= n} A010055(k); [corrected by M. F. Hasler, Jun 15 2021]
a(n) = A065515(n)*(A065515(n)-A065515(n-1)).
a(n) = A065515(n)*A069513(n). - M. F. Hasler, Jun 16 2021

Edited by M. F. Hasler, Jun 15 2021

A045616 Primes p such that 10^(p-1) == 1 (mod p^2).

Original entry on oeis.org

3, 487, 56598313

Offset: 1

Helmut Richter, Dec 11 1999

Primes p such that the decimal fraction 1/p has same period length as 1/p^2, i.e., the multiplicative order of 10 modulo p is the same as the multiplicative order of 10 modulo p^2. [extended by Felix Fröhlich, Feb 05 2017]
No further terms below 1.172*10^14 (as of Feb 2020, cf. Fischer's table).
56598313 was announced in the paper by Brillhart et al. - Helmut Richter, May 17 2004
A265012(A049084(a(n))) = 1. - Reinhard Zumkeller, Nov 30 2015
From Jianing Song, Jun 21 2025: (Start)
The ring of integers of Q(10^(1/k)) is Z[10^(1/k)] if and only if k does not have a prime factor in this sequence. See Theorem 5.3 of the paper of Keith Conrad. For example, we have:
  (1 + 10^(1/3) + 10^(2/3))/3 is an algebraic integer, but it is not in Z[10^(1/3)];
  (1 + 10^(486/487) + 10^(2*486/487) + ... + 10^(486*486/487))/487 is an algebraic integer, but it is not in Z[10^(1/487)];
  (1 + 10^(56598312/56598313) + 10^(2*56598312/56598313) + ... + 10^(56598312*56598312/56598313))/56598313 is an algebraic integer, but it is not in Z[10^(1/56598313)]. (End)

J. Brillhart, J. Tonascia, and P. Weinberger, On the Fermat quotient, pp. 213-222 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
Richard K. Guy, Unsolved Problems in Number Theory, Springer, 2004, A3.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Keith Conrad, The ring of integers in a radical extension.
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2).
Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp. 74 (2005), 927-936.
Peter L. Montgomery, New solutions of a^(p-1) == 1 (mod p^2), Math. Comp. 61 (1993), 361-363.
Math Overflow, Is the smallest primitive root modulo p a primitive root modulo p^2?, Jun 09 2010.
Helmut Richter, The period length of the decimal expansion of a fraction.
Helmut Richter, The Prime Factors Of 10^486-1.
Siqiong Yao and Akira Toyohara, The length of the repeating decimal, arXiv:2507.01295 [math.NT], 2025. See p. 19.
Samuel Yates, The Mystique of Repunits, Math. Mag. 51 (1978), 22-28.

Cf. A001220, A014127, A123692, A212583, A123693, A111027, A128667, A234810, A242741, A128668, A244260, A090968, A242982, A128669, A039951.

Cf. A265012, A049084, A000040.

import Math.NumberTheory.Moduli (powerMod)
a045616 n = a045616_list !! (n-1)
a045616_list = filter
               (\p -> powerMod 10 (p - 1) (p ^ 2) == 1) a000040_list'
-- Reinhard Zumkeller, Nov 30 2015

A045616Q = PrimeQ@# && PowerMod[10, # - 1, #^2] == 1 &; Select[Range[1000000], A045616Q] (* JungHwan Min, Feb 04 2017 *)
Select[Prime[Range[34*10^5]],PowerMod[10,#-1,#^2]==1&] (* Harvey P. Dale, Apr 10 2018 *)

lista(nn) = forprime(p=2, nn, if (Mod(10, p^2)^(p-1)==1, print1(p, ", "))); \\ Michel Marcus, Aug 16 2015

A064553 a(1) = 1, a(prime(i)) = i + 1 for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16, 8, 18, 9, 16, 15, 12, 10, 24, 16, 14, 27, 20, 11, 24, 12, 32, 18, 16, 20, 36, 13, 18, 21, 32, 14, 30, 15, 24, 36, 20, 16, 48, 25, 32, 24, 28, 17, 54, 24, 40, 27, 22, 18, 48, 19, 24, 45, 64, 28, 36, 20, 32, 30, 40, 21, 72, 22, 26

Offset: 1

Reinhard Zumkeller, Sep 21 2001

a(n) <= n for all n and a(x) = x iff x = 2^i * 3^j for i, j >= 0: a(A003586(n)) = A003586(n) for n > 0. By definition a is completely multiplicative and also surjective. a(p) < a(q) for primes p < q.
Completely multiplicative with a(prime(i)) = i + 1. - Charles R Greathouse IV, Sep 07 2012
a(A080688(n,k)) = A080444(n,k) = n for k=1..A001055(n). - Reinhard Zumkeller, Oct 01 2012

			a(5) = a(prime(3)) = 3 + 1 = 4; a(14) = a(2*7) = a(prime(1)* prime(4)) = (1+1)*(4+1) = 10.

T. D. Noe, Table of n, a(n) for n = 1..8000
T. D. Noe, Plot of A064553
Index to divisibility sequences
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A000040, A003961, A003963, A049084, A020639, A064554, A064555, A001055, A003586, A064557, A064558, A027746, A027748, A124010, A181819.

A left inverse of A290641.

a064553 1 = 1
a064553 n = product $ map ((+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Feb 17 2012, Jan 28 2011

A064553 := proc(n)
    local a,f,p,e ;
    a := 1 ;
    for f in ifactors(n)[2] do
        p :=op(1,f) ;
        e :=op(2,f) ;
        a := a*(numtheory[pi](p)+1)^e ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 07 2012

nn=100; a=Table[0, {nn}]; a[[1]]=1; Do[If[PrimeQ[i], a[[i]]=PrimePi[i]+1, p=FactorInteger[i][[1,1]]; a[[i]] = a[[p]]*a[[i/p]]], {i, 2, nn}]; a (* T. D. Noe, Dec 12 2004, revised Sep 27 2011 *)
Array[Apply[Times, Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[ #]] /. p_ /; PrimeQ@ p :> PrimePi@ p + 1] &, 74] (* Michael De Vlieger, Aug 22 2017 *)

A064553(n)={n=factor(n);n[,1]=apply(f->1+primepi(f),n[,1]);factorback(n)} \\ M. F. Hasler, Aug 28 2012

(define (A064553 n) (if (= 1 n) n (* (+ 1 (A055396 n)) (A064553 (A032742 n))))) ;; Antti Karttunen, Aug 22 2017

a(A000040(n)) = n+1.
Let the prime factorization of n be p1^e1...pk^ek, then a(n) = (pi(p1)+1)^e1...(pi(pk)+1)^ek, where pi(p) is the index of prime p. - T. D. Noe, Dec 12 2004
From Antti Karttunen, Aug 22 2017: (Start)
a(n) = A003963(A003961(n)).
a(A181819(n)) = A000005(n).
a(A290641(n)) = n. (End)

Displayed values double-checked with new PARI code by M. F. Hasler, Aug 28 2012

A286469 a(n) = maximum of {the index of least prime dividing n} and {the maximal gap between indices of the successive primes in the prime factorization of n}.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 2, 5, 1, 6, 3, 2, 1, 7, 1, 8, 2, 2, 4, 9, 1, 3, 5, 2, 3, 10, 1, 11, 1, 3, 6, 3, 1, 12, 7, 4, 2, 13, 2, 14, 4, 2, 8, 15, 1, 4, 2, 5, 5, 16, 1, 3, 3, 6, 9, 17, 1, 18, 10, 2, 1, 3, 3, 19, 6, 7, 2, 20, 1, 21, 11, 2, 7, 4, 4, 22, 2, 2, 12, 23, 2, 4, 13, 8, 4, 24, 1, 4, 8, 9, 14, 5, 1, 25, 3, 3, 2, 26, 5, 27, 5, 2, 15, 28, 1, 29, 2, 10, 3

Offset: 1

Antti Karttunen, May 14 2017

nonn

This gives the maximal gap between the indices of successive prime factors p_i <= p_j <= ... <= p_k of n = p_i * p_j * ... * p_k when the index of the least prime factor p_i (A055396) is considered as the initial gap from the "level zero".

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A051903, A055396, A122111, A286470, A286621.

from sympy import primepi, isprime, primefactors, divisors
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def x(n): return 1 if n==1 else divisors(n)[-2]
def a286470(n): return 0 if n==1 or len(primefactors(n))==1 else max(a055396(x(n)) - a055396(n), a286470(x(n)))
def a(n): return max(a055396(n), a286470(n)) # Indranil Ghosh, May 17 2017

(define (A286469 n) (max (A055396 n) (A286470 n)))

a(n) = max(A055396(n), A286470(n)).
a(n) = A051903(A122111(n)).
For all i, j: A286621(i) = A286621(j) => a(i) = a(j). [Because of the above formula.]

Definition corrected May 17 2017

A039709 a(n) = n-th prime modulo 11.

Original entry on oeis.org

2, 3, 5, 7, 0, 2, 6, 8, 1, 7, 9, 4, 8, 10, 3, 9, 4, 6, 1, 5, 7, 2, 6, 1, 9, 2, 4, 8, 10, 3, 6, 10, 5, 7, 6, 8, 3, 9, 2, 8, 3, 5, 4, 6, 10, 1, 2, 3, 7, 9, 2, 8, 10, 9, 4, 10, 5, 7, 2, 6, 8, 7, 10, 3, 5, 9, 1, 7, 6, 8, 1, 7, 4, 10, 5, 9, 4, 1, 5, 2, 1, 3, 2, 4, 10, 3

Offset: 1

Clark Kimberling, Dec 11 1999

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000040, A039703, A039705, A039710-A039715, A049084, A137977, A137978.

[p mod 11: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

seq(ithprime(n) mod 11, n=1..100); # Nathaniel Johnston, Jun 29 2011

Mod[Prime@ Range@ 86, 11] (* Robert G. Wilson v *)

primes(1000)%11 \\ Charles R Greathouse IV, Mar 13 2011

a(A049084(A137977(n-1))) = even; a(A049084(A137978(n-1))) = odd. - Reinhard Zumkeller, Feb 25 2008

Sum_k={1..n} a(k) ~ (11/2)*n. - Amiram Eldar, Dec 11 2024

A073490 Number of prime gaps in factorization of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 03 2002

A137723(n) is the smallest number of the first occurring set of exactly n consecutive numbers with at least one prime gap in their factorization: a(A137723(n)+k)>0 for 0<=kA137723(n)-1)=a(A137723(n)+n)=0. - Reinhard Zumkeller, Feb 09 2008

			84 = 2*2*3*7 with one gap between 3 and 7, therefore a(84) = 1;
110 = 2*5*11 with two gaps: between 2 and 5 and between 5 and 11, therefore a(110) = 2.

R. Zumkeller, Table of n, a(n) for n = 1..50000

Cf. A073491, A073492, A073493, A073494, A073495.

Cf. A137721, A137722, A027748, A049084.

a073490 1 = 0
a073490 n = length $ filter (> 1) $ zipWith (-) (tail ips) ips
   where ips = map a049084 $ a027748_row n
-- Reinhard Zumkeller, Jul 04 2012

A073490 := proc(n)
    local a,plist ;
    plist := sort(convert(numtheory[factorset](n),list)) ;
    a := 0 ;
    for i from 2 to nops(plist) do
        if op(i,plist) <> nextprime(op(i-1,plist)) then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A073490(n),n=1..110) ; # R. J. Mathar, Oct 27 2019

gaps[n_Integer/;n>0]:=If[n===1, 0, Complement[Prime[PrimePi[Rest[ # ]]-1], # ]&[First/@FactorInteger[n]]]; Table[Length[gaps[n]], {n, 1, 120}] (* Wouter Meeussen, Oct 30 2004 *)
pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Table[Total[pa @@@ Partition[First /@ FactorInteger[n], 2, 1]], {n, 120}] (* Jayanta Basu, Jul 01 2013 *)

from sympy import primefactors, nextprime
def a(n):
    pf = primefactors(n)
    return sum(p2 != nextprime(p1) for p1, p2 in zip(pf[:-1], pf[1:]))
print([a(n) for n in range(1, 121)]) # Michael S. Branicky, Oct 14 2021

a(n) = A073484(A007947(n)).
a(A000040(n))=0; a(A000961(n))=0; a(A006094(n))=0; a(A002110(n))=0; a(A073485(n))=0.
a(A073486(n))>0; a(A073487(n)) = 1; a(A073488(n))=2; a(A073489(n))=3.
a(n)=0 iff A073483(n) = 1.
a(A097889(n)) = 0. - Reinhard Zumkeller, Nov 20 2004
0 <= a(m*n) <= a(m) + a(n) + 1. A137794(n) = 0^a(n). - Reinhard Zumkeller, Feb 11 2008

More terms from Franklin T. Adams-Watters, May 19 2006

A080148 Positions of primes of the form 4*k+3 (A002145) among all primes (A000040).

Original entry on oeis.org

2, 4, 5, 8, 9, 11, 14, 15, 17, 19, 20, 22, 23, 27, 28, 31, 32, 34, 36, 38, 39, 41, 43, 46, 47, 48, 49, 52, 54, 56, 58, 61, 63, 64, 67, 69, 72, 73, 75, 76, 81, 83, 85, 86, 90, 91, 92, 93, 94, 95, 96, 99, 101, 103, 105, 107, 109, 111, 114, 115, 117, 118, 120, 124, 125, 128

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

It appears that a(n) = k such that binomial(prime(k),3) mod 2 = 1. See Maple code. - Gary Detlefs, Dec 06 2011
The above is correct (work mod 4). - Charles R Greathouse IV, Dec 06 2011
The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

Zak Seidov, Table of n, a(n) for n = 1..10000

Almost complement of A080147 (1 is excluded from both).

Cf. A000040, A002145, A049084.

pos_of_primes_k_mod_n(300,3,4); # Given in A080147.
A080148 := proc(n)
        numtheory[pi](A002145(n)) ;
end proc:
seq(A080148(n),n=1..40) ; # R. J. Mathar, Dec 08 2011

Flatten[Position[Prime[Range[200]],?(IntegerQ[(#-3)/4]&)]] (* _Harvey P. Dale, Jun 06 2011 *)
Select[Range[135], Mod[Prime[#], 4] == 3 &] (* Amiram Eldar, Mar 01 2021 *)

i=0;forprime(p=2,1e3,i++;if(p%4==3,print1(i", "))) \\ Charles R Greathouse IV, Dec 06 2011

a(n) = A049084(A002145(n)). - R. J. Mathar, Oct 06 2008

A112049 a(n) = position of A112046(n) in A000040.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

A112051 gives the first positions of distinct new values in this sequence, that seem also to be the positions of the first occurrence of each n, and thus the positions of the records. Compare also to A084921. - Antti Karttunen, May 26 2017

Indranil Ghosh (terms 1..1000) & Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A049084, A112046, A112051, A112060, A165471, A268829, A286465, A286466.

Cf. A286579 (ordinal transform).

a112046[n_]:=Block[{i=1},While[JacobiSymbol[i, 2n + 1]==1, i++]; i];a049084[n_]:=If[PrimeQ[n], PrimePi[n], 0]; Table[a049084[a112046[n]], {n, 102}] (* Indranil Ghosh, May 11 2017 *)

A112049(n) = for(i=1, (2*n), if((kronecker(i, (n+n+1)) < 1), return(primepi(i)))); \\ Antti Karttunen, May 26 2017

from sympy import jacobi_symbol as J, isprime, primepi
def a049084(n):
    return primepi(n) if isprime(n) else 0
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a049084(a112046(n))
print([a(n) for n in range(1, 103)]) # Indranil Ghosh, May 11 2017

a(n) = A049084(A112046(n)).

Unnecessary fallback-clause removed from the name by Antti Karttunen, May 26 2017

A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 5, 1, 0, 5, 9, 7, 7, 1, 0, 6, 7, 25, 11, 11, 1, 0, 7, 15, 11, 49, 13, 13, 1, 0, 8, 11, 35, 13, 121, 17, 17, 1, 0, 9, 27, 13, 77, 17, 169, 19, 19, 1, 0, 10, 25, 125, 17, 143, 19, 289, 23, 23, 1, 0, 11, 21, 49, 343, 19, 221, 23, 361, 29, 29, 1, 0

Offset: 0

Antti Karttunen, May 12 2014

Each row i is a multiplicative function, being in essence "the i-th power" of A003961, i.e., A(i,j) = A003961^i (j). Zeroth power gives an identity function, A001477, which occurs as the row zero.
The terms in the same column have the same prime signature.
The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

			The top-left corner of the array:
  0,   1,   2,   3,   4,   5,   6,   7,   8, ...
  0,   1,   3,   5,   9,   7,  15,  11,  27, ...
  0,   1,   5,   7,  25,  11,  35,  13, 125, ...
  0,   1,   7,  11,  49,  13,  77,  17, 343, ...
  0,   1,  11,  13, 121,  17, 143,  19,1331, ...
  0,   1,  13,  17, 169,  19, 221,  23,2197, ...
...
A(2,6) = A003961(A003961(6)) = p_{1+2} * p_{2+2} = p_3 * p_4 = 5 * 7 = 35, because 6 = 2*3 = p_1 * p_2.

Antti Karttunen, Table of n, a(n) for n = 0..10439; Antidiagonals n = 0..143, flattened

Taking every second column from column 2 onward gives array A246278 which is a permutation of natural numbers larger than 1.
Transpose: A242379.
Row 0: A001477, Row 1: A003961 (from 1 onward), Row 2: A357852 (from 1 onward), Row 3: A045968 (from 7 onward), Row 4: A045970 (from 11 onward).
Column 2: A000040 (primes), Column 3: A065091 (odd primes), Column 4: A001248 (squares of primes), Column 6: A006094 (products of two successive primes), Column 8: A030078 (cubes of primes).
Excluding column 0, a subtable of A297845.
Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

A(0,j) = j, A(i,0) = 0, A(i > 0, j > 0) = A003961(A(i-1,j)).

For j > 0, A(i,j) = A297845(A000040(i+1),j) = A297845(j,A000040(i+1)). - Peter Munn, Sep 02 2025

A256617 Numbers having exactly two distinct prime factors, which are also adjacent prime numbers.

Original entry on oeis.org

6, 12, 15, 18, 24, 35, 36, 45, 48, 54, 72, 75, 77, 96, 108, 135, 143, 144, 162, 175, 192, 216, 221, 225, 245, 288, 323, 324, 375, 384, 405, 432, 437, 486, 539, 576, 648, 667, 675, 768, 847, 864, 875, 899, 972, 1125, 1147, 1152, 1215, 1225, 1296, 1458, 1517, 1536, 1573, 1715, 1728, 1763, 1859, 1875, 1944

Offset: 1

Reinhard Zumkeller, Apr 05 2015

nonn

			.   n | a(n)                      n | a(n)
. ----+------------------       ----+------------------
.   1 |   6 = 2 * 3              13 |  77 = 7 * 11
.   2 |  12 = 2^2 * 3            14 |  96 = 2^5 * 3
.   3 |  15 = 3 * 5              15 | 108 = 2^2 * 3^3
.   4 |  18 = 2 * 3^2            16 | 135 = 3^3 * 5
.   5 |  24 = 2^3 * 3            17 | 143 = 11 * 13
.   6 |  35 = 5 * 7              18 | 144 = 2^4 * 3^2
.   7 |  36 = 2^2 * 3^2          19 | 162 = 2 * 3^4
.   8 |  45 = 3^2 * 5            20 | 175 = 5^2 * 7
.   9 |  48 = 2^4 * 3            21 | 192 = 2^6 * 3
.  10 |  54 = 2 * 3^3            22 | 216 = 2^3 * 3^3
.  11 |  72 = 2^3 * 3^2          23 | 221 = 13 * 17
.  12 |  75 = 3 * 5^2            24 | 225 = 3^2 * 5^2 .

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

Subsequence of A007774.
Subsequences: A006094, A033845, A033849, A033851.
Cf. A000040, A001222, A020639, A006530, A049084, A083553, A151800.

import Data.Set (singleton, deleteFindMin, insert)
a256617 n = a256617_list !! (n-1)
a256617_list = f (singleton (6, 2, 3)) $ tail a000040_list where
   f s ps@(p : ps'@(p':_))
     | m < p * p' = m : f (insert (m * q, q, q')
                          (insert (m * q', q, q') s')) ps
     | otherwise  = f (insert (p * p', p, p') s) ps'
     where ((m, q, q'), s') = deleteFindMin s

Select[Range[2000], MatchQ[FactorInteger[#], {{p_, }, {q, }} /; q == NextPrime[p]]&] (* _Jean-François Alcover, Dec 31 2017 *)

is(n) = if(omega(n)!=2, return(0), my(f=factor(n)[, 1]~); if(f[2]==nextprime(f[1]+1), return(1))); 0 \\ Felix Fröhlich, Dec 31 2017

list(lim)=my(v=List(),c=sqrtnint(lim\=1,3),d=nextprime(c+1),p=2); forprime(q=3,d, for(i=1,logint(lim\q,p), my(t=p^i); while((t*=q)<=lim, listput(v,t))); p=q); forprime(q=d+1,lim\precprime(sqrtint(lim)), listput(v,p*q); p=q); Set(v) \\ Charles R Greathouse IV, Apr 12 2020

from sympy import primefactors, nextprime
A256617_list = []
for n in range(1,10**5):
    plist = primefactors(n)
    if len(plist) == 2 and plist[1] == nextprime(plist[0]):
        A256617_list.append(n) # Chai Wah Wu, Aug 23 2021

A001222(a(n)) = 2.
A006530(a(n)) = A151800(A020639(n)) = A000040(A049084(A020639(a(n)))+1).
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/A083553(n) = Sum_{n>=1} 1/((prime(n)-1)*(prime(n+1)-1)) = 0.7126073495... - Amiram Eldar, Dec 23 2020

cp's OEIS Frontend

A095874 a(n) = k if n = A000961(k) (powers of primes), a(n) = 0 if n is not in A000961.

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A045616 Primes p such that 10^(p-1) == 1 (mod p^2).

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A064553 a(1) = 1, a(prime(i)) = i + 1 for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.

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A286469 a(n) = maximum of {the index of least prime dividing n} and {the maximal gap between indices of the successive primes in the prime factorization of n}.

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A039709 a(n) = n-th prime modulo 11.

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A073490 Number of prime gaps in factorization of n.

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