A378194 Rectangular array, read by descending antidiagonals: row n shows the integers m such that the number of primes of the form 4k+3 (including multiplicities) that divide m is n-1.
1, 2, 3, 4, 6, 9, 5, 7, 18, 27, 8, 11, 21, 54, 81, 10, 12, 33, 63, 162, 243, 13, 14, 36, 99, 189, 486, 729, 16, 15, 42, 108, 297, 567, 1458, 2187, 17, 19, 45, 126, 324, 891, 1701, 4374, 6561, 20, 22, 49, 135, 378, 972, 2673, 5103, 13122, 19683, 25, 23, 57, 147, 405, 1134, 2916, 8019, 15309, 39366, 59049, 26, 24, 66, 171, 441, 1215, 3402, 8748, 24057, 45927, 118098, 177147
Offset: 1
Examples
Corner:
1 2 4 5 8 10 13 16 17
3 6 7 11 12 14 15 19 22
9 18 21 33 36 42 45 49 57
27 54 63 99 108 126 135 147 171
81 162 189 297 324 378 405 441 513
243 486 567 891 972 1134 1215 1323 1539
729 1458 1701 2673 2916 3402 3645 3969 4617
2187 4374 5103 8019 8748 10206 10935 11907 13851
Crossrefs
Programs
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Maple
A378194 := proc(n, k) option remember; local a; if k = 0 then 0; else for a from procname(n, k-1)+1 do if A065339(a) = n-1 then return a; end if; end do; end if; end proc: seq(seq( A378194(n, d-n), n=1..d-1), d=2..10) ; # R. J. Mathar, Jan 28 2025
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Mathematica
u = Map[Map[#[[1]] &, #] &, GatherBy[ SortBy[Map[{#, 1 + Count[Map[IntegerQ[(# - 3)/4] && PrimeQ[#] &, Flatten[Map[ConstantArray[#[[1]], #[[2]]] &, FactorInteger[#]]]], True]} &, Range[24000]], #[[2]] &], #[[2]] &]]; r[m_] := Take[u[[m]], 10]; w[m_, n_] := r[m][[n]]; Grid[Table[w[m, n], {m, 1, 8}, {n, 1, 9}]] (* array *) Table[w[n - k + 1, k], {n, 8}, {k, n, 1, -1}] // Flatten (* sequence *) (* Peter J. C. Moses, Nov 19 2024 *)
Extensions
Definition corrected. - R. J. Mathar, Jan 28 2025
Comments