A060735 -id:A060735 - cp's OEIS frontend

A147572 Numbers with exactly 5 distinct prime divisors {2,3,5,7,11}.

2310, 4620, 6930, 9240, 11550, 13860, 16170, 18480, 20790, 23100, 25410, 27720, 32340, 34650, 36960, 41580, 46200, 48510, 50820, 55440, 57750, 62370, 64680, 69300, 73920, 76230, 80850, 83160, 92400, 97020, 101640, 103950, 110880, 113190, 115500, 124740, 127050

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575

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

Cf. A051038, A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 16/77, AppendTo[a, x]], {x, 1, 100000}]; a
Select[Range[130000],FactorInteger[#][[All,1]]=={2,3,5,7,11}&] (* Harvey P. Dale, Oct 04 2020 *)

from sympy import integer_log, prevprime
def A147572(n):
    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
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,11)
    return 2310*bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

a(n) = 2310 * A051038(n). - Amiram Eldar, Mar 10 2020

Sum_{n>=1} 1/a(n) = 1/480. - Amiram Eldar, Nov 12 2020

More terms from Amiram Eldar, Mar 10 2020

A168264 For all sufficiently high values of k, d(n^k) > d(m^k) for all m < n. (Let k, m, and n represent positive integers only.)

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 60, 120, 180, 210, 420, 840, 1260, 1680, 2310, 4620, 9240, 13860, 18480, 27720, 30030, 60060, 120120, 180180, 240240, 360360, 510510, 1021020, 2042040, 3063060, 4084080, 6126120, 9699690, 19399380, 38798760, 58198140

Offset: 1

Matthew Vandermast, Nov 23 2009

nonn

d(n) is the number of divisors of n (A000005(n)).

			Since the exponents in 1680's prime factorization are (4,1,1,1), the k-th power of 1680 has (4k+1)(k+1)^3 = 4k^4 + 13k^3 + 15k^2 + 7k + 1 divisors. Comparison with the analogous formulas for all smaller members of A025487 shows the following:
a) No number smaller than 1680 has a positive coefficient in its "power formula" for any exponent larger than k^4.
b) The only power formula with a k^4 coefficient as high as 4 is that for 1260 (4k^4 + 12k^3 + 13k^2 + 6k + 1).
c) The k^3 coefficient for 1680 is higher than for 1260.
So for all sufficiently high values of k, d(1680^k) > d(m^k) for all m < 1680.

Anonymous?, Polynomial Calculator
Eric Weisstein's World of Mathematics, Distinct Prime Factors
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

Subsequence of A025487, A060735, A116998. Includes A002110, A168262, A168263.

A168263 For any m < n, and for all values of k, d(n^k) > d(m^k). (Let k, m, and n represent positive integers only.)

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 60, 120, 180, 840, 1260, 1680, 27720

Offset: 1

Matthew Vandermast, Nov 23 2009

d(n) is the number of divisors of n (A000005(n)).
All members must be highly composite numbers (A002182) with at least as many distinct prime factors as any smaller positive integer (A116998). (See Formula and Example sections.) It turns out that these two conditions are jointly sufficient.
Ramanujan proved that a) for any prime p, there exist a finite number of highly composite numbers with p as its largest prime factor; and b) in the canonical prime factorization of a highly composite number with largest prime factor p, the exponents for all primes > p are never smaller than they are in the factorization of A003418(p). (See formula 54 of the Ramanujan paper.)
It follows that, if the intersection of A003418 and A116998 is finite, so is the intersection of A002182 and A116998. For proof that the former intersection is finite, see A168262.
By using the given formula for the number of divisors, it is possible to define a canonical polynomial p_n(k) for every natural number n. For example, because 60 = (2^2)(3^1)(5^1), we define p_60(k) = (1+2k)(1+k)(1+k). The present sequence is defined only by examining whether p_n(k) achieves a record for natural numbers k, but the question could also be asked whether p_n(k) achieves a record for all k > 0. This stricter requirement does not hold for a(7)-a(13) at various positive values of k < 1, but it does hold for a(1)-a(6). The present sequence is "full", so a(1)-a(6) are the only numbers to satisfy the stronger property. - Hal M. Switkay, Aug 17 2025

			1) 1680 has more divisors than any smaller positive integer; thus for all m < n, d(1680^1) > d(m^1).
2) Since the exponents in 1680's prime factorization are (4,1,1,1), the k-th power of 1680 has (4k+1)(k+1)^3 = 4k^4 + 13k^3 + 15k^2 + 7k + 1 divisors. Comparison with the analogous formulas for all smaller members of A025487 shows the following:
a) No number smaller than 1680 has a positive coefficient in its "power formula" for any exponent larger than k^4.
b) The only power formula with a k^4 coefficient as high as 4 is that for 1260 (4k^4 + 12k^3 + 13k^2 + 6k + 1).
c) The k^3 coefficient for 1680 is higher than for 1260.
So for all sufficiently high values of k, d(1680^k) > d(m^k) for all m < 1680.
3) Careful comparison of 1680's "power formula" with the analogous formulas for smaller members of A025487 shows that no intermediate value of k can exist for which d(m^k) >= d(1680^k) if m < 1680.

S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

Anonymous?, Polynomial Calculator
S. Ramanujan, Highly Composite Numbers (p. 15) (note especially pp. 11-15)
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

Intersection of A002182 and A116998. Also, intersection of A002182 and A060735, and of A002182 and A168264. (A168264 is a subsequence of A060735, which is a subsequence of A116998.)

If the canonical factorization of n into prime powers is Product p^e(p), then the formula for the number of divisors of the k-th power of n is Product_p (ek + 1). (See also A146289, A146290.)

For two positive integers m and n with different prime signatures, let j be the largest exponent of k for which m and n have different coefficients, after the above formula for each integer is expanded as a polynomial. Let m_j and n_j denote the corresponding coefficients. d(n^k) > d(m^k) for all sufficiently high values of k if and only if n_j > m_j.

A087732 Smaller of twin primes of the form P=jP(i)#-1 and P=jP(i)#+1 with 0 < j < P(i+1), where P(i) denotes i-th prime and P(i)# the i-th primorial number A002110(i).

Original entry on oeis.org

3, 5, 11, 17, 29, 59, 149, 179, 419, 1049, 2309, 9239, 11549, 25409, 180179, 270269, 300299, 330329, 390389, 420419, 4084079, 8678669, 106696589, 892371479, 2454021569, 3569485919, 4238764529, 4461857399, 4908043139, 6023507489

Offset: 1

Pierre CAMI, Sep 29 2003

nonn

Probably an infinite sequence. Using the UB874 program (UBASIC) I found the first 123 primes of the sequence for i <= 382. I think I have a proof that the sequence is infinite.

			17=3*P(2)#-1 and 19=3*P(2)#+1 are twin primes, so 17 is in the sequence, corresponding to i=2, j=3. Again, 182*2633#-1 and 182*2633#+1 are prime twins, with j=182, i=382. These are 1111-digit twin primes.
The above prime is a(124). - _Robert G. Wilson v_, Jul 22 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..104 (first 128 terms from Robert G. Wilson v) (shortened by _N. J. A. Sloane_, Jan 13 2019)

Cf. A000040, A002110, A086916, A087700, A087731, A088676, A060735.

f[n_] := Range[Prime[n + 1] - 1] Times @@ Prime@ Range@ n; s = Select[ Union@ Flatten@ Join[ Array[f, 10] - 1, Array[f, 11, 0] + 1], PrimeQ@# &]; s[[Select[ Range[-1 + Length@ s], s[[#]] + 2 == s[[# + 1]] &]]] (* Robert G. Wilson v, Jul 22 2015 *)

do(lastprime)=my(v=List(),P=1,p=2); forprime(q=3,nextprime(lastprime\1+1), P*=p; for(j=1,q-1, if(isprime(j*P-1)&&isprime(j*P+1), listput(v, j*P-1))); p=q); Vec(v) \\ Charles R Greathouse IV, Jul 22 2015

Edited by Jud McCranie, Oct 06 2003

Corrected by T. D. Noe, Nov 15 2006

A147577 Numbers with exactly 4 distinct odd prime divisors {3,5,7,11}.

Original entry on oeis.org

1155, 3465, 5775, 8085, 10395, 12705, 17325, 24255, 28875, 31185, 38115, 40425, 51975, 56595, 63525, 72765, 86625, 88935, 93555, 114345, 121275, 139755, 144375, 155925, 169785, 190575, 202125, 218295, 259875, 266805, 280665, 282975, 317625

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

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

Cf. A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 32/77, AppendTo[a, x]], {x, 1, 1000000}]; a
Select[Range[350000],EulerPhi[#]/#==32/77&] (* Harvey P. Dale, Mar 25 2016 *)

from sympy import integer_log
def A147577(n):
    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
    def f(x):
        c = n+x
        for i11 in range(integer_log(x,11)[0]+1):
            for i7 in range(integer_log(x11:=x//11**i11,7)[0]+1):
                for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):
                    c -= integer_log(x7//5**i5,3)[0]+1
        return c
    return 1155*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Sum_{n>=1} 1/a(n) = 1/480. - Amiram Eldar, Dec 22 2020

A147578 Numbers with exactly 5 distinct odd prime divisors {3,5,7,11,13}.

Original entry on oeis.org

15015, 45045, 75075, 105105, 135135, 165165, 195195, 225225, 315315, 375375, 405405, 495495, 525525, 585585, 675675, 735735, 825825, 945945, 975975, 1126125, 1156155, 1216215, 1366365, 1486485, 1576575, 1756755, 1816815, 1876875, 2027025, 2147145, 2207205, 2477475

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244;
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849;
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576;
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577;
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578;
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579;
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580;
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581.

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

Cf. A000010 (EulerPhi), A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 384/1001, AppendTo[a, x]], {x, 1, 1000000}]; a

Sum_{n>=1} 1/a(n) = 1/5760. - Amiram Eldar, Dec 22 2020

More terms from Amiram Eldar, Mar 11 2020

A147579 Numbers with exactly 6 distinct odd prime divisors {3,5,7,11,13,17}.

Original entry on oeis.org

255255, 765765, 1276275, 1786785, 2297295, 2807805, 3318315, 3828825, 4339335, 5360355, 6381375, 6891885, 8423415, 8933925, 9954945, 11486475, 12507495, 13018005, 14039025, 16081065, 16591575, 19144125, 19654635, 20675655, 21696675, 23228205, 25270245, 26801775

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

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

Cf. A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[255255 x] == 92160 x, AppendTo[a, 255255 x]], {x, 1, 100}]; a

Sum_{n>=1} 1/a(n) = 1/92160. - Amiram Eldar, Dec 22 2020

More terms from Amiram Eldar, Mar 11 2020

A147581 Numbers with exactly 8 distinct odd prime divisors {3,5,7,11,13,17,19,23}.

Original entry on oeis.org

111546435, 334639305, 557732175, 780825045, 1003917915, 1227010785, 1450103655, 1673196525, 1896289395, 2119382265, 2342475135, 2565568005, 2788660875, 3011753745, 3681032355, 3904125225, 4350310965, 5019589575, 5465775315, 5688868185, 6135053925, 6358146795

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Numbers k such that phi(k)/k = m
( Family of sequences for successive n odd primes )
m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244
m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849
m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576
m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577
m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578
m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579
m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580
m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

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

Cf. A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[111546435 x] == 36495360 x, AppendTo[a, 111546435 x]], {x, 1, 100}]; a

from sympy import integer_log
def A147581(n):
    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
    def f(x):
        c = n+x
        for i23 in range(integer_log(x,23)[0]+1):
            for i19 in range(integer_log(x23:=x//23**i23,19)[0]+1):
                for i17 in range(integer_log(x19:=x23//19**i19,17)[0]+1):
                    for i13 in range(integer_log(x17:=x19//17**i17,13)[0]+1):
                        for i11 in range(integer_log(x13:=x17//13**i13,11)[0]+1):
                            for i7 in range(integer_log(x11:=x13//11**i11,7)[0]+1):
                                for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):
                                    c -= integer_log(x7//5**i5,3)[0]+1
        return c
    return 111546435*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Sum_{n>=1} 1/a(n) = 1/36495360. - Amiram Eldar, Dec 22 2020

More terms from Amiram Eldar, Mar 11 2020

A351566 Radix of the second least significant nonzero digit in the primorial base expansion of n, or 1 if there is no such digit.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 5, 3, 5, 3, 1, 5, 5, 3, 5, 3, 1, 5, 5, 3, 5, 3, 1, 5, 5, 3, 5, 3, 1, 7, 7, 3, 7, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 1, 7, 7, 3, 7, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 1, 7, 7, 3, 7, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3

Offset: 0

Antti Karttunen, Apr 01 2022

The terms larger than one are given by the k-th prime (A000040), where k is the position of the second least significant nonzero digit in the primorial base expansion of n, counted from the right. See the example.

			For n = 13, its primorial base representation (see A049345) is "201" as 13 = 2*A002110(2) + 1*A002110(0). The one-based index of the second least significant nonzero digit ("2"), when counted from the right, is 3, therefore a(13) = A000040(3) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..60060
Index entries for sequences related to primorial base.

Cf. A060735 (gives the positions of ones after the initial one at a(0)=1).

Cf. A000040, A002110, A049345, A053669, A119288, A276086, A351567.

a[n_] := Module[{k = n, p = 2, s = {}, r, i}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; i = Position[s, ?(# > 0 &)] // Flatten; If[Length[i] < 2, 1, Prime[i[[2]]]]]; Array[a, 100, 0] (* _Amiram Eldar, Mar 13 2024 *)

A119288(n) = if(1>=omega(n), 1, (factor(n))[2,1]);
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351566(n) = A119288(A276086(n));

a(n) = A119288(A276086(n)).
For all n, a(n) > A351567(n).
If a(n) > 1, then a(n) > A053669(n).

A147573 Numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13}.

Original entry on oeis.org

30030, 60060, 90090, 120120, 150150, 180180, 210210, 240240, 270270, 300300, 330330, 360360, 390390, 420420, 450450, 480480, 540540, 600600, 630630, 660660, 720720, 750750, 780780, 810810, 840840, 900900, 960960, 990990, 1051050, 1081080, 1171170, 1201200, 1261260

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575
Although 39270 has exactly 6 distinct prime divisors (39270=2*3*5*7*11*17), it is not in this sequence because the 6 distinct prime divisors may only comprise 2, 3, 5, 7, 11, and 13. - Harvey P. Dale, Oct 11 2014

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

Subsequence of A067885 and of A080197.

Cf. A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 192/1001, AppendTo[a, x]], {x, 1, 100000}]; a

is(n)=if(n%30030, return(0)); my(g=30030); while(g>1, n/=g; g=gcd(n,30030)); n==1 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = 30030 * A080197(n). - Charles R Greathouse IV, Sep 14 2015

Sum_{n>=1} 1/a(n) = 1/5760. - Amiram Eldar, Nov 12 2020

More terms from Amiram Eldar, Mar 10 2020

cp's OEIS Frontend

A147572 Numbers with exactly 5 distinct prime divisors {2,3,5,7,11}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A168264 For all sufficiently high values of k, d(n^k) > d(m^k) for all m < n. (Let k, m, and n represent positive integers only.)

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A168263 For any m < n, and for all values of k, d(n^k) > d(m^k). (Let k, m, and n represent positive integers only.)

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A087732 Smaller of twin primes of the form P=j*P(i)#-1 and P=j*P(i)#+1 with 0 < j < P(i+1), where P(i) denotes i-th prime and P(i)# the i-th primorial number A002110(i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A147577 Numbers with exactly 4 distinct odd prime divisors {3,5,7,11}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A147578 Numbers with exactly 5 distinct odd prime divisors {3,5,7,11,13}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A147579 Numbers with exactly 6 distinct odd prime divisors {3,5,7,11,13,17}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A147581 Numbers with exactly 8 distinct odd prime divisors {3,5,7,11,13,17,19,23}.

Views

Author

A087732 Smaller of twin primes of the form P=jP(i)#-1 and P=jP(i)#+1 with 0 < j < P(i+1), where P(i) denotes i-th prime and P(i)# the i-th primorial number A002110(i).