A054640 -id:A054640 - cp's OEIS frontend

A272446 Numbers n such that A054640(n) is not divisible by n.

5, 11, 13, 17, 23, 26, 29, 43, 47, 59, 61, 67, 73, 89, 101, 103, 107, 109, 113, 127, 146, 149, 151, 163, 167, 178, 179, 181, 191, 193, 223, 233, 241, 251, 257, 277, 311, 349, 353, 382, 401, 409, 419, 421, 433, 461, 466, 479, 487, 491, 509, 541, 557, 571, 573, 631, 641, 643, 659, 673, 719

Offset: 1

Altug Alkan, Apr 29 2016

nonn

For n < 1000, there are 8 nonprime terms that are 26, 146, 178, 382, 466, 573, 802, 838.
4252 is the first term that has 3 prime divisors.
Note that 4252 is of the form 4p where p is prime, as are the next 42 such terms; the 44th, 213438, is 2*3*35573. Similarly, the first terms with 4 prime divisors are 1000024, 5921528, 6060344, 7355576, 10427512, 11727704, all of the form 8p with p prime. - Charles R Greathouse IV, Jan 06 2023

			5 is a term because A054640(5) = 6912 is not divisible by 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002110, A054640.

lista(nn) = for(n=1, nn, if(prod(i=1, n, prime(i)+1) % n != 0, print1(n, ", ")));

is(n)=prod(i=1, n, prime(i)+1)%n!=0 \\ Charles R Greathouse IV, Apr 29 2016

is(n)=my(m=Mod(1,n)); forprime(p=2,prime(n), m*=p+1; if(m==0, return(0))); 1 \\ Charles R Greathouse IV, Apr 29 2016

is(n,f=factor(n),r=prime(n))=for(i=1,#f~, my(p=f[i,1],e=f[i,2]); forprimestep(q=p-1,r,p, e-=valuation(q+1,p); if(e<=0, break)); if(e>0, return(1))); 0 \\ Charles R Greathouse IV, Jan 03 2023

A374852 a(n) = (1/3)*A054640(n) for n >= 1.

Original entry on oeis.org

1, 4, 24, 192, 2304, 32256, 580608, 11612160, 278691840, 8360755200, 267544166400, 10166678323200, 427000489574400, 18788021541273600, 901825033981132800, 48698551834981171200, 2921913110098870272000, 181158612826129956864000, 12318785672176837066752000

Offset: 1

Clark Kimberling, Aug 05 2024

nonn

a(n+1)/a(n) is an integer for n>=0, so (a(n)) is a divisibility sequence.

Cf. A000040, A374848.

s[n_] := 1; t[n_] := Prime[n];
u[n_] := Product[s[k] + t[n + 1 - k], {k, 1, n}]
(1/3) Table[u[n], {n, 1, 25}]

a(n) = (1/3)(s**t)(n) for n>=1, where s = (1,1,1,...), t = A000040 (the primes), and ** denotes obverse convolution, as in A374848.

a(n) = 2n*a(n-1) for n >= 2.

A005867 a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1).

Original entry on oeis.org

1, 1, 2, 8, 48, 480, 5760, 92160, 1658880, 36495360, 1021870080, 30656102400, 1103619686400, 44144787456000, 1854081073152000, 85287729364992000, 4434961926979584000, 257227791764815872000, 15433667505888952320000

Offset: 0

N. J. A. Sloane

Local minima of Euler's phi function. - Walter Nissen
Number of potential primes in a modulus primorial(n+1) sieve. - Robert G. Wilson v, Nov 20 2000
Let p=prime(n) and let p# be the primorial (A002110), then it can be shown that any p# consecutive numbers have exactly a(n-1) numbers whose lowest prime factor is p. For a proof, see the "Proofs Regarding Primorial Patterns" link. For example, if we let p=7 and consider the interval [101,310] containing 210 numbers, we find the 8 numbers 119, 133, 161, 203, 217, 259, 287, 301. - Dennis Martin (dennis.martin(AT)dptechnology.com), Jul 16 2006
From Gary W. Adamson, Apr 21 2009: (Start)
Equals (-1)^n * (1, 1, 1, 2, 8, 48, ...) dot (-1, 2, -3, 5, -7, 11, ...).
a(6) = 480 = (1, 1, 1, 2, 8, 48) dot (-1, 2, -3, 5, -7, 11) = (-1, 2, -3, 10, -56, 528). (End)
It can be proved that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*Sum_{i=0..T(sqrt(N))} A005867(i)/A002110(i). This can show for example that at least 0.16*N numbers are primes less than N for 29^2 > N > 23^2. - Ben Paul Thurston, Aug 23 2010
First column of A096294. - Eric Desbiaux, Jun 20 2013
Conjecture: The g.f. for the prime(n+1)-rough numbers (A000027, A005408, A007310, A007775, A008364, A008365, A008366, A166061, A166063) is x*P(x)/(1-x-x^a(n)+x^(a(n)+1)), where P(x) is an order a(n) polynomial with symmetric coefficients (i.e., c(0)=c(n), c(1)=c(n-1), ...). - Benedict W. J. Irwin, Mar 18 2016
a(n)/A002110(n+1) (primorial(n+1)) is the ratio of natural numbers whose smallest prime factor is prime(n+1); i.e., prime(n+1) coprime to A002110(n). So the ratio of even numbers to natural numbers = 1/2; odd multiples of 3 = 1/6; multiples of 5 coprime to 6 (A084967) = 2/30 = 1/15; multiples of 7 coprime to 30 (A084968) = 8/210 = 4/105; etc. - Bob Selcoe, Aug 11 2016
The 2-adic valuation of a(n) is A057773(n), being sum of the 2-adic valuations of the product terms here. - Kevin Ryde, Jan 03 2023
For n > 1, a(n) is the number of prime(n+1)-rough numbers in [1, primorial(prime(n))]. - Alexandre Herrera, Aug 29 2023

			a(3): the mod 30 prime remainder set sieve representation yields the remainder set: {1, 7, 11, 13, 17, 19, 23, 29}, 8 elements.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..99
Larry Deering, The Black Key Sieve, Box 275, Bellport NY 11713-0275, 1998.
Alphonse de Polignac, Six propositions arithmologiques déduites du crible d'Ératosthène, Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale, Série 1, Tome 8 (1849), pp. 423-429. See p. 425.
Frank Ellermann, Illustration for A002110, A005867, A038110, A060753.
Ken Hicks and Kevin Ward, Series and Product Relations Made from Primes, arXiv:2108.03268 [math.NT], 2021.
Dennis Martin, Proofs Regarding Primorial Patterns [via Internet Archive Wayback-machine]
Dennis Martin, Proofs Regarding Primorial Patterns [Cached copy, with permission of the author]
Francis E. Masat, Letter to N. J. A. Sloane with attachment: "A note on prime number sequences" (unpublished manuscript), Apr. 1991.
Travis Near, Improving MATLAB's isprime performance without arbitrary-precision arithmetic, arXiv:2108.04791 [cs.MS], 2021.
John K. Sellers, Distribution of twin primes in repeating sequences of prime factors, arXiv:2108.00288 [math.GM], 2021. See Table 1 p. 11.
Andrew V. Sutherland, Order Computations in Generic Groups, Ph. D. Dissertation, Math. Dept., M.I.T., 2007.

Cf. A000010, A002110, A006093, A054640, A058254, A055768, A070918, A101301, A058251, A060753.
Cf. A057773 (2-adic valuation).
Column 1 of A281890.
Cf. A016035, A345974.

a005867 n = a005867_list !! n
a005867_list = scanl (*) 1 a006093_list
-- Reinhard Zumkeller, May 01 2013

A005867 := proc(n)
    mul(ithprime(j)-1,j=1..n) ;
end proc: # Zerinvary Lajos, Aug 24 2008, R. J. Mathar, May 03 2017

Table[ Product[ EulerPhi[ Prime[ j ] ], {j, 1, n} ], {n, 1, 20} ]
RecurrenceTable[{a[0]==1,a[n]==(Prime[n]-1)a[n-1]},a,{n,20}] (* Harvey P. Dale, Dec 09 2013 *)
EulerPhi@ FoldList[Times, 1, Prime@ Range@ 18] (* Michael De Vlieger, Mar 18 2016 *)

for(n=0, 22, print1(prod(k=1,n, prime(k)-1), ", "))

a(n) = phi(product of first n primes) = A000010(A002110(n)).
a(n) = Product_{k=1..n} (prime(k)-1) = Product_{k=1..n} A006093(n).
Sum_{n>=0} a(n)/A002110(n+1) = 1. - Bob Selcoe, Jan 09 2015
a(n) = A002110(n)-((1/A000040(n+1) - A038110(n+1)/A038111(n+1))*A002110(n+1)). - Jamie Morken, Mar 27 2019
a(n) = |Sum_{k=0..n} A070918(n,k)|. - Alois P. Heinz, Aug 18 2019
a(n) = A058251(n)/A060753(n+1). - Jamie Morken, Apr 25 2022
a(n) = A002110(n) - A016035(A002110(n)) - 1 for n >= 1. - David James Sycamore, Sep 07 2024
Sum_{n>=0} 1/a(n) = A345974. - Amiram Eldar, Jun 26 2025

Offset changed to 0, Name changed, and Comments and Examples sections edited by T. D. Noe, Apr 04 2010

A374848 Obverse convolution A000045**A000045; see Comments.

Original entry on oeis.org

0, 1, 2, 16, 162, 3600, 147456, 12320100, 2058386904, 701841817600, 488286500625000, 696425232679321600, 2038348954317776486400, 12259459134020160144810000, 151596002479762016373851690400, 3855806813438155578522841251840000

Offset: 0

Clark Kimberling, Jul 31 2024

nonn

The obverse convolution of sequences
  s = (s(0), s(1), ...) and t = (t(0), t(1), ...)
is introduced here as the sequence s**t given by
s**t(n) = (s(0)+t(n)) * (s(1)+t(n-1)) * ... * (s(n)+t(0)).
Swapping * and + in the representation s(0)*t(n) + s(1)*t(n-1) + ... + s(n)*t(0)
of ordinary convolution yields s**t.
If x is an indeterminate or real (or complex) variable, then for every sequence t of real (or complex) numbers, s**t is a sequence of polynomials p(n) in x, and the zeros of p(n) are the numbers -t(0), -t(1), ..., -t(n).
Following are abbreviations in the guide below for triples (s, t, s**t):
F = (0,1,1,2,3,5,...) = A000045, Fibonacci numbers
L = (2,1,3,4,7,11,...) = A000032, Lucas numbers
P = (2,3,5,7,11,...) = A000040, primes
T = (1,3,6,10,15,...) = A000217, triangular numbers
C = (1,2,6,20,70, ...) = A000984, central binomial coefficients
LW = (1,3,4,6,8,9,...) = A000201, lower Wythoff sequence
UW = (2,5,7,10,13,...) = A001950, upper Wythoff sequence
[ ] = floor
In the guide below, sequences s**t are identified with index numbers Axxxxxx; in some cases, s**t and Axxxxxx differ in one or two initial terms.
Table 1. s = A000012 = (1,1,1,1...) = (1);
t = A000012; 1                 s**t = A000079; 2^(n+1)
t = A000027; n                 s**t = A000142; (n+1)!
t = A000040, P                 s**t = A054640
t = A000040, P           (1/3) s**t = A374852
t = A000079, 2^n               s**t = A028361
t = A000079, 2^n         (1/3) s**t = A028362
t = A000045, F                 s**t = A082480
t = A000032, L                 s**t = A374890
t = A000201, LW                s**t = A374860
t = A001950, UW                s**t = A374864
t = A005408, 2*n+1             s**t = A000165, 2^n*n!
t = A016777, 3*n+1             s**t = A008544
t = A016789, 3*n+2             s**t = A032031
t = A000142, n!                s**t = A217757
t = A000051, 2^n+1             s**t = A139486
t = A000225, 2^n-1             s**t = A006125
t = A032766, [3*n/2]           s**t = A111394
t = A034472, 3^n+1             s**t = A153280
t = A024023, 3^n-1             s**t = A047656
t = A000217, T                 s**t = A128814
t = A000984, C                 s**t = A374891
t = A279019, n^2-n             s**t = A130032
t = A004526, 1+[n/2]           s**t = A010551
t = A002264, 1+[n/3]           s**t = A264557
t = A002265, 1+[n/4]           s**t = A264635
Sequences (c)**L, for c=2..4: A374656 to A374661
Sequences (c)**F, for c=2..6: A374662, A374662, A374982 to A374855
The obverse convolutions listed in Table 1 are, trivially, divisibility sequences. Likewise, if s = (-1,-1,-1,...) instead of s = (1,1,1,...), then s**t is a divisibility sequence for every choice of t; e.g. if s = (-1,-1,-1,...) and t = A279019, then s**t = A130031.
Table 2. s = A000027 = (0,1,2,3,4,5,...) = (n);
t = A000027, n                 s**t = A007778, n^(n+1)
t = A000290, n^2               s**t = A374881
t = A000040, P                 s**t = A374853
t = A000045, F                 s**t = A374857
t = A000032, L                 s**t = A374858
t = A000079, 2^n               s**t = A374859
t = A000201, LW                s**t = A374861
t = A005408, 2*n+1             s**t = A000407, (2*n+1)! / n!
t = A016777, 3*n+1             s**t = A113551
t = A016789, 3*n+2             s**t = A374866
t = A000142, n!                s**t = A374871
t = A032766, [3*n/2]           s**t = A374879
t = A000217, T                 s**t = A374892
t = A000984, C                 s**t = A374893
t = A038608, n*(-1)^n          s**t = A374894
Table 3. s = A000290 = (0,1,4,9,16,...) = (n^2);
t = A000290, n^2               s**t = A323540
t = A002522, n^2+1             s**t = A374884
t = A000217, T                 s**t = A374885
t = A000578, n^3               s**t = A374886
t = A000079, 2^n               s**t = A374887
t = A000225, 2^n-1             s**t = A374888
t = A005408, 2*n+1             s**t = A374889
t = A000045, F                 s**t = A374890
Table 4. s = t;
s = t = A000012, 1             s**s = A000079; 2^(n+1)
s = t = A000027, n             s**s = A007778, n^(n+1)
s = t = A000290, n^2           s**s = A323540
s = t = A000045, F             s**s = this sequence
s = t = A000032, L             s**s = A374850
s = t = A000079, 2^n           s**s = A369673
s = t = A000244, 3^n           s**s = A369674
s = t = A000040, P             s**s = A374851
s = t = A000201, LW            s**s = A374862
s = t = A005408, 2*n+1         s**s = A062971
s = t = A016777, 3*n+1         s**s = A374877
s = t = A016789, 3*n+2         s**s = A374878
s = t = A032766, [3*n/2]       s**s = A374880
s = t = A000217, T             s**s = A375050
s = t = A005563, n^2-1         s**s = A375051
s = t = A279019, n^2-n         s**s = A375056
s = t = A002398, n^2+n         s**s = A375058
s = t = A002061, n^2+n+1       s**s = A375059
If n = 2*k+1, then s**s(n) is a square; specifically,
s**s(n) = ((s(0)+s(n))*(s(1)+s(n-1))*...*(s(k)+s(k+1)))^2.
If n = 2*k, then s**s(n) has the form 2*s(k)*m^2, where m is an integer.
Table 5. Others
s = A000201, LW      t = A001950, UW         s**t = A374863
s = A000045, F       t = A000032, L          s**t = A374865
s = A005843, 2*n     t = A005408, 2*n+1      s**t = A085528, (2*n+1)^(n+1)
s = A016777, 3*n+1   t = A016789, 3*n+2      s**t = A091482
s = A005408, 2*n+1   t = A000045, F          s**t = A374867
s = A005408, 2*n+1   t = A000032, L          s**t = A374868
s = A005408, 2*n+1   t = A000079, 2^n        s**t = A374869
s = A000027, n       t = A000142, n!         s**t = A374871
s = A005408, 2*n+1   t = A000142, n!         s**t = A374872
s = A000079, 2^n     t = A000142, n!         s**t = A374874
s = A000142, n!      t = A000045, F          s**t = A374875
s = A000142, n!      t = A000032, L          s**t = A374876
s = A005408, 2*n+1   t = A016777, 3*n+1      s**t = A352601
s = A005408, 2*n+1   t = A016789, 3*n+2      s**t = A064352
Table 6. Arrays of coefficients of s(x)**t(x), where s(x) and t(x) are polynomials
s(x)              t(x)            s(x)**t(x)
 n                 x              A132393
n^2                x              A269944
x+1               x+1             A038220
x+2               x+2             A038244
 x                x+3             A038220
nx                x+1             A094638
 1              x^2+x+1           A336996
n^2 x             x+1             A375041
n^2 x            2x+1             A375042
n^2 x             x+2             A375043
2^n x             x+1             A375044
2^n              2x+1             A375045
2^n              x+2              A375046
x+1              F(n)             A375047
x+1             x+F(n)            A375048
x+F(n)          x+F(n)            A375049

			a(0) = 0 + 0 = 0
a(1) = (0+1) * (1+0) = 1
a(2) = (0+1) * (1+1) * (1+0) = 2
a(3) = (0+2) * (1+1) * (1+1) * (2+0) = 16
As noted above, a(2*k+1) is a square for k>=0. The first 5 squares are 1, 16, 3600, 12320100, 701841817600, with corresponding square roots 1, 4, 60, 3510, 837760.
If n = 2*k, then s**s(n) has the form 2*F(k)*m^2, where m is an integer and F(k) is the k-th Fibonacci number; e.g., a(6) = 2*F(3)*(192)^2.

Cf. A000045, A374850-to-A374881, A374884-to-A374894, A375041-to-A375049.

a:= n-> (F-> mul(F(n-j)+F(j), j=0..n))(combinat[fibonacci]):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 02 2024

s[n_] := Fibonacci[n]; t[n_] := Fibonacci[n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]

a(n)=prod(k=0, n, fibonacci(k) + fibonacci(n-k)) \\ Andrew Howroyd, Jul 31 2024

a(n) ~ c * phi^(3*n^2/4 + n) / 5^((n+1)/2), where c = QPochhammer(-1, 1/phi^2)^2/2 if n is even and c = phi^(1/4) * QPochhammer(-phi, 1/phi^2)^2 / (phi + 1)^2 if n is odd, and phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 01 2024

A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 66, 77, 110, 154, 165, 231, 330, 385, 462, 770, 1155, 2310, 13, 26, 39, 65, 78, 91, 130, 143, 182, 195, 273, 286, 390, 429, 455, 546, 715, 858, 910, 1001, 1365, 1430, 2002, 2145, 2730, 3003, 4290, 5005, 6006, 10010, 15015, 30030

Offset: 0

Gus Wiseman, Dec 02 2020

Also Heinz numbers of subsets of {1..n} that contain n if n>0, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A019565 in its triangle form, with each row's terms in increasing order. - Peter Munn, Feb 26 2021
From David James Sycamore, Jan 09 2025: (Start)
Alternative definition, with offset = 1: a(1) = 1. For n>1 if a(n-1) = A002110(k), a(n) = prime(k+1). Otherwise a(n) is the smallest novel squarefree number whose prime factors have already occurred as previous terms.
Permutation of A005117, Squarefree version A379746. (End)

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70  105  210

Alois P. Heinz, Rows n = 0..14, flattened
Michael De Vlieger, Plot p | a(n) at (x,y) = (n,pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function related to the order of a(n) in A019565.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function showing 1 in gray, primes in red, primorials in bright green, even squarefree semiprimes in yellow, odd squarefree semiprimes in light green, thereafter, progressively deeper green related to omega(a(n)) = m until m >= 6.

A011782 gives row lengths.
A339360 gives row sums.
A008578 (shifted) is column k = 1.
A100484 is column k = 2.
A001748 is column k = 3.
A002110 is column k = 2^(n-1).
A070826 is column k = 2^(n-1) - 1.
A209862 takes prime indices to binary indices in these terms.
A246867 groups squarefree numbers by Heinz weight, with row sums A147655.
A261144 divides the n-th row by prime(n), with row sums A054640.
A339116 is the restriction to semiprimes, with row sums A339194.
A005117 lists squarefree numbers, ordered lexicographically by prime factors: A019565.
A006881 lists squarefree semiprimes.
A072047 counts prime factors of squarefree numbers.
A319246 is the sum of prime indices of the n-th squarefree number.
A329631 lists prime indices of squarefree numbers, reversed: A319247.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014342, A014466, A019565, A098350, A112798, A320656, A326882, A338901, A379770.
Cf. A379746.

T:= proc(n) option remember; `if`(n=0, 1, (p-> map(
      x-> x*p, {seq(T(i), i=0..n-1)})[])(ithprime(n)))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Jan 08 2025

Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]],{n,5}]

For n > 1, T(n,k) = prime(n) * A261144(n-1,k).

a(n) = A019565(A379770(n)). - Michael De Vlieger, Jan 08 2025

Row n=0 (term 1) prepended by Alois P. Heinz, Jan 08 2025

A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210, 1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42

Offset: 1

Jean-François Alcover, Nov 26 2015

If we define a triangle whose n-th row consists of all squarefree numbers whose prime factors are all less than prime(k), we get this same triangle except starting with a row {1}, with offset 1. - Gus Wiseman, Aug 24 2021

			Triangle begins:
1, 2;                        squarefree and 2-smooth
1, 2, 3, 6;                  squarefree and 3-smooth
1, 2, 3, 5, 6, 10, 15, 30;
1, 2, 3, 5, 6,  7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
...

Jean-François Alcover, Table of n, a(n) for n = 1..2046 (first 10 rows)
A. Hildebrand and G. Tenenbaum, Integers without large prime factors, Journal de théorie des nombres de Bordeaux (1993) Volume:5, Issue:2, p. 411-484.
Eric Weisstein's MathWorld, Smooth number.
Wikipedia, Smooth number

Cf. A000079 (2-smooth), A003586 (3-smooth), A051037 (5-smooth), A002473 (7-smooth), A018336 (7-smooth & squarefree), A051038 (11-smooth), A087005 (11-smooth & squarefree), A080197 (13-smooth), A087006 (13-smooth & squarefree), A087007 (17-smooth & squarefree), A087008 (19-smooth & squarefree).
Row lengths are A000079.
Rightmost terms (or column k = 2^n) are A002110.
Rows are partial unions of rows of A019565.
Row n is A027750(A002110(n)), i.e., divisors of primorials.
Row sums are A054640.
Column k = 2^n-1 is A070826.
Multiplying row n by prime(n+1) gives A339195, row sums A339360.
A005117 lists squarefree numbers.
A056239 adds up prime indices, row sums of A112798.
A072047 counts prime factors of squarefree numbers.
A246867 groups squarefree numbers by Heinz weight, row sums A147655.
A329631 lists prime indices of squarefree numbers, sums A319246.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
Cf. A000040, A001221, A006881, A071403, A209862, A319247.

b:= proc(n) option remember; `if`(n=0, [1],
      sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
    end:
T:= n-> b(n)[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Nov 28 2015

primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten

T(n-1,k) = A339195(n,k)/prime(n). - Gus Wiseman, Aug 24 2021

A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

Original entry on oeis.org

1, 2, 6, 30, 180, 1260, 12600, 138600, 1801800, 25225200, 378378000, 6432426000, 122216094000, 2566537974000, 56463835428000, 1298668214844000, 33765373585944000, 979195833992376000, 29375875019771280000

Offset: 1

Leroy Quet, Oct 07 2005

nonn

Do all terms belong to A242031 (weakly decreasing prime signature)? - Gus Wiseman, May 14 2021

			Since the first 6 squarefree positive integers are 1, 2, 3, 5, 6, 7, the 6th term of the sequence is 1*2*3*5*6*7 = 1260.
From _Gus Wiseman_, May 14 2021: (Start)
The sequence of terms together with their prime signatures begins:
             1: ()
             2: (1)
             6: (1,1)
            30: (1,1,1)
           180: (2,2,1)
          1260: (2,2,1,1)
         12600: (3,2,2,1)
        138600: (3,2,2,1,1)
       1801800: (3,2,2,1,1,1)
      25225200: (4,2,2,2,1,1)
     378378000: (4,3,3,2,1,1)
    6432426000: (4,3,3,2,1,1,1)
  122216094000: (4,3,3,2,1,1,1,1)
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..417

A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A072047 applies Omega to each squarefree number.
A246867 groups squarefree numbers by Heinz weight (row sums: A147655).
A261144 groups squarefree numbers by smoothness (row sums: A054640).
A319246 gives the sum of prime indices of each squarefree number.
A329631 lists prime indices of squarefree numbers (reversed: A319247).
Cf. A001221, A002110, A014466, A070826, A338901, A339360.

Rest[FoldList[Times,1,Select[Range[40],SquareFreeQ]]] (* Harvey P. Dale, Jun 14 2011 *)

m=30;k=1;for(n=1,m,if(issquarefree(n),print1(k=k*n,",")))

More terms from Klaus Brockhaus, Oct 08 2005

A339360 Sum of all squarefree numbers with greatest prime factor prime(n).

Original entry on oeis.org

1, 2, 9, 60, 504, 6336, 89856, 1645056, 33094656, 801239040, 24246190080, 777550233600, 29697402470400, 1250501433753600, 55083063155097600, 2649111037319577600, 143390180403000115200, 8619643674791667302400, 534710099148093259776000, 36412881178052121329664000

Offset: 0

Gus Wiseman, Dec 04 2020

nonn

			The initial terms are:
   1 = 1,
   2 = 2,
   9 = 3 + 6,
  60 = 5 + 10 + 15 + 30.

Robert Israel, Table of n, a(n) for n = 0..349

A010036 takes prime indices here to binary indices, row sums of A209862.
A048672 takes prime indices to binary indices in squarefree numbers.
A054640 divides the n-th term by prime(n), row sums of A261144.
A072047 counts prime factors of squarefree numbers.
A339194 is the restriction to semiprimes, row sums of A339116.
A339195 has this as row sums.
A002110 lists primorials.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A056239 is the sum of prime indices of n (Heinz weight).
A246867 groups squarefree numbers by weight, with row sums A147655.
A319246 is the sum of prime indices of the n-th squarefree number.
A319247 lists reversed prime indices of squarefree numbers.
A329631 lists prime indices of squarefree numbers.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014466, A098350, A168472, A282935, A302590, A326878, A338901.

f:= proc(n) local i;
  `if`(n=0, 1, ithprime(n)) *mul(1+ithprime(i),i=1..n-1)
end proc:
map(f, [$0..20]); # Robert Israel, Dec 08 2020

Table[Sum[Times@@Prime/@stn,{stn,Select[Subsets[Range[n]],MemberQ[#,n]&]}],{n,10}]

For n >= 1, a(n) = A054640(n-1) * prime(n).

a(0)=1 prepended by Alois P. Heinz, Jan 08 2025

A061217 Number of zeros in the concatenation n(n-1)(n-2)(n-3)...321.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 12, 13, 14

Offset: 1

Amarnath Murthy, Apr 22 2001

The number of zeros necessary to write down all the numbers 1, 2, ..., n. Thus, the partial sums of A055641 are given by a(n)+1 (for n>=1). - Hieronymus Fischer, Jun 12 2012

			a(30) = 3 since number of zeros in 302928272625242322212019181716151413121110987654321 is 3. (This example implies offset = 1.)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Tanya Khovanova and Gregory Marton, Archive Labeling Sequences, arXiv:2305.10357 [math.HO], 2023.

Cf. A055640, A055641, A102669-A102685, A117804.

a061217 n = a061217_list !! (n-1)
a061217_list = scanl1 (+) $ map a055641 [1..]
-- Reinhard Zumkeller, Oct 27 2013

Table[Count[Flatten[IntegerDigits/@Table[x-n,{n,0,x-1}]],0],{x,110}] (* Harvey P. Dale, Aug 10 2011 *)

a(n) = my(m=logint(n,10)); (m+1)*(n+1) - (10^(m+1)-1)/9 + (1/2) * sum(j=1, m+1, (n\10^j * (2*n+2 - (1 + n\10^j) * 10 ^ j) - floor(n/10^j+9/10) * (2*n+2 + ((4/5 - floor(n / 10^j + 9 / 10))*10^j)))) \\ adapted from formula by Hieronymus Fischer \\ David A. Corneth, Jan 23 2019

From Hieronymus Fischer, Jun 12 2012: (Start)
a(n) = (m+1)*(n+1) - (10^(m+1)-1)/9 + (1/2)*Sum_{j=1..m+1} (floor(n/10^j)*(2*n + 2 - (1 + floor(n/10^j))*10^j) - floor(n/10^j + 9/10)*(2*n + 2 + (4/5 - floor(n/10^j + 9/10))*10^j)), where m=floor(log_10(n)).
a(n) = A117804(n+1) - (n+1)*A054640(n) + (1/2)*Sum_{j=1..m+1} ((floor(n/10^j + 9/10)^2 - floor(n/10^j)^2)*10^j - (4/5*floor(n/10^j + 9/10) + floor(n/10^j))*10^j), where m=floor(log_10(n)).
a(10^m-1) = m*10^(m-1) - (10^m-1)/9.
(This is the total number of zeros occurring in all the numbers 1..10^m-1 or numbers with <= m places excluding zero.)
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (1-x^10^j)*x^10^(j+1)/(1-x^10^(j+1)). (End)

Corrected and extended by Patrick De Geest, Jun 05 2001

Offset changed to 1 by Hieronymus Fischer, Jun 12 2012

A070918 Triangle of T(n,k) coefficients of polynomials with first n prime numbers as roots.

Original entry on oeis.org

1, -2, 1, 6, -5, 1, -30, 31, -10, 1, 210, -247, 101, -17, 1, -2310, 2927, -1358, 288, -28, 1, 30030, -40361, 20581, -5102, 652, -41, 1, -510510, 716167, -390238, 107315, -16186, 1349, -58, 1, 9699690, -14117683, 8130689, -2429223, 414849, -41817, 2451, -77, 1

Offset: 0

Rick L. Shepherd, May 20 2002

Analog of the Stirling numbers of the first kind (A008275): The Stirling numbers (beginning with the 2nd row) are the coefficients of the polynomials having exactly the first n natural numbers as roots. This sequence (beginning with first row) consists of the coefficients of the polynomials having exactly the first n prime numbers as roots.

			Row 4 of this sequence is 210, -247, 101, -17, 1 because (x-prime(1))(x-prime(2))(x-prime(3))(x-prime(4)) = (x-2)(x-3)(x-5)(x-7) = x^4 - 17*x^3 + 101*x^2 - 247*x + 210.
Triangle begins:
        1;
       -2,      1;
        6,     -5,       1;
      -30,     31,     -10,      1;
      210,   -247,     101,    -17,      1;
    -2310,   2927,   -1358,    288,    -28,    1;
    30030, -40361,   20581,  -5102,    652,  -41,   1;
  -510510, 716167, -390238, 107315, -16186, 1349, -58, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A008275 (Stirling numbers of first kind).
Cf. A005867 (absolute values of row sums).
Cf. A054640 (sum of absolute values of terms in rows).
Cf. A000040, A078456.

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(mul(x-ithprime(i), i=1..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 18 2019

Table[CoefficientList[Expand[Times@@(x-Prime[Range[n]])],x],{n,0,10}]// Flatten (* Harvey P. Dale, Feb 12 2020 *)

p=1; for(k=1,10,p=p*(x-prime(k)); for(n=0,k,print1(polcoeff(p,n),",")))

From Alois P. Heinz, Aug 18 2019: (Start)
T(n,k) = [x^k] Product_{i=1..n} (x-prime(i)).
Sum_{k=0..n} |T(n,k)| = A054640(n).
|Sum_{k=0..n} T(n,k)| = A005867(n).
|Sum_{k=0..n} k * T(n,k)| = A078456(n). (End)

First term T(0,0)=1 prepended by Alois P. Heinz, Aug 18 2019

cp's OEIS Frontend

A272446 Numbers n such that A054640(n) is not divisible by n.

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A374852 a(n) = (1/3)*A054640(n) for n >= 1.

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A005867 a(0) = 1; for n > 0, a(n) = (prime(n)-1)*a(n-1).

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A374848 Obverse convolution A000045**A000045; see Comments.

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A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

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A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

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A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

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