A132274
a(1)=1; a(n+1) = Sum_{k=1..n} (k-th integer from among those positive integers which are coprime to a(n+1-k)).
Original entry on oeis.org
1, 1, 3, 6, 10, 19, 27, 41, 51, 66, 78, 101, 119, 145, 167, 197, 219, 247, 272, 306, 335, 371, 403, 443, 477, 521, 559, 609, 647, 693, 737, 789, 834, 886, 940, 996, 1055, 1118, 1176, 1243, 1306, 1385, 1450, 1523, 1596, 1676, 1749, 1844, 1914, 2010, 2092, 2188
Offset: 1
The integers coprime to a(1)=1 are 1,2,3,4,5,6,... The 5th of these is 5. The integers coprime to a(2)=1 are 1,2,3,4,5... The 4th of these is 4. The integers coprime to a(3)=3 are 1,2,4,5,7,... The 3rd of these is 4. The integers coprime to a(4)=6 are 1,5,7,11,... The 2nd of these is 5. And the integers coprime to a(5)=10 are 1,3,7,9,11,... The first of these is 1. So a(6) = 5 + 4 + 4 + 5 + 1 = 19.
-
A132274 := proc(n) option remember; local a,k,an1k,kcoud,c ; if n = 1 then 1; else a :=0 ; for k from 1 to n-1 do an1k := procname(n-k) ; kcoud := 0 ; for c from 1 do if gcd(c,an1k) = 1 then kcoud := kcoud+1 ; fi; if kcoud = k then a := a+c ; break; fi; od: od: a; fi; end: seq(A132274(n),n=1..60) ; # R. J. Mathar, Jul 20 2009
-
A132274[n_] := A132274[n] = Module[{a, k, an1k, kcoud, c}, If[n == 1, 1, a = 0; For[k = 1, k <= n-1, k++, an1k = A132274[n-k]; kcoud = 0; For[c = 1, True, c++, If[GCD[c, an1k] == 1, kcoud++]; If[kcoud == k, a = a+c; Break[]]]]; a]];
Table[A132274[n], {n, 1, 60}] (* Jean-François Alcover, Jan 28 2024, after R. J. Mathar *)
A132275
a(1)=1. a(n+1) = sum{k=1 to n} (a(k)th integer from among those positive integers which are coprime to a(n+1-k)).
Original entry on oeis.org
1, 1, 2, 4, 8, 17, 37, 81, 177, 387, 847, 1856, 4066, 8910, 19524, 42783, 93760, 205475, 450282, 986770, 2162473, 4738974, 10385267, 22758885, 49875175, 109299427, 239525260, 524909877, 1150318695, 2520876742, 5524399079, 12106496388, 26530895539, 58141380910
Offset: 1
To compute a(5) we add the first integer coprime to a(4), the first integer coprime to a(3), the 2nd integer coprime to a(2) and the 4th integer coprime to a(1), which is the first integer in {1,3,4,5,..}, the first integer in {1,2,3,4,...}, the 2nd integer in {1,2,3,4,...} and the 4th integer in {1,2,3,4,..} = 1+1+2+4=8.
-
A132275 := proc(n) option remember; local a,k,an1k,kcoud,c ; if n = 1 then 1; else a :=0 ; for k from 1 to n-1 do an1k := procname(n-k) ; kcoud := 0 ; for c from 1 do if gcd(c,an1k) = 1 then kcoud := kcoud+1 ; fi; if kcoud = procname(k) then a := a+c ; break; fi; od: od: a; fi; end:
seq(A132275(n),n=1..18) ; # R. J. Mathar, Jul 20 2009
with(numtheory): fc:= proc(t,p) option remember; local m, j, h, pp; if p=1 then t else pp:= phi(p); m:= iquo(t,pp); j:= m*pp; h:= m*p-1; while jAlois P. Heinz, Aug 05 2009
-
fc[t_, p_] := fc[t, p] = Module[{m, j, h, pp}, If[p==1, t, pp = EulerPhi[p]; m = Quotient[t, pp]; j = m*pp; h = m*p-1; While[jJean-François Alcover, Mar 21 2017, after Alois P. Heinz *)
A130767
a(n) = product{k=1 to n} (k-th integer from among those positive integers which are coprime to (n+1-k)).
Original entry on oeis.org
1, 2, 9, 40, 420, 2700, 56595, 419328, 8820900, 88488400, 2327925600, 38767286880, 1912404574080, 21612951360000, 644047087612500, 10985391056640000, 634391869996684800, 14046187624838328960, 764077915447610400000, 15840110879873280000000, 755098009918296312668400
Offset: 1
The integers coprime to 1 are: 1,2,3,4,5,6,... The 5th of these is 5. The integers coprime to 2 are: 1,3,5,7,9,... The 4th of these is 7. The integers coprime to 3 are: 1,2,4,5,7,... The 3rd of these is 4. The integers coprime to 4 are: 1,3,5,... The 2nd of these is 3. And the integers coprime to 5 are: 1,2,3,4,6,... The first of these is 1. So a(5) = 5 * 7 * 4 * 3 * 1 = 420.
-
cop(k, j) = {my(nbc = 0, i = 0); while (nbc != j, i++; if (gcd(i,k)==1, nbc++)); i;}
a(n) = {my(vc = vector(n, k, cop(k, n-k+1))); prod(k=1, n, vc[k]);} \\ Michel Marcus, Mar 14 2018
A130802
a(1) = 1; a(n+1) = Sum_{k=1..n} (a(k)-th integer from among those positive integers which are coprime to (n+1-k)).
Original entry on oeis.org
1, 1, 2, 4, 9, 20, 47, 110, 260, 614, 1448, 3421, 8081, 19092, 45107, 106567, 251768, 594816, 1405285, 3320066, 7843851, 18531547, 43781846, 103437135, 244376187, 577352823, 1364029309, 3222597827, 7613573030, 17987504932, 42496516727, 100400469160
Offset: 1
The integers coprime to 5 are 1, 2, 3, 4, 6, ... The a(1)-th=1st of these is 1. The integers coprime to 4 are 1, 3, 5, ... The a(2)-th=1st of these is 1. The integers coprime to 3 are 1, 2, 4, 5, ... The a(3)-th=2nd of these is 2. The integers coprime to 2 are 1, 3, 5, 7, 9, ... The a(4)-th=4th of these is 7. And the integers coprime to 1 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... The a(5)-th=9th of these is 9. So a(6) = 1 + 1 + 2 + 7 + 9 = 20.
-
with(numtheory): fc:= proc(t,p) option remember; local m, j, h, pp; if p=1 then t else pp:= phi(p); m:= iquo(t,pp); j:= m*pp; h:= m*p-1; while jAlois P. Heinz, Aug 05 2009
-
fc[t_, p_] := fc[t, p] = Module[{m, j, h, pp}, If[p == 1, t, pp = EulerPhi[p]; m = Quotient[t, pp]; j = m pp; h = m p - 1; While[j < t, h++; If[GCD[p, h] == 1, j++]]; h]];
a[n_] := a[n] = If [n == 1, 1, Sum[fc[a[k], (n - k)], {k, 1, n - 1}]];
Array[a, 35] (* Jean-François Alcover, Nov 19 2020, after Alois P. Heinz *)
Showing 1-4 of 4 results.
Comments