A077320 -id:A077320 - cp's OEIS frontend

A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

2, 3, 4, 5, 6, 8, 7, 10, 9, 16, 11, 14, 15, 12, 32, 13, 22, 21, 20, 18, 64, 17, 26, 33, 28, 25, 24, 128, 19, 34, 39, 44, 35, 30, 27, 256, 23, 38, 51, 52, 55, 42, 40, 36, 512, 29, 46, 57, 68, 65, 66, 49, 45, 48, 1024, 31, 58, 69, 76, 85, 78, 77, 56, 50, 54, 2048, 37, 62, 87, 92

Offset: 1

Leroy Quet, Jan 27 2007

This sequence is a permutation of the integers >= 2.
Since the table has been entered by rising instead of falling antidiagonals, the sequence represents the transpose, with columns instead of rows: cf. the "table" link, section "infinite square array". - M. F. Hasler, Oct 22 2019
Start with table headed by primes in the first row, then list beneath each prime(k) the ordered prime(k)-smooth numbers. Read the table by falling antidiagonals to get the terms of this sequence. - David James Sycamore, Jun 23 2024

			Array begins: (rows here appear as columns in the "table" display of the sequence)
   2,  4,  8, 16, 32, 64, 128, 256, 512, ... (A000079)
   3,  6,  9, 12, 18, 24,  27,  36,  48, ... (A065119)
   5, 10, 15, 20, 25, 30,  40,  45,  50, ... (A080193)
   7, 14, 21, 28, 35, 42,  49,  56,  63, ... (A080194)
  11, 22, 33, 44, 55, 66,  77,  88,  99, ... (A080195)
  13, 26, 39, 52, 65, 78,  91, 104, 117, ... (A080196)
The 3rd row, for example, contains the positive integers where the 3rd prime, 5, is the largest prime divisor. That is, each integer in this row is divisible by 5 and may be divisible by 2 or 3 as well, but none of the integers in this row are divisible by primes larger than 5. (So for example, 35 = 5*7 is excluded from the 3rd row.)

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A083140, A006530, A000040 (1st col), A033286 (main diag), A077320.

Cf. A000079, A003586, A051037, A002473, A051038, A080197, A080681.

lpf[n_] := FactorInteger[n][[ -1, 1]];f[n_, m_] := f[n, m] = Block[{k},k = If[m == 1, Prime[n], f[n, m - 1] + 1];While[lpf[k] != Prime[n], k++ ];k];Table[f[ d - m + 1, m], {d, 12}, {m, d}] // Flatten (* Ray Chandler, Feb 09 2007 *)

T=List(); r=c=1; for(n=1,99, #TT[r][1], ); print1(T[r][c]","); r-- && c++ || r=c+c=1) \\ M. F. Hasler, Oct 22 2019

Extended by Ray Chandler, Feb 09 2007

A196421 a(n) = prime(n)*T(n), where T = A000217.

Original entry on oeis.org

2, 9, 30, 70, 165, 273, 476, 684, 1035, 1595, 2046, 2886, 3731, 4515, 5640, 7208, 9027, 10431, 12730, 14910, 16863, 19987, 22908, 26700, 31525, 35451, 38934, 43442, 47415, 52545, 62992, 69168, 76857, 82705, 93870, 100566, 110371, 120783, 130260, 141860

Offset: 1

Harvey P. Dale, Oct 15 2011

nonn

This sequence is mentioned in A077320. - Omar E. Pol, Mar 12 2012

			The 4th prime is 7, the 4th triangular number is 10, therefore a(4) = 7*10 = 70.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217.
Row sums of triangle A077320. - Omar E. Pol, Mar 12 2012
Subsequence of A085783. - Michel Marcus, May 15 2018

With[{nn=60},Prime[Range[nn]]Accumulate[Range[nn]]]

a(n)=prime(n)*binomial(n+1,2) \\ Charles R Greathouse IV, Nov 22 2011

from sympy import prime
def a(n): return prime(n) * n*(n+1)//2
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Sep 01 2022

a(n) ~ 0.5 n^3 log n. - Charles R Greathouse IV, Nov 22 2011

a(n) = A000040(n)*A000217(n). - Omar E. Pol, Mar 12 2012

cp's OEIS Frontend

A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

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