

A125624


Array read by antidiagonals: nth row contains the positive integers with their largest prime factor equal to the nth prime.


8



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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


LINKS



EXAMPLE

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.)


MATHEMATICA

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 *)


PROG

(PARI) T=List(); r=c=1; for(n=1, 99, #T<r && listput(T, List(prime(r))); #T[r]<c && listput(T[r], T[r][c1]) && while(vecmax(factor(T[r][c]+=T[r][1])[, 1])>T[r][1], ); print1(T[r][c]", "); r && c++  r=c+c=1) \\ M. F. Hasler, Oct 22 2019


CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



