A126256 -id:A126256 - cp's OEIS frontend

A126257 Number of distinct new terms in row n of Pascal's triangle.

1, 0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 5, 7, 8, 9, 9, 9, 8, 11, 11, 12, 12, 13, 13, 13, 14, 15, 15, 16, 16, 17, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 21, 23, 23, 24, 24, 25, 25, 26, 26, 27, 26, 26, 28, 29, 29, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 35, 36, 36, 37, 37, 38

Offset: 0

Nick Hobson, Dec 24 2006

Partial sums are in A126256.

n occurs a(n) times in A265912. - Reinhard Zumkeller, Dec 18 2015

			Row 6 of Pascal's triangle is: 1, 6, 15, 20, 15, 6, 1. Of these terms, only 15 and 20 do not appear in rows 0-5. Hence a(6)=2.

Nick Hobson, Table of n, a(n) for n = 0..1000
Nick Hobson, Python program for this sequence

Cf. A007318, A062854, A126254, A126255, A126256.

Cf. A034868, A265912.

import Data.List.Ordered (minus, union)
a126257 n = a126257_list !! n
a126257_list = f [] a034868_tabf where
   f zs (xs:xss) = (length ys) : f (ys `union` zs) xss
                   where ys = xs `minus` zs
-- Reinhard Zumkeller, Dec 18 2015

lim=77; z=listcreate(1+lim^2\4); print1(1, ", "); r=1; for(a=1, lim, for(b=1, a\2, s=Str(binomial(a, b)); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); print1(1+#z-r, ", "); r=1+#z)

def A126257(n):
    if n:
        s, c = (1,), {1}
        for i in range(n-1):
            c.update(set(s:=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1))+(1,)))
        return len(set((1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1))+(1,))-c)
    return 1 # Chai Wah Wu, Oct 17 2023

A126254 Number of distinct terms i^j for 1 <= i,j <= n.

Original entry on oeis.org

1, 3, 7, 11, 19, 28, 40, 50, 60, 76, 96, 115, 139, 163, 189, 207, 239, 270, 306, 340, 378, 417, 461, 503, 539, 585, 621, 670, 726, 779, 839, 881, 941, 1003, 1067, 1113, 1185, 1254, 1326, 1397, 1477, 1553, 1637, 1717, 1799, 1884, 1976, 2063, 2135, 2225

Offset: 1

Nick Hobson, Dec 24 2006

An easy upper bound is n(n-1)+1 = A002061(n).

			a(4) = 11, as there are 11 distinct terms in 1^1=1, 1^2=1, 1^3=1, 1^4=1, 2^1=2, 2^2=4, 2^3=8, 2^4=16, 3^1=3, 3^2=9, 3^3=27, 3^4=81, 4^1=4, 4^2=16, 4^3=64, 4^4=256.

Nick Hobson and John Silberholz, Table of n, a(n) for n = 1..10000 [Terms 1 through 1000 were computed by N. Hobson; terms 1001 through 10000 by John Silberholz, Feb 23 2015]
John Silberholz, Combinatorial approach to calculate sequence

Cf. A027424, A061786, A126255, A126256, A126257.

seq(nops({seq(seq(i^j, i=1..n),j=1..n)}),n=1..100); # Robert Israel, Feb 23 2015

lim=50; z=listcreate(lim*(lim-1)+1); for(m=1, lim, for(i=1, m, x=factor(i); x[, 2]*=m; s=Str(x); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); t=factor(m); for(j=1, m, x=t; x[, 2]=j*t[, 2]; s=Str(x); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); print1(#z, ", "))

def A126254(n): return len({i**j for i in range(1,n+1) for j in range(1,n+1)}) # Chai Wah Wu, Oct 17 2023

A126254 <- function(limit) {  if (limit == 1) { return(1) } ; num.powers <- c(1, rep(0, limit-1)) ; handled <- c(T, rep(F, limit-1)) ; for (base in 2:ceiling(sqrt(limit))) { if (!handled[base]) { num.handle <- floor(log(limit, base)) ; handled[base^(1:num.handle)] <- T ; num.powers[base] <- length(unique(as.vector(outer(1:num.handle, 1:limit)))) }} ; num.powers[!handled] <- limit ; sum(num.powers) } ; A126254(50) # John Silberholz, Feb 23 2015

A126255 Number of distinct terms i^j for 2 <= i,j <= n.

Original entry on oeis.org

1, 4, 8, 15, 23, 34, 44, 54, 69, 88, 106, 129, 152, 177, 195, 226, 256, 291, 324, 361, 399, 442, 483, 519, 564, 600, 648, 703, 755, 814, 856, 915, 976, 1039, 1085, 1156, 1224, 1295, 1365, 1444, 1519, 1602, 1681, 1762, 1846, 1937, 2023, 2095, 2184, 2279

Offset: 2

Nick Hobson, Dec 24 2006

An easy upper bound is (n-1)^2 = A000290(n-1).

			a(4) = 8 as there are 8 distinct terms in 2^2=4, 2^3=8, 2^4=16, 3^2=9, 3^3=27, 3^4=81, 4^2=16, 4^3=64, 4^4=256.

Eric M. Schmidt, Table of n, a(n) for n = 2..10000
Project Euler, Problem 29: Distinct powers.

Cf. A027424, A061786, A126254, A126256, A126257.

SetAttributes[a, {Listable, NumericFunction}]
a[n_ /; n < 2] := "error"
a[2] := 1
a[n_Integer?IntegerQ /; n > 2] :=
 Length[DeleteDuplicates[
   Distribute[f[Range[2, n], Range[2, n]], List,
     f] /. {f ->
      Power}]](*By using Distribute instead of Outer I avoid having to use Flatten on Outer*)
a[Range[2, 100]]
(* Peter Cullen Burbery, Aug 15 2023 *)

lim=51; z=listcreate((lim-1)^2); for(m=2, lim, for(i=2, m, x=factor(i); x[, 2]*=m; s=Str(x); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); t=factor(m); for(j=2, m, x=t; x[, 2]=j*t[, 2]; s=Str(x); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); print1(#z, ", "))

def A126255(n): return len({i**j for i in range(2,n+1) for j in range(2,n+1)}) # Chai Wah Wu, Oct 17 2023

A199425 Number of distinct primes in rows 0 through n of the triangle in A199333.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 8, 10, 13, 17, 20, 24, 30, 35, 42, 49, 57, 63, 71, 80, 90, 99, 110, 121, 133, 145, 157, 170, 184, 197, 212, 227, 242, 258, 275, 292, 310, 327, 345, 364, 384, 404, 425, 446, 467, 489, 512, 535, 558, 581, 606, 630, 656, 682, 709, 736, 764

Offset: 0

Reinhard Zumkeller, Nov 09 2011

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..150

Cf. A199424, A126256.

import Data.List (union)
a199425 n = a199425_list !! n
a199425_list = f [] a199333_tabl where
   f ps (ts:tts) =  (length ps') : f ps' tts where
     ps' = ps `union` (take ((length ts - 1) `div` 2) $ tail ts)

A258318 Number of distinct numbers in rows 0 through n of triangle A258197.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 9, 12, 16, 20, 24, 29, 34, 41, 44, 50, 57, 66, 74, 83, 91, 102, 112, 124, 135, 148, 161, 174, 187, 201, 214, 230, 246, 263, 279, 296, 313, 331, 349, 369, 388, 408, 428, 450, 471, 494, 516, 540, 563, 588, 612, 637, 662, 689, 715, 741, 769

Offset: 0

Reinhard Zumkeller, May 26 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A253197, A126256.

import Data.Set (singleton, insert, size)
a258318 n = a258318_list !! n
a258318_list = f 2 a258197_tabl $ singleton 0 where
   f k (xs:xss) zs = g (take (div k 2) xs) zs where
     g []     ys = size ys : f (k + 1) xss ys
     g (x:xs) ys = g xs (insert x ys)

cp's OEIS Frontend

A126257 Number of distinct new terms in row n of Pascal's triangle.

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A126254 Number of distinct terms i^j for 1 <= i,j <= n.

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A126255 Number of distinct terms i^j for 2 <= i,j <= n.

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Mathematica

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A199425 Number of distinct primes in rows 0 through n of the triangle in A199333.

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Haskell

A258318 Number of distinct numbers in rows 0 through n of triangle A258197.

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Haskell