A226779 -id:A226779 - cp's OEIS frontend

A257103 Composite numbers n such that n'=(n+4)', where n' is the arithmetic derivative of n.

21, 56, 1862, 2526, 1352797, 2201422, 3206062, 34844422, 42400318, 8586830293, 20967471193, 23194695022, 43790421673, 45041812729, 48438881254, 101060544853, 177839630854, 180939891343, 198419358598, 305550990673, 354694083622, 661663823662, 785220739279

Offset: 1

Paolo P. Lava, Apr 17 2015

nonn

If the limitation of being composite is removed we also have the lesser of cousin prime pairs (A023200).

a(45) > 5*10^13. - Hiroaki Yamanouchi, Aug 27 2015

			21' = (21 + 4)' = 25' = 10;
56' = (56 + 4)' = 60' = 92.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..44

Cf. A003415, A023200, A226779.

with(numtheory); P:= proc(q,h) local a,b,n,p;
for n from 1 to q do if not isprime(n) then a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=(n+h)*add(op(2,p)/op(1,p),p=ifactors(n+h)[2]);
if a=b then print(n); fi; fi; od; end: P(10^9,4);

a[n_] := If[Abs@n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; Select[Range@ 10000, And[CompositeQ@ #, a@ # == a[# + 4]] &] (* Michael De Vlieger, Apr 22 2015, after Michael Somos at A003415 *)

a(8)-a(10) from Lars Blomberg, May 06 2015

a(11)-a(23) from Hiroaki Yamanouchi, Aug 27 2015

A293252 Numbers k such that k = x + y, k' = x' + y' and k'' = x'' + y'', where k' and k'' are the first and second arithmetic derivatives of k.

Original entry on oeis.org

3, 778, 1331, 1575, 1589, 3111, 5368, 14060, 17649, 17714, 23232, 33813, 34353, 36234, 52936, 53391, 66375, 74544, 80938, 88945, 93475, 94905, 97470, 98434, 156816, 180804, 207754, 229502, 238830, 267120, 274065, 357318, 367921, 400500, 406700, 411872, 418037

Offset: 1

Paolo P. Lava, Oct 04 2017

nonn

A226779(n) + 1 are terms of the sequence: for these numbers the relation stands for any following derivative because n = 1 + (n-1), n' = 0 + (n-1)' and n' = (n-1)' by definition. Apart 3, no other prime p can be in the sequence because p = x + y implies p' = 1 = x' + y' that is impossible (for 3 we have 3 = 1 + 2 and 3' = 1 = 1' + 2' = 0 + 1). Similarly, x and y cannot be both primes.

Is there any number that admits two or more different partitions?

			1331 = 198 + 1133, 1331' = 363 = 198' + 1133' = 249 + 114, 1331'' = 187 = 198'' + 1133'' = 86 + 101.

Paolo P. Lava, List of k, x, and y

Cf. A003415, A226779.

with(numtheory): P:=proc(q) local a,b,c,k,n,p; for n from 1 to q do
for k from 1 to trunc(n/2) do a:=k*add(op(2,p)/op(1,p),p=ifactors(k)[2]);
b:=(n-k)*add(op(2,p)/op(1,p),p=ifactors(n-k)[2]); c:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); if c=a+b then a:=a*add(op(2,p)/op(1,p),p=ifactors(a)[2]); b:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]); c:=c*add(op(2,p)/op(1,p),p=ifactors(c)[2]);
if c=a+b then print(n); break; fi; fi; od; od; end: P(10^5);

f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Select[Range[2000], Function[k, Count[IntegerPartitions[k, {2}], ?(And[f@ k == f@ #1 + f@ #2, Nest[f, k, 2] == Nest[f, #1, 2] + Nest[f, #2, 2]] & @@ # &)] > 0]] (* _Michael De Vlieger, Oct 08 2017 *)

a(25)-a(37) from Giovanni Resta, Oct 05 2017

A257105 Composite numbers n such that n'=(n+8)', where n' is the arithmetic derivative of n.

Original entry on oeis.org

132, 476, 2108, 16748, 27548, 28676, 99524, 100076, 239948, 308228, 344129, 573476, 601676, 822908, 860276, 883268, 1673228, 3274010, 4959476, 7548956, 8916044, 9048428, 9215348, 9643169, 9833588, 10011908, 14773676, 17119436, 18529964, 19459028, 21335948, 21739148

Offset: 1

Paolo P. Lava, Apr 17 2015

nonn

If the limitation of being composite is removed we also have the numbers p such that if p is prime then p + 8 is prime too (A023202).

			132' = (132 + 8)' = 140' = 188;
476' = (476 + 8)' = 484' = 572.

Cf. A003415, A023202, A226779.

with(numtheory); P:= proc(q,h) local a,b,n,p;
for n from 1 to q do if not isprime(n) then a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=(n+h)*add(op(2,p)/op(1,p),p=ifactors(n+h)[2]);
if a=b then print(n); fi; fi; od; end: P(10^9,8);

a[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]];
Select[Range@ 100000, And[CompositeQ@ #, a@# == a[# + 8]] &] (* Michael De Vlieger, Apr 22 2015, after Michael Somos at A003415 *)

A257107 Composite numbers n such that n'=(n+12)', where n' is the arithmetic derivative of n.

Original entry on oeis.org

16, 65, 88, 209, 11009, 38009, 680609, 2205209, 2860198, 3515609, 4347209, 5365387, 5809361, 10595009, 12006209, 31979009, 83255059, 89019209, 152915402, 169130009, 172147423, 225869899, 244766009, 247590209, 258084209, 325622009, 357777209, 377330609

Offset: 1

Paolo P. Lava, Apr 17 2015

nonn

If the limitation of being composite is removed we also have the numbers p such that if p is prime then p + 12 is prime too (A046133).

			16' = (16 + 12)' = 28' = 32;
65' = (65 + 12)' = 77' = 18.

Cf. A003415, A046133, A226779.

with(numtheory); P:= proc(q,h) local a,b,n,p;
for n from 1 to q do if not isprime(n) then a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=(n+h)*add(op(2,p)/op(1,p),p=ifactors(n+h)[2]);
if a=b then print(n); fi; fi; od; end: P(10^9,12);

a[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]];
Select[Range@ 100000, And[CompositeQ@ #, a@# == a[# + 12]] &] (* Michael De Vlieger, Apr 22 2015, after Michael Somos at A003415 *)

a(16)-a(28) from Lars Blomberg, May 06 2015

cp's OEIS Frontend

A257103 Composite numbers n such that n'=(n+4)', where n' is the arithmetic derivative of n.

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A293252 Numbers k such that k = x + y, k' = x' + y' and k'' = x'' + y'', where k' and k'' are the first and second arithmetic derivatives of k.

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A257105 Composite numbers n such that n'=(n+8)', where n' is the arithmetic derivative of n.

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Author

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Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A257107 Composite numbers n such that n'=(n+12)', where n' is the arithmetic derivative of n.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions