A000420 -id:A000420 - cp's OEIS frontend

A153649 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+1)prime(j)T(n-2, k-1) with j=4, read by rows.

2, 7, 7, 2, 94, 2, 2, 341, 341, 2, 2, 413, 3972, 413, 2, 2, 485, 16320, 16320, 485, 2, 2, 557, 31260, 171660, 31260, 557, 2, 2, 629, 48792, 774120, 774120, 48792, 629, 2, 2, 701, 68916, 1917012, 7556340, 1917012, 68916, 701, 2, 2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 2

Offset: 1

Roger L. Bagula, Dec 30 2008

			Triangle begins as:
  2;
  7,   7;
  2,  94,     2;
  2, 341,   341,       2;
  2, 413,  3972,     413,        2;
  2, 485, 16320,   16320,      485,        2;
  2, 557, 31260,  171660,    31260,      557,       2;
  2, 629, 48792,  774120,   774120,    48792,     629,     2;
  2, 701, 68916, 1917012,  7556340,  1917012,   68916,   701,   2;
  2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 2;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), this sequence (1,1,4), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), A153657 (2,7,10).

Cf. A000420 (powers of 7).

f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,p,q,j)
  if n eq 2 then return NthPrime(j);
  elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
  elif (k eq 1 or k eq n) then return 2;
  else return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*NthPrime(j)*T(n-2,k-1,p,q,j);
  end if; return T;
end function;
[T(n,k,1,1,4): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021

T[n_, k_, p_, q_, j_]:= T[n,k,p,q,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,p,q,j] + T[n-1,k-1,p,q,j] + (p*j+q)*Prime[j]*T[n-2,k-1,p,q,j] ]]];
Table[T[n,k,1,1,4], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 04 2021 *)

@CachedFunction
def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n,k,p,q,j):
    if (n==2): return nth_prime(j)
    elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
    elif (k==1 or k==n): return 2
    else: return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*nth_prime(j)*T(n-2,k-1,p,q,j)
flatten([[T(n,k,1,1,4) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 04 2021

T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+1)*prime(j)*T(n-2, k-1) with j=4.
From G. C. Greubel, Mar 04 2021: (Start)
T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 -4, T(4,2,p,q,j) = T(4,3,p,q,j) = prime(j)^2 -2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p,q,j) = (1,1,4).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(1,1,4), = 2*A000420(n-1). (End)

Edited by G. C. Greubel, Mar 04 2021

A008568 Digits of powers of 7.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Irregular table with row length sequence A210062. - Jason Kimberley, Nov 26 2012

The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). - Amiram Eldar, Mar 23 2025

			Triangle begins:
  1;
  7;
  4, 9;
  3, 4, 3;
  2, 4, 0, 1;
  1, 6, 8, 0, 7;
  1, 1, 7, 6, 4, 9;
  8, 2, 3, 5, 4, 3;
  ...

Seiichi Manyama, Rows n = 0..100, flattened
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.

Cf. A000420, A210062.

Cf. A000455, A008564, A008565, A008566, A008567, A008569, A008570, A008571, A008572, A008573, A010054.

Flatten[IntegerDigits[7^Range[0,20]]]  (* Harvey P. Dale, Mar 03 2011 *)

A087752 Powers of 49.

Original entry on oeis.org

1, 49, 2401, 117649, 5764801, 282475249, 13841287201, 678223072849, 33232930569601, 1628413597910449, 79792266297612001, 3909821048582988049, 191581231380566414401, 9387480337647754305649, 459986536544739960976801, 22539340290692258087863249, 1104427674243920646305299201

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Oct 02 2003

Same as Pisot sequences E(1, 49), L(1, 49), P(1, 49), T(1, 49). Essentially same as Pisot sequences E(49, 2401), L(49, 2401), P(49, 2401), T(49, 2401). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 49-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (49).

Bisection of A000420.
Cf. A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48).
Cf. A005843, A008776.

[49^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

49^Range[0,20] (* or *) Join[{1},NestList[49#&,49,20]] (* Harvey P. Dale, May 10 2019 *)

makelist(49^n,n,0,20); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=49^n \\ M. F. Hasler, Apr 19 2015
```

G.f.: 1/(1-49*x). - Philippe Deléham, Nov 24 2008
From Vincenzo Librandi, Nov 21 2010: (Start)
a(n) = 49^n.
a(n) = 49*a(n-1), a(0)=1. (End)
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(49*x).
a(n) = A000420(A005843(n)). (End)

Edited by M. F. Hasler, Apr 19 2015

A009972 Powers of 28.

Original entry on oeis.org

1, 28, 784, 21952, 614656, 17210368, 481890304, 13492928512, 377801998336, 10578455953408, 296196766695424, 8293509467471872, 232218265089212416, 6502111422497947648, 182059119829942534144, 5097655355238390956032, 142734349946674946768896, 3996561798506898509529088

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 28), L(1, 28), P(1, 28), T(1, 28). Essentially same as Pisot sequences E(28, 784), L(28, 784), P(28, 784), T(28, 784). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 28-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (28).

Cf. A000079, A000302, A000420, A001023, A008776.

[28^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010

28^Range[0, 13] (* Alonso del Arte, Feb 28 2015 *)
NestList[28#&,1,20] (* Harvey P. Dale, Jan 19 2019 *)

a(n)=28^n \\ Charles R Greathouse IV, Sep 28 2015

[lucas_number1(n,28,0) for n in range(1, 17)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-28*x). - Philippe Deléham, Nov 24 2008
a(n) = 28^n; a(n) = 28*a(n-1), n > 0, a(0) = 1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(28*x).
a(n) = A000079(n)*A001023(n) = A000302(n)*A000420(n). (End)

A038255 Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).

Original entry on oeis.org

1, 6, 1, 36, 12, 1, 216, 108, 18, 1, 1296, 864, 216, 24, 1, 7776, 6480, 2160, 360, 30, 1, 46656, 46656, 19440, 4320, 540, 36, 1, 279936, 326592, 163296, 45360, 7560, 756, 42, 1, 1679616, 2239488, 1306368, 435456, 90720, 12096, 1008

Offset: 0

N. J. A. Sloane

T(n,k) = A013613(n,n-k), 0 <= k <= n. - Reinhard Zumkeller, Nov 21 2013

			1
6, 1
36, 12, 1
216, 108, 18, 1
1296, 864, 216, 24, 1
7776, 6480, 2160, 360, 30, 1
46656, 46656, 19440, 4320, 540, 36, 1
279936, 326592, 163296, 45360, 7560, 756, 42, 1
1679616, 2239488, 1306368, 435456, 90720, 12096, 1008, 48, 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A038207.

Cf. A000420 (row sums), A013613 (mirrored), A110440, A007318, A000400.

a038255 n k = a038255_tabl !! n !! k
a038255_row n = a038255_tabl !! n
a038255_tabl = map reverse a013613_tabl
-- Reinhard Zumkeller, Nov 21 2013

for i from 0 to 8 do seq(binomial(i, j)*6^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

Table[Binomial[n,m]6^(n-m),{n,0,10},{m,0,n}]//Flatten (* Harvey P. Dale, Dec 25 2019 *)

G.f.: 1/(1 - 6*x - x*y). - Ilya Gutkovskiy, Apr 21 2017

A222462 T(n,k) = number of n X k 0..7 arrays with no entry increasing mod 8 by 7 rightwards or downwards, starting with upper left zero.

Original entry on oeis.org

1, 7, 7, 49, 301, 49, 343, 12943, 12943, 343, 2401, 556549, 3418807, 556549, 2401, 16807, 23931607, 903055069, 903055069, 23931607, 16807, 117649, 1029059101, 238535974201, 1465295106499, 238535974201, 1029059101, 117649, 823543

Offset: 1

R. H. Hardin, Feb 21 2013

1/8 the number of 8-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
......1.............7..................49........................343
......7...........301...............12943.....................556549
.....49.........12943.............3418807..................903055069
....343........556549...........903055069..............1465295106499
...2401......23931607........238535974201...........2377584520856755
..16807....1029059101......63007686842527........3857863258420747009
.117649...44249541343...16643060295393343.....6259760185235726701945
.823543.1902730277749.4396153388210813341.10157072698503130798653535
...
Some solutions for n=3, k=4:
..0..4..2..3....0..0..0..4....0..4..6..1....0..4..0..4....0..2..6..2
..0..0..5..6....0..0..4..6....0..0..1..5....0..0..6..0....0..0..2..3
..0..0..0..1....0..0..5..1....0..0..3..5....0..0..0..1....0..0..3..5

Andrew Howroyd, Table of n, a(n) for n = 1..276 (terms 1..83 from R. H. Hardin)

Columns 1-5 are A000420(n-1), 7*43^(n-1), A222459, A222460, A222461.
Main diagonal is A068258.
Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A198914 (unlabeled 8 colorings).

T(n, k) = 7 * (720*A198914(n,k) - 360*A198982(n,k) - 240*A198906(n,k) - 90*A198715(n,k) - 24*A207997(n,k) - 5) for n*k > 1. - Andrew Howroyd, Jun 27 2017
Empirical for column k:
k=1: a(n) = 7*a(n-1).
k=2: a(n) = 43*a(n-1).
k=3: a(n) = 270*a(n-1) - 1547*a(n-2).
k=4: a(n) = 1689*a(n-1) - 108775*a(n-2) + 1672631*a(n-3).
k=5: a(n) = 10754*a(n-1) - 8060499*a(n-2) + 2219242223*a(n-3) - 245682627864*a(n-4) + 5798947687589*a(n-5) + 448113231493438*a(n-6) - 2763020698450992*a(n-7).

A379300 Number of prime indices of n that are composite.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 25 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 39 are {2,6}, so a(39) = 1.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 3.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 3.

Positions of first appearances are A000420.
Positions of zero are A302540, counted by A034891 (strict A036497).
Positions of one are A379301, counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A023895, A113646, A175804, A204389, A320629, A376680, A378373, A378456.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],CompositeQ]],{n,100}]

Totally additive with a(prime(k)) = A066247(k).

A075502 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).

Original entry on oeis.org

1, 7, 1, 49, 21, 1, 343, 343, 42, 1, 2401, 5145, 1225, 70, 1, 16807, 74431, 30870, 3185, 105, 1, 117649, 1058841, 722701, 120050, 6860, 147, 1, 823543, 14941423, 16235562, 4084101, 360150, 13034, 196, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(7*z) - 1)*x/7) - 1.

			[1]; [7,1]; [49,21,1]; ...; p(3,x) = x * (49 + 21*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*      1
*      7        1
*     49       21        1
*    343      343       42       1
*   2401     5145     1225      70      1
*  16807    74431    30870    3185    105     1
* 117649  1058841   722701  120050   6860   147   1
* 823543 14941423 16235562 4084101 360150 13034 196 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000420, A075921-A075925, A076002. Row sums are A075506.

Cf. A075501, A075503.

Flatten[Table[7^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(7^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (7^(n-m)) * stirling2(n, m).
a(n, m) = 7*m*a(n-1, m) + a(n-1, m-1), n>=m>=1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*7)^(n-m))/(m-1)! for n >= m >= 1, else 0.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-7*k*x), m >= 1.
E.g.f. for m-th column: (((exp(7*x)-1)/7)^m)/m!, m >= 1.

A339687 a(n) = Sum_{d|n} 7^(d-1).

Original entry on oeis.org

1, 8, 50, 351, 2402, 16864, 117650, 823894, 5764851, 40356016, 282475250, 1977343950, 13841287202, 96889128064, 678223075300, 4747562333837, 33232930569602, 232630519768872, 1628413597910450, 11398895225729502, 79792266297729700, 558545864365759264

Offset: 1

Ilya Gutkovskiy, Dec 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column 7 of A308813.
Cf. A000420, A320072.
Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), this sequence (q=7), A339688 (q=8), A339689 (q=9).

A339687:= func< n | (&+[7^(d-1): d in Divisors(n)]) >;
[A339687(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Table[Sum[7^(d - 1), {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[x^k/(1 - 7 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, 7^(d-1)); \\ Michel Marcus, Dec 13 2020

def A339687(n): return sum(7^(k-1) for k in (1..n) if (k).divides(n))
[A339687(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

G.f.: Sum_{k>=1} x^k / (1 - 7*x^k).
G.f.: Sum_{k>=1} 7^(k-1) * x^k / (1 - x^k).
a(n) ~ 7^(n-1). - Vaclav Kotesovec, Jun 05 2021

A379310 Number of nonsquarefree prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 27 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 39 are {2,6}, so a(39) = 0.
The prime indices of 70 are {1,3,4}, so a(70) = 1.
The prime indices of 98 are {1,4,4}, so a(98) = 2.
The prime indices of 294 are {1,2,4,4}, a(294) = 2.
The prime indices of 1911 are {2,4,4,6}, so a(1911) = 2.
The prime indices of 2548 are {1,1,4,4,6}, so a(2548) = 2.

Positions of first appearances are A000420.
Positions of zero are A302478, counted by A073576 (strict A087188).
No squarefree parts: A379307, counted by A114374 (strict A256012).
One squarefree part: A379316, counted by A379308 (strict A379309).
A000040 lists the primes, differences A001223.
A005117 lists the squarefree numbers, differences A076259.
A008966 is the characteristic function for the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061398 counts squarefree numbers between primes, zeros A068360.
A377038 gives k-th differences of squarefree numbers.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.
- A330944 nonprime, see A000586, A000607, A076610, A330945.
- A379300 composite, see A023895, A034891, A036497, A302540, A379301.
- A379311 old prime, see A204389, A320629, A379312-A379315.
Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A087436, A112929, A120327, A378086.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[prix[n],Not@*SquareFreeQ]],{n,100}]

Totally additive with a(prime(k)) = A107078(k) = 1 - A008966(k).

cp's OEIS Frontend

A153649 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+1)*prime(j)*T(n-2, k-1) with j=4, read by rows.

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Magma

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Sage

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A008568 Digits of powers of 7.

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Mathematica

A087752 Powers of 49.

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Magma

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Maxima

PARI

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A009972 Powers of 28.

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PARI

Sage

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A038255 Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).

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A222462 T(n,k) = number of n X k 0..7 arrays with no entry increasing mod 8 by 7 rightwards or downwards, starting with upper left zero.

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A379300 Number of prime indices of n that are composite.

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A153649 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+1)prime(j)T(n-2, k-1) with j=4, read by rows.