A230543
Numbers n that form a Pythagorean quadruple with n', n'' and sqrt(n^2 + n'^2 + n''^2), where n' and n'' are the first and the second arithmetic derivative of n.
Original entry on oeis.org
512, 1203, 3456, 6336, 23328, 42768, 157464, 249753, 288684, 400000, 722718, 1062882, 1948617, 2700000, 4950000, 18225000, 33412500, 105413504, 123018750, 225534375, 312500000, 408918816
Offset: 1
If n = 6336 then n' = 23808, n'' = 103936 and sqrt(n^2 + n'^2 + n''^2) = 106816.
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with(numtheory): P:= proc(q) local a1, a2, n, p;
for n from 2 to q do a1:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
a2:=a1*add(op(2,p)/op(1,p),p=ifactors(a1)[2]);
if type(sqrt(n^2+a1^2+a2^2),integer) then print(n);
fi; od; end: P(10^10);
A249105
Numbers that form a Pythagorean 5-tuple with their first three arithmetic derivatives.
Original entry on oeis.org
4, 27, 1808, 3125, 12204, 12707, 82377, 269827, 823543, 1412500, 7089739, 9534375, 46873785, 78192979, 372241436
Offset: 1
First three arithmetic derivatives of 1808 are 3632, 7280, 17616 and sqrt(1808^2 + 3632^2 + 7280^2 + 17616^2) = 19488.
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with(numtheory);
Dr:=proc(w) local x,p; x:=w*add(op(2,p)/op(1,p),p=ifactors(w)[2]); end:
P:=proc(q,h) local a,b,k,n; for n from 2 to q do a:=n; b:=n^2;
for k from 1 to h do a:=Dr(a); b:=b+a^2; od; if type(sqrt(b),integer) then print(n);
fi; od; end: P(10^9,3);
A249106
Numbers that form a Pythagorean 6-tuple with their first four arithmetic derivatives.
Original entry on oeis.org
19164, 129357, 14971875, 45316123, 434325391
Offset: 1
First four arithmetic derivatives of 19164 are 25564, 31848, 58412, 61916 and sqrt(19164^2 + 25564^2 + 31848^2 + 58412^2 + 61916^2) = 96336.
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with(numtheory);
Dr:=proc(w) local x,p; x:=w*add(op(2,p)/op(1,p),p=ifactors(w)[2]); end:
P:=proc(q,h) local a,b,k,n; for n from 2 to q do a:=n; b:=n^2;
for k from 1 to h do a:=Dr(a); b:=b+a^2; od; if type(sqrt(b),integer) then print(n);
fi; od; end: P(10^9,4);
A249107
Numbers that form a Pythagorean 7-tuple with their first five arithmetic derivatives.
Original entry on oeis.org
4031, 10823, 416959, 496939, 1354980, 9146115, 38949392, 44472866, 262908396, 380264131
Offset: 1
First five arithmetic derivatives of 4031 are 168, 332, 336, 832, 2560 and sqrt(4031^2 + 168^2 + 332^2 + 336^2 + 832^2 + 2560^2) = 4873.
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with(numtheory);
Dr:=proc(w) local x,p; x:=w*add(op(2,p)/op(1,p),p=ifactors(w)[2]); end:
P:=proc(q,h) local a,b,k,n; for n from 2 to q do a:=n; b:=n^2;
for k from 1 to h do a:=Dr(a); b:=b+a^2; od; if type(sqrt(b),integer) then print(n);
fi; od; end: P(10^9,5);
A249110
Numbers that form a Pythagorean 10-tuple with their first eight arithmetic derivatives.
Original entry on oeis.org
4, 27, 3125, 398747, 823543
Offset: 1
First eight arithmetic derivatives of 398747 are 1692, 2856, 5812, 5816, 8732, 9116, 9500, 15700 and sqrt(398747^2 + 1692^2 + 2856^2 + 5812^2 + 5816^2 + 8732^2 + 9116^2 + 9500^2 + 15700^2) = 399467.
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with(numtheory);
Dr:=proc(w) local x,p; x:=w*add(op(2,p)/op(1,p),p=ifactors(w)[2]); end:
P:=proc(q,h) local a,b,k,n; for n from 2 to q do a:=n; b:=n^2;
for k from 1 to h do a:=Dr(a); b:=b+a^2; od; if type(sqrt(b),integer) then print(n);
fi; od; end: P(10^9,8);
A211176
Numbers n which are the hypotenuse of a Pythagorean triple with n' as a leg, where n' is the arithmetic derivative of n.
Original entry on oeis.org
125, 625, 23125, 142805, 210125, 371293, 7983625, 9370805, 25757525, 50062025, 120670225, 489766225, 881052625, 1471596725, 2307267625, 2489771125, 3145529225, 3474871553, 6975757441, 7977558641
Offset: 1
n = 23125, n' = 19125 and sqrt(n^2-n'^2) = 13000.
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with(numtheory); ListA211176:= proc(q)local a,n,p;
for n from 2 to q do a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
if n<>a and type(sqrt(n^2-a^2),integer) then print(n); fi;
od; end: ListA211176(10^9);
Showing 1-6 of 6 results.
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