A252738 -id:A252738 - cp's OEIS frontend

A252737 Row sums of irregular tables A005940, A163511, and A332977.

1, 2, 7, 28, 130, 702, 4384, 31516, 260068, 2445372, 25796360, 299286550, 3751803964, 50211590696, 712746859372, 10697637496288, 169490803535680, 2830925427778810, 49785906936838240, 921273098388684878, 17944637546960083042, 368472898102440537484, 7993616254370783660414, 183539682466936703629744

Offset: 0

Antti Karttunen, Dec 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Row sums of tables A005940, A163511, and A332977.
Cf. A252738 (row products).
Cf. A000079, A000225, A252745, A252746.

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-`if`(
      i=0, j, 1), j)*ithprime(j), j=1..`if`(i=0, n, i)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Mar 04 2020

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - If[i == 0, j, 1], j]* Prime[j], {j, 1, If[i == 0, n, i]}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jan 03 2022, after Alois P. Heinz *)

allocatemem(234567890);
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A252737print(up_to_n) = { my(s, i=0, n=0); for(n=0, up_to_n, if(0 == n, s = 1, if(1 == n, s = 2; lev = vector(1); lev[1] = 2, oldlev = lev; lev = vector(2*length(oldlev)); s = 0; for(i = 0, (2^(n-1))-1, lev[i+1] = if((i%2),A003961(oldlev[(i\2)+1]),2*oldlev[(i\2)+1]); s += lev[i+1]))); write("b252737.txt", n, " ", s)); };
A252737print(23); \\ Terms a(0) .. a(23) were computed with this program.

(define (A252737 n) (if (zero? n) 1 (add A163511 (A000079 (- n 1)) (A000225 n))))

(define (A252737 n) (if (zero? n) 1 (add (COMPOSE A005940 1+) (A000079 (- n 1)) (A000225 n))))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))
(define (COMPOSE . funlist) (cond ((null? funlist) (lambda (x) x)) (else (lambda (x) ((car funlist) ((apply COMPOSE (cdr funlist)) x))))))

a(0) = 1; for n>1: a(n) = Sum_{k = A000079(n-1) .. A000225(n)} A163511(k) = Sum_{k = 2^(n-1) .. (2^n)-1} A163511(k).

A267096 a(n) = Product_{i=0..n} prime(i+2)^binomial(n,i).

Original entry on oeis.org

3, 15, 525, 1414875, 41985913344375, 433555011900329243987584396875, 3514495551481947615680580256869117013417604971088496013610671875

Offset: 0

Antti Karttunen, Feb 06 2016

nonn

			Terms are obtained by exponentiating the odd primes in range [3 .. prime(2+n)] with the binomial coefficients obtained from row n of Pascal's triangle (A007318) and then multiplying the factors together:
            3^1
         3^1 * 5^1
      3^1 * 5^2 * 7^1
   3^1 * 5^3 * 7^3 * 11^1
3^1 * 5^4 * 7^6 * 11^4 * 13^1
etc.

Index entries for triangles and arrays related to Pascal's triangle

Second column (or diagonal from right) in A066117.

Cf. A000040, A003961, A007188, A007318, A252738, A276804.

(define (A267096 n) (mul (lambda (k) (expt (A000040 (+ 2 k)) (A007318tr n k))) 0 n)) ;; Where A007318tr gives binomial coefficients, as in A007318.
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (1+ i) (* res (intfun i)))))))

a(n) = Product_{i=0..n} prime(i+2)^C(n,i).

a(n) = A003961(A007188(n)).

A332977 Triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m satisfying Omega(m) + pi(gpf(m)) - [m<>1] = n; n>=0, 1<=k<=A011782(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 7, 10, 12, 15, 16, 18, 25, 27, 11, 14, 20, 21, 24, 30, 32, 35, 36, 45, 49, 50, 54, 75, 81, 125, 13, 22, 28, 33, 40, 42, 48, 55, 60, 63, 64, 70, 72, 77, 90, 98, 100, 105, 108, 121, 135, 147, 150, 162, 175, 225, 243, 245, 250, 343, 375, 625

Offset: 0

Alois P. Heinz, Mar 04 2020

Integer m > 0 is listed in row n if the index of the largest prime factor of m (or 0 for empty prime factor set) plus the cardinality of the other prime factors of m (counted with multiplicity) equals n.
Row n+k-1 contains prime(n)^k (for all n, k >= 1).
The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1, 2, 3, 4, 5, 6, 10, ... and inverse permutation A332990.
This is a variant with sorted rows of A005940 (offset differs) or A163511.

			Triangle T(n,k) begins:
   1;
   2;
   3,  4;
   5,  6,  8,  9;
   7, 10, 12, 15, 16, 18, 25, 27;
  11, 14, 20, 21, 24, 30, 32, 35, 36, 45, 49, 50, 54, 75, 81, 125;
  ...

Alois P. Heinz, Rows n = 0..16, flattened
Wikipedia, Iverson bracket
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences computed from indices in prime factorization

Columns k=1-2 give: A008578(n+1), A100484(n-1) for n>1.
Last elements of rows give A332979.
Row sums give A252737.
Product of row elements give A252738.
Row lengths give A011782.
Cf. A000027, A000040, A000720 (pi), A001222 (Omega), A006530 (GPF), A060576 ([n<>1]), A061395 (pi(gpf(n))), A215366, A332990.
Cf. A005940, A163511.

b:= proc(n, i) option remember; `if`(n=0, [1], sort([seq(map(x-> x*
      ithprime(j), b(n-`if`(i=0, j, 1), j))[], j=1..`if`(i=0, n, i))]))
    end:
T:= n-> b(n, 0)[]:
seq(T(n), n=0..7);

b[n_, i_] := b[n, i] = If[n == 0, {1}, Sort[Flatten[Table[#*
    Prime[j]& /@ b[n-If[i == 0, j, 1], j], {j, 1, If[i == 0, n, i]}]]]];
T[n_] := b[n, 0];
T /@ Range[0, 7] // Flatten (* Jean-François Alcover, Mar 30 2021, after Alois P. Heinz *)

A253788 Row products of irregular tables A252753 & A252755.

Original entry on oeis.org

1, 2, 12, 2160, 1905120000, 1034766105221882880000000, 139870330430486189977277369128961542673635737600000000000000

Offset: 0

Antti Karttunen, Jan 13 2015

nonn

Cf. A253787 (the corresponding sums).

Cf. A000079, A000225, A252738, A252753, A252755.

(define (A253788 n) (if (zero? n) 1 (mul A252753 (A000079 (- n 1)) (A000225 n))))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))

a(0) = 1; for n>1: a(n) = Product_{k = A000079(n-1) .. A000225(n)} A252753(k) = Product_{k = 2^(n-1) .. (2^n)-1} A252753(k).

A276804 Second column T[.,2] of array T = A255483: T[0,j] = prime(j), T[i+1,j] = T[i,j]*T[i,j+1]/gcd(T[i,j],T[i,j+1])^2, i >= 0, j >= 1.

Original entry on oeis.org

3, 15, 21, 1155, 39, 3315, 5187, 111546435, 87, 13485, 22533, 1575169365, 48633, 6022953885, 12684118629, 961380175077106319535, 183, 61305, 90951, 24466273755, 187941, 88836891585, 157950690807, 133754519645521334494935, 536007, 573342567585

Offset: 0

M. F. Hasler, Sep 17 2016

nonn

By construction all terms are divisible by 3, and the n-th term a(n-1) is divisible by prime(n+1). We have a(n)/3 = (1, 5, 7, 385, 13, 1105, 1729, 37182145, 29, 4495, ...). Neither the sequence of primes appearing here, (5, 7, 13, 29, 61, ...), nor its complement in the primes, ([2, 3,] 11, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 67, ...), seem to be listed in the OEIS.

This is also the multiplicative encoding of Pascal's triangle in Z_2 (A047999), shifted by prefixing an initial 0 to the n-th row; e.g., n=2 => 1,0,1 => 0,1,0,1 => 2^0 * 3^1 * 5^0 * 7^1 = a(2).

Cf. A255483 (the square array T), A123098 (first column of T), A003961.

Cf. A007913, A252738, A267096.

A276804(n)=prod(j=0, n, if(bitand(n-j, j), 1, prime(j+2)))

a(n) = A003961(A123098(n)).
a(n) = Prod_{j=0..n} prime(j+2)^(!(n-j & j)), where ! is "not" (=0 for nonzero and 1 for zero) and & is bitwise AND.
a(n) = A007913(A267096(n)) = A007913(A252738(n+2)). - Antti Karttunen, Sep 18 2016

cp's OEIS Frontend

A252737 Row sums of irregular tables A005940, A163511, and A332977.

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A267096 a(n) = Product_{i=0..n} prime(i+2)^binomial(n,i).

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A332977 Triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m satisfying Omega(m) + pi(gpf(m)) - [m<>1] = n; n>=0, 1<=k<=A011782(n).

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A253788 Row products of irregular tables A252753 & A252755.

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A276804 Second column T[.,2] of array T = A255483: T[0,j] = prime(j), T[i+1,j] = T[i,j]*T[i,j+1]/gcd(T[i,j],T[i,j+1])^2, i >= 0, j >= 1.

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