A005867 -id:A005867 - cp's OEIS frontend

A328574 a(1) = 0, and, for n >= 2, numbers n whose primorial base expansion doesn't contain any nonleading zeros.

0, 1, 3, 5, 9, 11, 15, 17, 21, 23, 27, 29, 39, 41, 45, 47, 51, 53, 57, 59, 69, 71, 75, 77, 81, 83, 87, 89, 99, 101, 105, 107, 111, 113, 117, 119, 129, 131, 135, 137, 141, 143, 147, 149, 159, 161, 165, 167, 171, 173, 177, 179, 189, 191, 195, 197, 201, 203, 207, 209, 249, 251, 255, 257, 261, 263, 267, 269, 279, 281, 285

Offset: 1

Antti Karttunen, Oct 20 2019

After the initial zero, numbers n for which A276086(n) produces an even number with no gaps in its prime factorization.
Numbers n such that A276086(n) is in A055932; numbers for which A328475(n) is equal to A328572(n) = A003557(A276086(n)).
The number of positive terms below prime(m)# = A002110(m) is Sum_{k=1..m} A005867(k). - Amiram Eldar, Feb 16 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Cf. A002110, A003557, A005867, A055932, A276086, A328475, A328572.
Positions of 1's in A328573, positions of 0's in A329027, cf. also A328840.
Cf. A227157 for analogous sequence.

max = 4; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; Join[{0}, Select[Range[nmax], FreeQ[IntegerDigits[#, MixedRadix[bases]], 0] &]] (* Amiram Eldar, Feb 16 2021 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA055932(n) = { my(f=factor(n)[, 1]~); f==primes(#f); }; \\ From A055932
isA328574(n) = isA055932(A276086(n));

A328475(n) = { my(m=1, p=2, y=1); while(n, if(n%p, m *= p^((n%p)-y), y=0); n = n\p; p = nextprime(1+p)); (m); };
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
isA328574(n) = (A328475(n) == A328572(n));

Primary definition changed, the old definition moved to comment section by Antti Karttunen, Nov 03 2019

A055768 Number of distinct primes dividing phi of n-th primorial number.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 12 2000

nonn

			For primorials with 10, 100, or 1000 prime factors, their totients have only 5, 32 or 241 prime divisors, corresponding to a(10), a(100), and a(1000).

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

Cf. A002110, A000010, A001221, A055769.

a055768 = a001221 . a005867  -- Reinhard Zumkeller, May 01 2013

Table[PrimeNu@ EulerPhi[Product[Prime@ i, {i, n}]], {n, 78}] (* or *)
With[{nn = 78}, PrimeNu@ FoldList[LCM @@ {#1, #2} &, Prime@ Range@ nn - 1]] (* Michael De Vlieger, Jul 14 2017 *)

a(n)=omega(lcm(apply(p->p-1, primes(n)))) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = A001221(A000010(A002110(n))) = A001221(A005867(n)).

a(n) < n. - Charles R Greathouse IV, Sep 02 2015

A096294 Triangle T(n,k) read by rows: for n >=0 and n >= k >=0, the fraction of positive integers with exactly k of the first n primes as divisors is T(n,k)/A002110(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 8, 14, 7, 1, 48, 92, 56, 13, 1, 480, 968, 652, 186, 23, 1, 5760, 12096, 8792, 2884, 462, 35, 1, 92160, 199296, 152768, 54936, 10276, 1022, 51, 1, 1658880, 3679488, 2949120, 1141616, 239904, 28672, 1940, 69, 1, 36495360, 82607616

Offset: 0

Matthew Vandermast, Jun 24 2004

Sum of entries in n-th row is A002110(n), the product of the first n primes (primorial numbers, first definition).
From Peter Munn, Apr 10 2017: (Start)
T(n,k) is a count of those integers in any interval of A002110(n) integers that have exactly k of the first n primes as divisors. The count is the same for each such interval because each of the first n primes is a factor of an integer m if and only if it is a factor of m + A002110(n).
A284411(m) is least p=prime(n) such that 2*Sum_{k=0..m-1} T(n,k) < A002110(n).
(End)

			Triangle begins:
1
1 1
2 3 1
8 14 7 1
48 92 56 13 1
480 968 652 186 23 1

First column is A005867; second column is A078456. See also A096180.

Cf. A194156, A284411.

primo(n) = prod(k=1, n, prime(k));
row(n) = {v = vector(n+1); for (k=1, primo(n), f = factor(k)[,1]; v[1+sum(j=1, #f, primepi(f[j])<=n)]++;); v;} \\ Michel Marcus, Apr 29 2017

A098592 Number of primes between n30 and (n+1)30.

Original entry on oeis.org

10, 7, 7, 6, 5, 6, 5, 6, 5, 5, 4, 6, 5, 4, 6, 5, 5, 2, 5, 5, 5, 6, 4, 4, 4, 5, 3, 6, 4, 4, 4, 4, 4, 5, 5, 4, 6, 3, 3, 4, 5, 4, 4, 6, 2, 3, 3, 5, 4, 7, 2, 5, 4, 6, 3, 4, 4, 3, 4, 4, 3, 2, 7, 3, 3, 3, 5, 5, 3, 5, 3, 5, 2, 3, 4, 4, 5, 3, 4, 7, 3, 4, 3, 1, 5, 3, 3, 3, 4, 7, 5, 4, 3, 5, 3, 4, 4, 3, 4, 2, 4, 3, 5, 2, 2, 3

Offset: 0

Hugo Pfoertner, Sep 16 2004

Number of nonzero bits in A098591(n).
The number a(n) is < 8 except for n=0. - Pierre CAMI, Jun 02 2009
For references to positions where a(n) = 7 and related explanation, see A100418. - Peter Munn, Sep 06 2023

			a(1)=7 because there are 7 primes in the interval (30,60): 31,37,41,43,47,53,59.
a(26)=3 because the interval of length 30 following 26*30=780 contains 3 primes: 787, 797 and 809.

Dennis Martin, Proofs Regarding Primorial Patterns [Cached copy, with permission of the author].
Hugo Pfoertner, Patterns count table.

Cf. A000040 (prime numbers), A098591 (packed representation of the primes mod 30), A100418, A185641.

Cf. A038822, A094892.

FORTRAN
```
! See links given in A098591.
```

a(n) = primepi(30*(n+1)) - primepi(30*n); \\ Michel Marcus, Apr 04 2020

from sympy import primerange
def a(n): return len(list(primerange(n*30, (n+1)*30)))
print([a(n) for n in range(106)]) # Michael S. Branicky, Oct 07 2021

Edited by N. J. A. Sloane, Jun 12 2009 at the suggestion of R. J. Mathar

A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 3, 4, 4, 10, 4, 12, 6, 8, 4, 16, 4, 18, 8, 12, 10, 22, 6, 8, 12, 6, 12, 28, 8, 30, 5, 20, 16, 24, 8, 36, 18, 24, 12, 40, 12, 42, 20, 16, 22, 46, 8, 12, 8, 32, 24, 52, 6, 40, 18, 36, 28, 58, 16, 60, 30, 24, 6, 48, 20, 66, 32, 44, 24, 70, 12, 72, 36, 16

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(60) = a(2^2*3*5) = (2 - 1)*2 * (3 - 1)*1 * (5 - 1)*1 = 16.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000010, A000026, A000040, A002110, A003958, A005117, A005361, A005867, A006093, A007947, A023900, A048250, A059975, A068997, A081294 (numbers n such that a(n) is odd), A087656, A090624, A152649, A173557, A304117, A304408, A304409, A304411, A304412.

Cf. A001221, A076479.

seq(mul((p-1)*padic[ordp](n, p), p in numtheory[factorset](n)), n=1..100); # Ridouane Oudra, Jun 06 2025

a[n_] := Times @@ ((#[[1]] - 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 75}]
Table[EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 75}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*e)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A005361(n)*abs(A023900(n)) = A005361(n)*A173557(n) = A005361(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*k where p is a prime and k > 0.
a(n) = phi(n) if n is a squarefree (A005117), where phi() = A000010.
a(A002110(k)) = A005867(k).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - 4/p^2 + 3/p^3 + 1/p^4 - 1/p^5) = 0.2644703894... . - Amiram Eldar, Nov 30 2022
a(n) = (-1)^A001221(n) * (Sum_{d1|n} Sum_{d2|n} mu(d1)*gcd(d1,d2)). - Ridouane Oudra, Jun 06 2025

A345974 Decimal expansion of Sum_{k>=0} Product_{i=1..k} 1/(prime(i)-1).

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jun 30 2021

			2.64810175970147102337088414536064714927258700214033932076744547925274...

Peter Cameron, A new constant?, Blog Post, May 23 2021.
Peter Cameron, A new constant?, local copy of Blog Post, with permission.

Cf. A005867, A055932, A057335, A345965.

c:= sum(product(1/(ithprime(i)-1), i=1..k), k=0..infinity):
evalf(c, 140);  # Alois P. Heinz, Jun 30 2021

PARI
```
suminf(k=0,prod(i=1,k,1/(prime(i)-1)))
```

Equals Sum_{k>=0} 1/A005867(k) = Sum_{k>=1} 1/A055932(k) = Sum_{k>=1} 1/A057335(k). - Amiram Eldar, Jun 26 2025

A055769 Largest prime dividing phi of the n-th primorial.

Original entry on oeis.org

1, 2, 2, 3, 5, 5, 5, 5, 11, 11, 11, 11, 11, 11, 23, 23, 29, 29, 29, 29, 29, 29, 41, 41, 41, 41, 41, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 53, 83, 83, 89, 89, 89, 89, 89, 89, 89, 89, 113, 113, 113, 113, 113, 113, 113, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131, 131

Offset: 1

Labos Elemer, Jul 12 2000

nonn

			While the largest prime factors of 10th, 100th or 1000th primorials are 29, 541, 7919, those of their totients are 11, 251, 3911, respectively.

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

Cf. A000010, A002110, A005867, A055768.

a055769 = a006530 . a005867  -- Reinhard Zumkeller, May 01 2013

Map[FactorInteger[EulerPhi@ #][[-1, 1]] &, FoldList[#1 #2 &, Prime@ Range@ 66]] (* Michael De Vlieger, Oct 26 2017 *)

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=my(p=prime(n),q=1); while(2*q+1Charles R Greathouse IV, Dec 03 2014

a(n) = A006530(A000010(A002110(n))). [corrected by Amiram Eldar, Sep 18 2024]

a(n) = A006530(A005867(n)). - Reinhard Zumkeller, May 01 2013

A058255 Distinct values of lcm_{i=1..n} (p(i)-1), where p() are the primes.

Original entry on oeis.org

1, 2, 4, 12, 60, 240, 720, 7920, 55440, 1275120, 16576560, 480720240, 19709529840, 39419059680, 197095298400, 3350620072800, 177582863858400, 532748591575200, 19711697888282400, 59135093664847200

Offset: 1

Labos Elemer, Dec 06 2000

nonn

The prime A095365(n) is the least prime yielding an LCM of a(n). This sequence and A095365 are related to A095366. - T. D. Noe, Jun 04 2004

			For p = 29, 31, 37, 41, 43 these LCMs are equal to 55440 = 11*7! = lcm[1, 2, 4, 6, 10, 12, 16, 22, 28, 30, 36, 40, 42] = lcm[1, 2, 4, 6, 10, 12, 16, 22, 28]. The values was put on the stage only once. Repetitions skipped.

Michael De Vlieger, Table of n, a(n) for n = 1..360

A058254 with duplicates removed.
Cf. A002110, A005867, A003418, A054451, A000142, A000010.
Cf. A095366 (least k > 1 such that k divides 1^n + 2^n + ... + (k-1)^n).

Union@ Table[LCM @@ (Prime@ Range[1, n] - 1), {n, 38}] (* Michael De Vlieger, Dec 06 2018 *)

A059862 a(n) = Product_{i=3..n} (prime(i) - 3).

Original entry on oeis.org

1, 1, 2, 8, 64, 640, 8960, 143360, 2867200, 74547200, 2087321600, 70968934400, 2696819507200, 107872780288000, 4746402332672000, 237320116633600000, 13289926531481600000, 770815738825932800000, 49332207284859699200000, 3354590095370459545600000, 234821306675932168192000000

Offset: 1

Labos Elemer, Feb 28 2001

nonn

			For n = 6, a(6) = 640 because:
prime(1..6)-3 = (-1,0,2,4,8,10) -> (1,1,2,4,8,10)
and
1*1*2*4*8*10 = 640. [Example generalized and reformatted per observation of _Jon E. Schoenfield_ by _Harlan J. Brothers_, Jul 15 2018]

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, A8, A1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly, 66 (1959), 375-384.

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865.

a:= proc(n) option remember;
      `if`(n<3, 1, a(n-1)*(ithprime(n)-3))
    end:
seq(a(n), n=1..21);  # Alois P. Heinz, Nov 19 2021

Join[{1, 1}, Table[Product[Prime[i] - 3, {i, 3, n}], {n, 3, 19}]] (* Harlan J. Brothers, Jul 02 2018 *)
a[1] = 1; a[2] = 1; a[n_] := a[n] = a[n - 1] (Prime[n] - 3);
Table[a[n], {n, 19}] (* Harlan J. Brothers, Jul 02 2018 *)

a(n) = prod(i=3, n, prime(i) - 3); \\ Michel Marcus, Jul 15 2018

a(1) = a(2) = 1; a(n) = a(n-1) * (prime(n) - 3) for n >= 3. - David A. Corneth, Jul 15 2018

Name clarified, offset corrected by David A. Corneth, Jul 15 2018

A061671 Numbers n such that { x +- 2^k : 0 < k < 4 } are primes, where x = 210*n - 105.

Original entry on oeis.org

1, 77, 93, 209, 5197, 7695, 9307, 13442, 13524, 15445, 16192, 28600, 30970, 34228, 36388, 38391, 41625, 50127, 52795, 55546, 69146, 70538, 70642, 70747, 76314, 76642, 90079, 91416, 93496, 94288, 95773, 96415, 101530, 104049, 107559, 118031

Offset: 1

Frank Ellermann, Jun 16 2001

nonn

This sequence does not include the sextet (7,11,13,17,19,23). It is a proper subset of A014561 in a certain sense.

			16057, 16061, 16063, 16067, 16069, 16073 are prime and (16065+105)/210= 77= a(2).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, conjectures following th. 5

Harvey P. Dale, Table of n, a(n) for n = 1..900
F. Ellermann, Illustration for A002110, A005867, A038110, A060753

210 = 7*5*3*2 = A002110(4), cf. A014561.

Select[Range[1, 1000000], Union[PrimeQ[(210*# - 105) + {-8, -4, -2, 2, 4, 8}]] == {True} &]
Select[Range[120000],AllTrue[210#-105+{-8,-4,-2,2,4,8},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 05 2019 *)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 20 2001 and from Frank Ellermann, Nov 26 2001. Mathematica script from Peter Bertok (peter(AT)bertok.com), Nov 27 2001.

cp's OEIS Frontend

A328574 a(1) = 0, and, for n >= 2, numbers n whose primorial base expansion doesn't contain any nonleading zeros.

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A055768 Number of distinct primes dividing phi of n-th primorial number.

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A096294 Triangle T(n,k) read by rows: for n >=0 and n >= k >=0, the fraction of positive integers with exactly k of the first n primes as divisors is T(n,k)/A002110(n).

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A098592 Number of primes between n*30 and (n+1)*30.

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A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

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Maple

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A345974 Decimal expansion of Sum_{k>=0} Product_{i=1..k} 1/(prime(i)-1).

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Maple

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A055769 Largest prime dividing phi of the n-th primorial.

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Haskell

Mathematica

A098592 Number of primes between n30 and (n+1)30.