A006093 -id:A006093 - cp's OEIS frontend

A075157 Run lengths in the binary expansion of n gives the vector of exponents in prime factorization of a(n)+1, with the least significant run corresponding to the exponent of the least prime, 2; with one subtracted from each run length, except for the most significant run of 1's.

0, 1, 2, 3, 5, 4, 8, 7, 11, 14, 6, 9, 17, 24, 26, 15, 23, 44, 34, 29, 13, 10, 20, 19, 35, 74, 48, 49, 53, 124, 80, 31, 47, 134, 174, 89, 69, 76, 104, 59, 27, 32, 12, 21, 41, 54, 62, 39, 71, 224, 244, 149, 97, 120, 146, 99, 107, 374, 342, 249, 161, 624, 242, 63, 95, 404

Offset: 0

Antti Karttunen, Sep 13 2002

nonn

To make this a permutation of nonnegative integers, we subtract one from each run count except for the most significant run, e.g. a(11) = 9, as 11 = 1011 and 9+1 = 10 = 5^1 * 3^(1-1) * 2^(2-1).

Paul Tek (terms 0..10000) & Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences that are permutations of the natural numbers

Inverse of A075158.
Cf. A008578, A056539, A059900, A075162.
Cf. A000040, A000975, A006093, A030308, A007088, A075159, A278217, A286617.

import Data.List (group)
a075157 0 = 0
a075157 n = product (zipWith (^) a000040_list rs') - 1 where
   rs' = reverse $ r : map (subtract 1) rs
   (r:rs) = reverse $ map length $ group $ a030308_row n
-- Reinhard Zumkeller, Aug 04 2014

A005811(n) = hammingweight(bitxor(n, n>>1));  \\ This function from Gheorghe Coserea, Sep 03 2015
A286468(n) = { my(p=((n+1)%2), i=0, m=1); while(n>0, if(((n%2)==p), m *= prime(i), p = (n%2); i = i+1); n = n\2); m };
A075157(n) = if(!n,n,(prime(A005811(n))*A286468(n))-1);

(define (A075157 n) (if (zero? n) n (+ -1 (* (A000040 (A005811 n)) (fold-left (lambda (a r) (* (A003961 a) (A000079 (- r 1)))) 1 (binexp->runcount1list n))))))
(define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (1+ count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2)))))))
;; Or, using the code of A286468:
(define (A075157 n) (if (zero? n) n (- (* (A000040 (A005811 n)) (A286468 n)) 1)))

a(n) = A075159(n+1) - 1.
a(0) = 0; for n >= 1, a(n) = (A000040(A005811(n)) * A286468(n)) - 1.
Other identities. For all n >= 1:
a(A000975(n)) = A006093(n) = A000040(n)-1.

Entry revised, PARI-program added and the old incorrect Scheme-program replaced with a new one by Antti Karttunen, May 17 2017

A077065 Semiprimes of form prime - 1.

Original entry on oeis.org

4, 6, 10, 22, 46, 58, 82, 106, 166, 178, 226, 262, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 886, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2026, 2038, 2062, 2098, 2206, 2446, 2458, 2578, 2818, 2878, 2902

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

There are 670 semiprimes of form prime-1 below 10^5.

			A001358(16) = 46 = 2*23 is a term as 46 = A000040(15) - 1 = 47 - 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Intersection of A006093 and A001358.
Intersection of A006093 and A100484.
Cf. A077068, A232342, A064911.
Cf. A005384, A005385.
Cf. A000010, A023900, A194593.

a077065 n = a077065_list !! (n-1)
a077065_list = filter ((== 1) . a010051' . (`div` 2)) a006093_list
-- Reinhard Zumkeller, Nov 22 2013, Oct 27 2012

IsSemiprime:=func; [s: n in [2..500] | IsSemiprime(s) where s is NthPrime(n)-1]; // Vincenzo Librandi, Oct 17 2012

q:= n-> (n::even) and andmap(isprime, [n+1, n/2]):
select(q, [$1..5000])[];  # Alois P. Heinz, Jul 19 2023

Select[Range[6000],Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#+1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[Range[3000],PrimeOmega[#]==2&&PrimeQ[#+1]&] (* Harvey P. Dale, Oct 16 2012 *)
Select[ Prime@ Range@ 430 - 1, PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

[x-1|x<-primes(10^4),bigomega(x-1)==2] \\ Charles R Greathouse IV, Nov 22 2013

a(n) = A005385(n) - 1 = 2*A005384(n).
A010051(A006093(a(n))/2) = A064911(A006093(a(n))) = 1. - Reinhard Zumkeller, Nov 22 2013
a(n) = A077068(n) - A232342(n). - Reinhard Zumkeller, Dec 16 2013
a(n) = A000010(A194593(n+1)). - Torlach Rush, Aug 23 2018
A000010((a(n)*2)+2) = A023900((a(n)*2)+2). - Torlach Rush, Aug 23 2018

A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 7, 12, 6, 4, 1, 16, 2, 18, 4, 6, 10, 22, 11, 4, 12, 2, 6, 28, 25, 30, 1, 10, 16, 6, 36, 36, 18, 12, 40, 40, 6, 42, 10, 23, 22, 46, 19, 6, 4, 16, 12, 52, 2, 10, 35, 18, 28, 58, 47, 60, 30, 63, 1, 12, 10, 66, 16, 22, 49, 70, 41, 72, 36, 4, 18, 10, 12, 78, 80, 2

Offset: 0

Antti Karttunen, Oct 25 2014

nonn

A000225 gives the positions of ones.

A006093 seems to give all such k, that a(k) = k.

			              Binomial coeff.   Their product  Largest k!
                 A007318          A001142(n)   which divides
Row 0                1                    1        1!
Row 1              1   1                  1        1!
Row 2            1   2   1                2        2!
Row 3          1   3   3   1              9        1!
Row 4        1   4   6   4   1           96        4! (96 = 4*24)
Row 5      1   5  10  10   5   1       2500        2! (2500 = 1250*2)
Row 6    1   6  15  20  15   6   1   162000        6! (162000 = 225*720)

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..4096 from Antti Karttunen)

One more than A249150.
Cf. A249423 (numbers k such that a(k) = k+1).
Cf. A249429 (numbers k such that a(k) > k).
Cf. A249433 (numbers k such that a(k) < k).
Cf. A249434 (numbers k such that a(k) >= k).
Cf. A249424 (numbers k such that a(k) = (k-1)/2).
Cf. A249428 (and the corresponding values, i.e. numbers n such that A249151(2n+1) = n).
Cf. A249425 (record positions).
Cf. A249427 (record values).
Cf. A001142, A006093, A000225, A007917, A055881, A187059, A249346, A249421, A249430, A249431, A249432.

A249151(n) = { my(uplim,padicvals,b); uplim = (n+3); padicvals = vector(uplim); for(k=0, n, b = binomial(n, k); for(i=1, uplim, padicvals[i] += valuation(b, prime(i)))); k = 1; while(k>0, for(i=1, uplim, if((padicvals[i] -= valuation(k, prime(i))) < 0, return(k-1))); k++); };
\\ Alternative implementation:
A001142(n) = prod(k=1, n, k^((k+k)-1-n));
A055881(n) = { my(i); i=2; while((0 == (n%i)), n = n/i; i++); return(i-1); }
A249151(n) = A055881(A001142(n));
for(n=0, 4096, write("b249151.txt", n, " ", A249151(n)));

from itertools import count
from collections import Counter
from math import comb
from sympy import factorint
def A249151(n):
    p = sum((Counter(factorint(comb(n,i))) for i in range(n+1)),start=Counter())
    for m in count(1):
        f = Counter(factorint(m))
        if not f<=p:
            return m-1
        p -= f # Chai Wah Wu, Aug 19 2025

(define (A249151 n) (A055881 (A001142 n)))

a(n) = A055881(A001142(n)).

A023503 Greatest prime divisor of prime(n) - 1.

Original entry on oeis.org

2, 2, 3, 5, 3, 2, 3, 11, 7, 5, 3, 5, 7, 23, 13, 29, 5, 11, 7, 3, 13, 41, 11, 3, 5, 17, 53, 3, 7, 7, 13, 17, 23, 37, 5, 13, 3, 83, 43, 89, 5, 19, 3, 7, 11, 7, 37, 113, 19, 29, 17, 5, 5, 2, 131, 67, 5, 23, 7, 47, 73, 17, 31, 13, 79, 11, 7, 173, 29, 11, 179, 61, 31, 7, 191

Offset: 2

Clark Kimberling

nonn

Baker & Harman (1998) show that there are infinitely many n such that a(n) > prime(n)^0.677. This improves on earlier work of Goldfeld, Hooley, Fouvry, Deshouillers, Iwaniec, Motohashi, et al.
Fouvry shows that a(n) > prime(n)^0.6683 for a positive proportion of members of this sequence. See Fouvry and also Baker & Harman (1996) which corrected an error in the former work.
The record values are the Sophie Germain primes A005384. - Daniel Suteu, May 09 2017
Conjecture: every prime is in the sequence. Cf. A035095 (see my comment). - Thomas Ordowski, Aug 06 2017
a(n) is 2 for n in A159611, and is at most 3 for n in A174099. Conjecture: liminf a(n) = 3. - Jeppe Stig Nielsen, Jul 04 2020

T. D. Noe, Table of n, a(n) for n = 2..10000
R. C. Baker and G. Harman, The Brun-Titchmarsh theorem on average, Analytic Number Theory (Proceedings in honor of Heini Halberstam), Birkhäuser, Boston, 1996, pp. 39-103.
R. Baker and G. Harman, Shifted primes without large prime factors, Acta Arithmetica 83 (1998), pp. 331-361.
Étienne Fouvry, Théorème de Brun-Titchmarsh; application au théorème de Fermat, Invent. Math 79 (1985), 383-407.
D. M. Goldfeld, On the number of primes p for which p + a has a large prime factor, Mathematika 16 (1969), pp. 23-27.
R. R. Hall, Some properties of the sequence {p-1}, Acta Arith. 28 (1975/76), 101-105.
G. Harman, On the greatest prime factor of p-1 with effective constants

Cf. A006093, A006530.

Cf. A005384, A035095, A159611, A174099.

A023503 := proc(n)
    A006530(ithprime(n)-1) ;
end proc:
seq( A023503(n),n=2..80) ; # R. J. Mathar, Sep 07 2016

Table[FactorInteger[Prime[n] - 1][[-1, 1]], {n, 2, 100}] (* T. D. Noe, Jun 08 2011 *)

a(n) = vecmax(factor(prime(n)-1)[,1]); \\ Michel Marcus, Aug 15 2015

a(n) = A006530(A006093(n)). - Michel Marcus, Aug 15 2015

Comments, references, and links from Charles R Greathouse IV, Mar 04 2011

A096173 Numbers k such that k^3+1 is an odd semiprime.

Original entry on oeis.org

2, 4, 6, 16, 18, 22, 28, 42, 58, 60, 70, 72, 78, 100, 102, 106, 112, 148, 156, 162, 190, 210, 232, 280, 310, 330, 352, 358, 382, 396, 448, 456, 490, 568, 606, 672, 756, 786, 820, 826, 828, 856, 858, 876, 928, 970, 982, 1008, 1012, 1030, 1068, 1092, 1108, 1150

Offset: 1

Hugo Pfoertner, Jun 20 2004

nonn

Numbers n such that n^3 + 1 is a semiprime, because then n^3 + 1 must be odd, since n^3 + 1 = (n+1)*(n^2 - n + 1) is a semiprime only if n+1 is odd. - Jonathan Sondow, Feb 02 2014

Obviously, n + 1 is always a prime number. Sequence is intersection of A006093 and A055494. - Altug Alkan, Dec 20 2015

			a(1)=2 because 2^3+1=9=3*3, a(13)=100: 100^3+1=1000001=101*9901.

R. J. Mathar, Table of n, a(n) for n = 1..2086

Cf. A001358; A081256: largest prime factor of k^3+1; A096174: (k^3+1)/(k+1) is prime; A046315, A237037, A237038, A237039, A237040.

Cf. A006093, A055494.

[n: n in [1..2*10^3] | IsPrime(n+1) and IsPrime(n^2-n+1)]; // Vincenzo Librandi, Dec 21 2015

select(n -> isprime(n+1) and isprime(n^2-n+1), [seq(2*i,i=1..1000)]); # Robert Israel, Dec 20 2015

Select[Range[1200], PrimeQ[#^2 - # + 1] && PrimeQ[# + 1] &] (* Jonathan Sondow, Feb 02 2014 *)

for(n=1, 1e5, if(bigomega(n^3+1)==2, print1(n, ", "))); \\ Altug Alkan, Dec 20 2015

a(n) = 2*A237037(n) = (A237040(n)-1)^(1/3). - Jonathan Sondow, Feb 02 2014

Corrected by Zak Seidov, Mar 08 2006

A172367 Numbers k > 0 such that k+4 is a prime.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 19, 25, 27, 33, 37, 39, 43, 49, 55, 57, 63, 67, 69, 75, 79, 85, 93, 97, 99, 103, 105, 109, 123, 127, 133, 135, 145, 147, 153, 159, 163, 169, 175, 177, 187, 189, 193, 195, 207, 219, 223, 225, 229, 235, 237, 247, 253, 259, 265, 267, 273, 277, 279

Offset: 1

Juri-Stepan Gerasimov, Feb 01 2010

The subsequence of primes A023200 consists of the smallest primes p of cousin prime pairs (p, p+4), while the subsequence of nonprimes is A164384. - Bernard Schott, Oct 19 2021

			a(1) = 5 - 4 = 1, a(2) = 7 - 4 = 3.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..10000
Sameen Ahmed Khan, Primes in Geometric-Arithmetic Progression, arXiv:1203.2083v1 [math.NT], (Mar 09 2012).

Union of A023200 and A164384.

Cf. A000040, A006093, A040976, A086801.

Table[Prime[n]-4,{n,3,53}] (* Charles R Greathouse IV, Mar 12 2012 *)

a(n)=prime(n+2)-4 \\ Charles R Greathouse IV, Mar 12 2012

a(n) = prime(n+2) - 4.

A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

4, 9, 11, 30, 327, 445, 3512, 7789, 9361, 26519413

Offset: 1

Gus Wiseman, Nov 02 2024

Perfect powers (A001597) are numbers with a proper integer root, the complement of A007916.
Is this sequence finite?
The Redmond-Sun conjecture (see A308658) implies that this sequence is finite. - Pontus von Brömssen, Nov 05 2024

			Primes 9 and 10 are 23 and 29, and the interval (24,25,26,27,28) contains two perfect powers (25,27), so 9 is in the sequence.

For powers of 2 see A013597, A014210, A014234, A188951, A244508, A377467.
For no prime-powers we have A377286, ones in A080101.
For a unique prime-power we have A377287.
For squarefree numbers see A377430, A061398, A377431, A068360, A224363.
These are the positions of terms > 1 in A377432.
For a unique perfect power we have A377434.
For no perfect powers we have A377436.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect power <= n.
A131605 lists perfect powers that are not prime-powers.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053607, A304521.
A377468 gives the least perfect power > n.
Cf. A000720, A023055, A031218, A045542, A052410, A053706, A069623, A116086, A116455, A216765, A308658, A336416, A345531, A375740, A376560, A376561, A377057.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Count[Range[Prime[#]+1, Prime[#+1]-1],_?perpowQ]>1&]

from itertools import islice
from sympy import prime
from gmpy2 import is_power, next_prime
def A377466_gen(startvalue=1): # generator of terms >= startvalue
    k = max(startvalue,1)
    p = prime(k)
    while (q:=next_prime(p)):
        c = 0
        for i in range(p+1,q):
            if is_power(i):
                c += 1
                if c>1:
                    yield k
                    break
        k += 1
        p = q
A377466_list = list(islice(A377466_gen(),9)) # Chai Wah Wu, Nov 04 2024

a(n) = A000720(A116086(n)) = A000720(A116455(n)) for n <= 10. This would hold for all n if there do not exist more than two perfect powers between any two consecutive primes, which is implied by the Redmond-Sun conjecture. - Pontus von Brömssen, Nov 05 2024

a(10) from Pontus von Brömssen, Nov 04 2024

A008328 Number of divisors of prime(n)-1.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 6, 4, 6, 8, 9, 8, 8, 4, 6, 4, 12, 8, 8, 12, 8, 4, 8, 12, 9, 8, 4, 12, 10, 12, 8, 8, 8, 6, 12, 12, 10, 4, 6, 4, 18, 8, 14, 9, 12, 16, 8, 4, 12, 8, 8, 20, 8, 9, 4, 6, 16, 12, 16, 8, 6, 12, 8, 16, 6, 16, 20, 4, 12, 12, 4, 8, 12, 16, 4, 6, 18, 15, 16, 8, 24, 8

Offset: 1

N. J. A. Sloane

Also the number of irreducible factors of Phi(p,x)-1, for cyclotomic polynomial Phi(p,x) and prime p. The formula is Phi(p,x)-1 = x*Product_{n>1, n|p-1} Phi(n,x). - T. D. Noe, Oct 17 2003

T. D. Noe, Table of n, a(n) for n = 1..10000
Karl Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben, Monatshefte für Mathematik, Vol. 59 (1955), pp. 91-97.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial.

Cf. A000005, A006093, A066800, A103199.

for i from 1 to 500 do if isprime(i) then print(tau(i-1)); fi; od;
A008328 := proc(n)
    numtheory[tau](ithprime(n)-1) ;
end proc: # R. J. Mathar, Oct 30 2015

DivisorSigma[0,#-1]&/@Prime[Range[90]] (* Harvey P. Dale, Dec 08 2011 *)

a(n) = numdiv(prime(n)-1); \\ Michel Marcus, Feb 25 2021

a(n) = A000005(A006093(n)) = A066800(prime(n)). - R. J. Mathar, Oct 01 2017
From Amiram Eldar, Apr 16 2024: (Start)
Formulas from Prachar (1955):
Sum_{prime(n) < x} a(n) = x * log(log(x)) + B*x + O(x/log(x)), where B is a constant.
There is a constant c > 0 such that for infinitely many values of n we have a(n) > exp(c * log(prime(n))/log(log(prime(n)))). (End)

A058340 Primes p such that phi(x) = p-1 has only 2 solutions, namely x = p and x = 2p.

Original entry on oeis.org

11, 23, 29, 31, 47, 53, 59, 67, 71, 79, 83, 103, 107, 127, 131, 137, 139, 149, 151, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 263, 269, 271, 283, 293, 307, 311, 317, 331, 347, 359, 367, 373, 379, 383, 389, 419, 431, 439, 443, 463, 467, 479

Offset: 1

Labos Elemer, Dec 14 2000

nonn

Two solutions, p and 2p, exist for all odd primes p; primes in sequence have no other solutions.
Conjecture: if q > 7 is in A005385, then q is in the sequence. - Thomas Ordowski, Jan 04 2017
Proof of conjecture: q'=(q-1)/2 is an odd prime > 3.  If phi(x)=2q', which has 2-adic order 1 but is not a power of 2, there must be exactly one odd prime r dividing x.  We could also have a factor of 2 (but no higher power, which would contribute more 2's to phi(x)).  If x = r^e or 2r^e, then phi(x) = (r-1) r^(e-1).  For this to be 2q' one possibility is r-1 = 2 and r^(e-1)=q', but then q'=r=3, ruled out by q > 7.  The only other possibility is r-1=2q' and e=1, which makes r=q and x=q or 2q. - Robert Israel, Jan 04 2017
Information from Carl Pomerance: It is known that almost all primes (in the sense of relative asymptotic density) are in the sequence. - Thomas Ordowski, Jan 08 2017

			For p=2, phi(x)=1 has only two solutions, but they are 1 and 2, not 2 and 4, so 2 is not in the sequence.

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

Cf. A000010, A000040, A005385, A006093, A058339, A066071, A066072, A066073, A066074, A066075, A066076, A066077, A066080, A138537.

filter:= n -> isprime(n) and nops(numtheory:-invphi(n-1))=2:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Aug 12 2016

Take[Rest@ Keys@ Select[KeySelect[KeyMap[# + 1 &, PositionIndex@ Array[EulerPhi, 10^4]], PrimeQ], Length@ # == 2 &], 54] (* Michael De Vlieger, Dec 29 2017 *)

a(n) ~ n log . - Charles R Greathouse IV, Nov 18 2022

Edited by Ray Chandler, Jun 06 2008

A075158 Prime factorization of n+1 encoded with the run lengths of binary expansion.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 10, 7, 6, 11, 21, 8, 42, 20, 9, 15, 85, 12, 170, 23, 22, 43, 341, 16, 13, 84, 14, 40, 682, 19, 1365, 31, 41, 171, 18, 24, 2730, 340, 86, 47, 5461, 44, 10922, 87, 17, 683, 21845, 32, 26, 27, 169, 168, 43690, 28, 45, 80, 342, 1364, 87381, 39, 174762

Offset: 0

Antti Karttunen, Sep 13 2002

nonn

a(2n) = 1 or 2 mod 4 and a(2n+1) = 0 or 3 mod 4 for all n > 1

			a(1) = 1 as 2 = 2^1, a(2) = 2 (10 in binary) as 3 = 3^1 * 2^0, a(3) = 3 (11) as 4 = 2^2, a(4) = 5 (101) as 5 = 5^1 * 3^0 * 2^0, a(5) = 4 (100) as 6 = 3^1 * 2^1, a(8) = 6 (110) as 9 = 3^2 * 2^0, a(11) = 8 (1000) as 12 = 3^1 * 2^2, a(89) = 35 (100011) as 90 = 5^1 * 3^2 * 2^1, a(90) = 90 (1011010) as 91 = 13^1 * 11^0 * 7^1 * 5^0 * 3^0 * 2^0.
The binary expansion of a(n) begins from the left with as many 1's as is the exponent of the largest prime present in the factorization of n+1 and from then on follows runs of ej+1 zeros and ones alternatively, where ej are the corresponding exponents of the successively lesser primes (0 if that prime does not divide n+1).

Index entries for sequences that are permutations of the natural numbers

Inverse of A075157. a(n) = A075160(n+1)-1. a(A006093(n)) = A000975(n). Cf. A059884.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a075158 = fromJust . (`elemIndex` a075157_list)
-- Reinhard Zumkeller, Aug 04 2014

cp's OEIS Frontend

A075157 Run lengths in the binary expansion of n gives the vector of exponents in prime factorization of a(n)+1, with the least significant run corresponding to the exponent of the least prime, 2; with one subtracted from each run length, except for the most significant run of 1's.

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A077065 Semiprimes of form prime - 1.

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A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

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A023503 Greatest prime divisor of prime(n) - 1.

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Maple

Mathematica

PARI

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A096173 Numbers k such that k^3+1 is an odd semiprime.

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Magma

Maple

Mathematica

PARI

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A172367 Numbers k > 0 such that k+4 is a prime.

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Mathematica