A060175 -id:A060175 - cp's OEIS frontend

A067255 Irregular triangle read by rows: row n gives exponents in prime factorization of n.

0, 1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 0, 0, 1, 3, 0, 2, 1, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 4, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Jeppe Stig Nielsen, Feb 20 2002

Row lengths are given by A061395(n), n >= 2: [1, 2, 1, 3, 2, 4, 1, 2, ... ].

This sequence contains every finite sequence of nonnegative integers. - Franklin T. Adams-Watters, Jun 22 2005

			1 = 2^0
2 = 2^1
3 = 2^0 3^1
4 = 2^2
5 = 2^0 3^0 5^1
6 = 2^1 3^1
... and reading the exponents gives the sequence.
Since for example 99=2^0*3^2*5^0*7^0*11^1, we use this symbol for ninety-nine: 99: {0,2,0,0,1}. Concatenating all the symbols for 1,2,3,4,5,6,..., we get the sequence.

Reinhard Zumkeller, Rows n = 1..250 of triangle, flattened
Jeppe Stig Nielsen, See this explanation.

Cf. A133457.
Cf. A001222 (row sums), A061395 (lengths of rows n >= 2).
Cf. A007814 (left edge), A071178 (right edge).
Other versions: A054841 (rows reversed and concatenated into a decimal number), A060175 (square array), A082786 (regular triangle), A124010 (with 0's removed, excepting row 1), A143078 (another irregular triangle).

a067255 n k = a067255_tabf !! (n-1) !! (k-1)
a067255_row 1 = [0]
a067255_row n = f n a000040_list where
   f 1 _      = []
   f u (p:ps) = g u 0 where
     g v e = if m == 0 then g v' (e + 1) else e : f v ps
             where (v',m) = divMod v p
a067255_tabf = map a067255_row [1..]
-- Reinhard Zumkeller, Jun 11 2013

f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; Array[f, 29] // Flatten (* Michael De Vlieger, Mar 08 2019 *)

A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 3, 7, 8, 9, 2, 5, 11, 4, 3, 13, 2, 7, 3, 5, 16, 17, 2, 9, 19, 4, 5, 3, 7, 2, 11, 23, 8, 3, 25, 2, 13, 27, 4, 7, 29, 2, 3, 5, 31, 32, 3, 11, 2, 17, 5, 7, 4, 9, 37, 2, 19, 3, 13, 8, 5, 41, 2, 3, 7, 43, 4, 11, 9, 5, 2, 23, 47, 16, 3, 49, 2, 25, 3, 17, 4, 13, 53, 2, 27, 5, 11, 8, 7, 3

Offset: 1

Leroy Quet, Jul 07 2008

In other words, except for row 1, row n contains the unitary prime power divisors of n, sorted by the prime. - Franklin T. Adams-Watters, May 05 2011

A034684(n) = smallest term of n-th row; A028233(n) = T(n,1); A053585(n) = T(n,A001221(n)); A008475(n) = sum of n-th row for n > 1. - Reinhard Zumkeller, Jan 29 2013

			60 has the prime factorization 2^2 * 3^1 * 5^1, so row 60 is (4,3,5).
From _M. F. Hasler_, Oct 12 2018: (Start)
The table starts:
    n : largest prime powers dividing n
    1 :  1
    2 :  2
    3 :  3
    4 :  4
    5 :  5
    6 :  2, 3
    7 :  7
    8 :  8
    9 :  9
   10 :  2, 5
   11 : 11
   12 :  4, 3
   etc. (End)

Reinhard Zumkeller, Rows n=1..10000 of triangle, flattened
Eric Weisstein's World of Mathematics, Prime Factorization

A027748, A124010 are used in a formula defining this sequence.
Cf. A001221 (row lengths), A008475 (row sums), A028233 (column 1), A034684 (row minima), A053585 (right edge).
Cf. A060175, A141810, A213925.

a141809 n k = a141809_row n !! (k-1)
a141809_row 1 = [1]
a141809_row n = zipWith (^) (a027748_row n) (a124010_row n)
a141809_tabf = map a141809_row [1..]
-- Reinhard Zumkeller, Mar 18 2012

f[{x_, y_}] := x^y; Table[Map[f, FactorInteger[n]], {n, 1, 50}] // Grid (* Geoffrey Critzer, Apr 03 2015 *)

A141809_row(n)=if(n>1, [f[1]^f[2]|f<-factor(n)~], [1]) \\ M. F. Hasler, Oct 12 2018, updated Aug 19 2022

T(n,k) = A027748(n,k)^A124010(n,k) for n > 1, k = 1..A001221(n). - Reinhard Zumkeller, Mar 15 2012

A249344 A(n,k) = exponent of the largest power of n-th prime which divides k, square array read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 28 2014

Square array A(n,k), where n = row, k = column, read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... (transpose of array A060175).
A(n,k) is the (p_n)-adic valuation of k, where p_n is the n-th prime, A000040(n).
Each row is effectively a ruler function, s, with s(1) = 0. - Peter Munn, Apr 30 2022

			The top-left corner of the array:
  0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ...
  0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, ...
  0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, ...
  0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, ...
  ...
A(1,8) = 3, because 2^3 is the largest power of 2 (= p_1 = A000040(1)) that divides 8.
a(2,9) = 2, because 3^2 is the largest power of 3 (= p_2) that divides 9.
a(3,15) = 1, because 5^1 is the largest power of 5 (= p_3) that divides 15.

Transpose: A060175.
Row 1: A007814.
Row 2: A007949.
Row 3: A112765.
Row 4: A214411.
Cf. also A000040, A002260, A004736, A085604, A090622, A115627, A249421.
Completely additive sequences where more than one prime is mapped to 1, all other primes to 0: A065339, A083025, A087436, A169611.
Ruler functions, s, with s(1) = 0 that are not rows here: A122840, A122841, A235127, A244413.

A[n_, k_] := IntegerExponent[k, Prime[n]]; Table[A[k, n - k + 1], {n, 1, 15}, {k, 1, n}] // Flatten (* Amiram Eldar, Oct 01 2023 *)

a(n, k) = valuation(k, prime(n)); \\ Michel Marcus, Jun 24 2017

from sympy import prime
def a(n, k):
    p=prime(n)
    i=z=0
    while p**i<=k:
        if k%(p**i)==0: z=i
        i+=1
    return z
for n in range(1, 10): print([a(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 24 2017

(define (A249344 n) (A249344bi (A002260 n) (A004736 n)))
(define (A249344bi row col) (let ((p (A000040 row))) (let loop ((n col) (i 0)) (cond ((not (zero? (modulo n p))) i) (else (loop (/ n p) (+ i 1)))))))

Row n, as a sequence, is completely additive with A(n, prime(n)) = 1, A(n, prime(m)) = 0 for m <> n. - Peter Munn, Apr 30 2022

Sum_{k=1..m} A(n,k) ~ (1/(prime(n)-1)) * m. - Amiram Eldar, Oct 01 2023

A060176 Square array A(n,k) = the largest power of k-th prime which divides n, read by falling antidiagonals.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Mar 14 2001

Product of terms on row n is n.

			The top left corner of the array:
  n\k |  1   2   3   4   5   6   7   8
  ----+---------------------------------
   1  |  1,  1,  1,  1,  1,  1,  1,  1,
   2  |  2,  1,  1,  1,  1,  1,  1,  1,
   3  |  1,  3,  1,  1,  1,  1,  1,  1,
   4  |  4,  1,  1,  1,  1,  1,  1,  1,
   5  |  1,  1,  5,  1,  1,  1,  1,  1,
   6  |  2,  3,  1,  1,  1,  1,  1,  1,
   7  |  1,  1,  1,  7,  1,  1,  1,  1,
   8  |  8,  1,  1,  1,  1,  1,  1,  1,
   9  |  1,  9,  1,  1,  1,  1,  1,  1,
  10  |  2,  1,  5,  1,  1,  1,  1,  1,
  11  |  1,  1,  1,  1, 11,  1,  1,  1,
  12  |  4,  3,  1,  1,  1,  1,  1,  1,
  13  |  1,  1,  1,  1,  1, 13,  1,  1,
  14  |  2,  1,  1,  7,  1,  1,  1,  1,
  15  |  1,  3,  5,  1,  1,  1,  1,  1,
  16  | 16,  1,  1,  1,  1,  1,  1,  1,
  17  |  1,  1,  1,  1,  1,  1, 17,  1,
  18  |  2,  9,  1,  1,  1,  1,  1,  1,
  19  |  1,  1,  1,  1,  1,  1,  1, 19,
  ...
a(12,1) = 4 since 4 = 2^2 = prime(1)^2 divides 12 but 8 = 2^3 does not.

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals

Columns include A006519, A038500.

up_to = 105;
A060176sq(n,k) = (prime(k)^valuation(n,prime(k)));
A060176list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A060176sq(col,(a-(col-1))))); (v); };
v060176 = A060176list(up_to);
A060176(n) = v060176[n]; \\ Antti Karttunen, Jan 16 2025

A(n, k) = A000040(k)^A060175(n, k).

Edited by Antti Karttunen, Jan 16 2025

A249422 Transpose of square array A249421.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 3, 2, 17, 0, 0, 0, 0, 0, 0, 0, 2, 0, 10, 0, 0, 0, 0, 0, 0, 0, 6, 1, 14, 12, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 10, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 6, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 6, 13, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 8, 11

Offset: 1

Antti Karttunen, Oct 28 2014

See comments at A249421.

Cf. A249421, A060175, A001142.

(define (A249422 n) (A249421bi (A004736 n) (A002260 n))) ;;
Code for A249421bi given in A249421.

A334215 T(n, k) is the greatest positive integer m such that m^k divides n; square array T(n, k), n, k > 0 read by antidiagonals downwards.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 9, 1, 1, 1, 1, 1, 1, 1, 2, 3, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Rémy Sigrist, Apr 19 2020

			Square array starts:
  n\k|   1  2  3  4  5  6  7  8  9 10
  ---+-------------------------------
    1|   1  1  1  1  1  1  1  1  1  1
    2|   2  1  1  1  1  1  1  1  1  1
    3|   3  1  1  1  1  1  1  1  1  1
    4|   4  2  1  1  1  1  1  1  1  1
    5|   5  1  1  1  1  1  1  1  1  1
    6|   6  1  1  1  1  1  1  1  1  1
    7|   7  1  1  1  1  1  1  1  1  1
    8|   8  2  2  1  1  1  1  1  1  1
    9|   9  3  1  1  1  1  1  1  1  1
   10|  10  1  1  1  1  1  1  1  1  1
   11|  11  1  1  1  1  1  1  1  1  1
   12|  12  2  1  1  1  1  1  1  1  1
   13|  13  1  1  1  1  1  1  1  1  1
   14|  14  1  1  1  1  1  1  1  1  1
   15|  15  1  1  1  1  1  1  1  1  1
   16|  16  4  2  2  1  1  1  1  1  1

Cf. A000188, A051903, A053150, A053164, A060175, A261969, A334215, A334216.

T(n,k) = { my (f=factor(n)); prod (i=1, #f~, f[i,1]^(f[i,2]\k)) }

T(n, 1) = n.
T(n, 2) = A000188(n).
T(n, 3) = A053150(n).
T(n, 4) = A053164(n).
T(n, A051903(n)) = A261969(n).
T(n, k) = 1 for any k > A051903(n).
T(n^k, k) = n.

A298155 For any n >= 0 and k > 0, the prime(k)-adic valuation of a(n) equals the prime(k)-adic valuation of n + k (where prime(k) denotes the k-th prime).

Original entry on oeis.org

1, 6, 5, 28, 3, 2, 11, 4680, 1, 2, 357, 76, 5, 6, 23, 16, 9, 770, 1, 348, 403, 2, 75, 8, 7, 1998, 1, 340, 1353, 86, 19, 672, 235, 26, 9, 4, 1, 36570, 7, 88, 3, 2, 295, 2196, 17, 98, 39, 400, 1943, 114, 11, 8804, 68985, 2, 1, 24, 1, 790, 3, 364, 1909, 3366, 185

Offset: 0

Rémy Sigrist, Jan 13 2018

nonn

This sequence has similarities with A102370: here, for k > 0, a(n) and n + k have the same prime(k)-adic valuation, there, for k >= 0, A102370(n) and n + k have the same k-th binary digit (the least significant binary digit having index 0).
For any positive number, say k, we can use the Chinese remainder theorem to find a term that is a multiple of k; this term has index < k.
a(n) is even iff n is odd.
See A298161 for the indices of ones in the sequence.

			For n = 7:
- the 2-adic valuation of 7 + 1 is 3,
- the 3-adic valuation of 7 + 2 is 2,
- the 5-adic valuation of 7 + 3 is 1,
- the 7-adic valuation of 7 + 4 is 0,
- the 11-adic valuation of 7 + 5 is 0,
- the 13-adic valuation of 7 + 6 is 1,
- for k > 6, the prime(k)-adic valuation of 7 + k is 0,
- hence a(7) = 2^3 * 3^2 * 5^1 * 13^1 = 4680.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A000040, A002110, A007814, A007949, A060175, A073605, A102370, A112765, A214411, A298161.

f:= proc(n) local v, p, k;
  v:= 1: p:= 1:
  for k from 1 do
    p:= nextprime(p);
    if p > n+k then return v fi;
    v:= v * p^padic:-ordp(n+k,p)
  od
end proc:
map(f, [$0..100]); # Robert Israel, Jan 16 2018

f[n_] := Module[{v = 1, p = 1, k}, For[k = 1, True, k++, p = NextPrime[p]; If[p > n + k, Return[v]]; v *= p^IntegerExponent[n + k, p]]];
f /@ Range[0, 100] (* Jean-François Alcover, Jul 30 2020, after Maple *)

a(n) = my (v=1, k=0); forprime(p=1, oo, k++; if (n+k < p, break); v *= p^valuation(n+k,p)); return (v)

For any n >= 0:
- a(n) = Product_{ k > 0 } A000040(k)^A060175(n + k, k) (this product is well defined as only finitely many terms are > 1),
- A007814(a(n)) = A007814(n + 1),
- A007949(a(n)) = A007949(n + 2),
- A112765(a(n)) = A112765(n + 3),
- A214411(a(n)) = A214411(n + 4),
- gcd(n, a(n)) = 1.
For any n > 0:
- a(A073605(n)) is a multiple of A002110(n).

A298161 Nonnegative numbers n such that for any k > 0, n + k is not a multiple of prime(k) (where prime(k) denotes the k-th prime).

Original entry on oeis.org

0, 8, 18, 26, 36, 54, 56, 74, 84, 86, 134, 140, 156, 168, 170, 174, 194, 200, 216, 224, 236, 240, 246, 260, 300, 308, 324, 326, 366, 368, 386, 390, 414, 420, 440, 456, 464, 476, 494, 498, 518, 536, 560, 564, 576, 590, 594, 624, 630, 650, 660, 678, 698, 708

Offset: 1

Rémy Sigrist, Jan 14 2018

nonn

Equivalently, these are the numbers n >= 0 such that A298155(n) = 1.
Equivalently, these are the numbers n >= 0 such that the diagonal of A060175 starting at A060175(n+1, 1) contains only zeros.
All terms are even.
This sequence is a subsequence of A005843, A007494, A047207 and A047318.

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

Cf. A005843, A007494, A007814, A007949, A047207, A047318, A060175, A112765, A214411, A298155.

filter:= proc(n) local p, k;
  p:= 1:
  for k from 1 do
    p:= nextprime(p);
    if p > n+k then return true
    elif n+k mod p = 0 then return false
    fi
  od
end proc:
select(filter, [seq(i,i=0..1000,2)]); # Robert Israel, Jan 16 2018

A007814(a(n) + 1) = 0.
A007949(a(n) + 2) = 0.
A112765(a(n) + 3) = 0.
A214411(a(n) + 4) = 0.

A384003 Irregular triangle T(n,k), n >= 0, 0 <= k < 2^(n-1), where T(n,k) = Product_{j=0..n-1} prime(j+1)^((n-j)*d_j), where d_j is the bit with digit weight 2^j in the binary expansion of 2^(n-1)+k.

Original entry on oeis.org

1, 2, 3, 12, 5, 40, 45, 360, 7, 112, 189, 3024, 175, 2800, 4725, 75600, 11, 352, 891, 28512, 1375, 44000, 111375, 3564000, 539, 17248, 43659, 1397088, 67375, 2156000, 5457375, 174636000, 13, 832, 3159, 202176, 8125, 520000, 1974375, 126360000, 4459, 285376, 1083537

Offset: 0

Michael De Vlieger and Peter Munn, May 28 2025

This sequence can be seen as a structured ordering of numbers m that are not divisible by the square of their greatest prime factor and where every prime in the canonical factorization of m has the same sum of prime index and exponent. For example, prime(1)^3 * prime(3)^1 = 2^3 * 5 = 40. The ordering is lexicographic according to prime divisors listed in decreasing order, as used for A019565. Row n has the numbers whose greatest prime factor is prime(n).

			Table begins:
n\k  0    1    2     3    4     5     6       7
-----------------------------------------------
0:   1;
1:   2;
2:   3,  12;
3:   5,  40,  45,  360;
4:   7, 112, 189, 3024, 175, 2800, 4725, 75600;
     ...
Table showing prime power decomposition of a(n), where A067255(a(n)) represents prime(i)^j | a(n), with j in the i-th position, replacing 0 with "." for visibility:
 n     a(n)  A067255(a(n))
--------------------------
 0       1   .
 1       2   1
 2       3   .1
 3      12   21
 4       5   ..1
 5      40   3.1
 6      45   .21
 7     360   321
 8       7   ...1
 9     112   4..1
10     189   .3.1
11    3024   43.1
12     175   ..21
13    2800   4.21
14    4725   .321
15   75600   4321

Michael De Vlieger, Table of n, a(n) for n = 0..16384 (rows n = 0..14, flattened)
Michael De Vlieger, Log log scatterplot of a(n), n = 0..16384.
Michael De Vlieger, Plot prime(i)^j at (x,y) = (n,i), n = 0..2047, 16X vertical exaggeration, with a color function representing j = 1 in black, j = 2 in red, j = 3 in orange, ..., j = 14 in magenta.

Cf. A000040, A003961, A006939, A007947, A019565, A060175, A061395, A071178, A251720.

f[x_] := If[x == 1, {0}, Function[g, ReplacePart[Table[0, {PrimePi[f[[-1, 1]] ]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, g]]@ FactorInteger@ x]; Table[f[Reverse@ Range[Length[#]]*#] &@ Reverse@ IntegerDigits[n, 2], {n, 0, 120}]

T(0,0) = 1; T(1,0) = 2.
Otherwise, T(n,2k) = A003961(T(n-1,k)).
T(n,2k+1) = T(n,2k)*2^n.
T(n,0) = prime(n).
T(n,2^(n-1)-1) = A006939(n).
T(n,2^(n-2)) = A251720(n).
Using a(m) to denote a term of the linear sequence with offset 0: (Start)
A019565(m) = A007947(a(m)).
a(m) = T(n,k) = gcd(A019565(m)^n, A006939(n)).
Equivalently, for p = A000040(i), the i-th prime, p|a(m) iff p|A019565(m), in which case A060175(m,i) = j - i + 1, where j = PrimePi(gpf(A019565(m))) = A061395(A019565(m)).
(End)
For n > 0, A071178(T(n,k)) = 1.

Name edited by Peter Munn, Aug 30 2025

cp's OEIS Frontend

A067255 Irregular triangle read by rows: row n gives exponents in prime factorization of n.

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A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

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A249344 A(n,k) = exponent of the largest power of n-th prime which divides k, square array read by antidiagonals.

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A060176 Square array A(n,k) = the largest power of k-th prime which divides n, read by falling antidiagonals.

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A249422 Transpose of square array A249421.

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A334215 T(n, k) is the greatest positive integer m such that m^k divides n; square array T(n, k), n, k > 0 read by antidiagonals downwards.

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A298155 For any n >= 0 and k > 0, the prime(k)-adic valuation of a(n) equals the prime(k)-adic valuation of n + k (where prime(k) denotes the k-th prime).

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