A070562 -id:A070562 - cp's OEIS frontend

A070061 Least number of fecundity n (A070562).

0, 5, 25, 19, 23, 18, 9, 7, 4, 2, 1, 282, 1529, 1586, 1397, 898, 658, 538, 477, 529, 736, 586, 397, 366, 294, 246, 243, 187, 3237326, 3677393, 3586673, 3553787, 3515987, 22572473, 518376965, 516675965, 516963965, 41883474553, 41881554553, 41863638649, 35632297938395

Offset: 0

Lekraj Beedassy, May 06 2002

a(41) > 2.75*10^14. - Giovanni Resta, Jun 04 2013

			a(9)=2 since we have the 9-step chain 2 -> 4 -> 8 -> 16 -> 22 -> 26 -> 38 -> 62 -> 74 -> 102.

P. Tougne, Jeux Mathematiques column, Pour La Science (French edition of "Scientific American"), Vol. 82, Aug. 1984, Prob. 6, pp. 101, 104.

Cf. A070562, A070257, A003001.

f[n_] := Length@ FixedPointList[ # + Times @@ IntegerDigits@# &, n] - 2; t = Table[0, {50}]; k = 1; While[k < 2300000001, a = f@k; If[ t[[a + 1]] == 0, t[[a + 1]] = k; Print[{k, a}]]; k++ ]; t (* Robert G. Wilson v, Jun 27 2010 *)

Corrected and extended by Jason Earls, May 26 2002
a(34)-a(36) from Robert G. Wilson v, Jun 27 2010
a(37)-a(40) from Giovanni Resta, Jun 04 2013

A337789 Numbers k such that trajectory of k under repeated calculation of fecundity (x -> A070562(x)) eventually reaches 0.

Original entry on oeis.org

0, 1, 5, 10, 15, 18, 20, 21, 22, 24, 27, 30, 35, 40, 42, 44, 46, 48, 50, 51, 55, 59, 60, 63, 64, 66, 67, 69, 70, 74, 75, 77, 80, 83, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 115, 118, 120, 121, 122, 124, 127

Offset: 1

Robert Bilinski, Sep 21 2020

			5 is a term in the sequence because the fecundity of 5 is 1, the fecundity of 1 is 10 and the fecundity of 10 is 0.
7 is not a term in the sequence because the fecundity of 7 is 7 and therefore the fecundity will never become 0.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A070562, A070061, A070257.

fec:= proc(n) local k, x,t;
  x:= n;
  for k from 0 do
    t:= convert(convert(x,base,10),`*`);
    if t = 0 then return k fi;
    x:= x+t
  od
end proc:
filter:= proc(n) local v; option remember;
    v:= fec(n);
    if v = 0 then true
    elif v = n then false
    else procname(v)
    fi
end proc:
select(filter, [$0..1000]); # Robert Israel, Apr 12 2021

fec[n_] := Length @ FixedPointList[# + Times @@ IntegerDigits[#] &, n] - 2; Select[Range[0, 100], FixedPoint[fec, #] == 0 &] (* Amiram Eldar, Sep 22 2020 *)

from math import prod
from functools import lru_cache
def pd(n): return prod(map(int, str(n)))
def A070562(n):
  s = 0
  while pd(n) != 0: n, s = n + pd(n), s + 1
  return s
@lru_cache(maxsize=None)
def ok(n):
  fn = A070562(n)
  if fn == 0: return True
  if fn == n: return False
  return ok(fn)
print(list(filter(ok, range(128)))) # Michael S. Branicky, Apr 12 2021

More terms from Amiram Eldar, Sep 22 2020

Offset changed by Robert Israel, Apr 12 2021

A070257 Fecundity of n sets a new record.

Original entry on oeis.org

1, 187, 3237326, 3515987, 22572473, 516675965, 516963965, 41863638649, 35632297938395

Offset: 1

Jason Earls, May 09 2002

a(9) > 10^11. - Donovan Johnson, Jun 23 2011

a(10) > 2.75*10^14. - Giovanni Resta, Jun 04 2013

Cf. A070562, A070061.

a(5)-a(7) from Donovan Johnson, Jul 29 2009
a(8) from Donovan Johnson, Jun 23 2011
a(9) from Giovanni Resta, Jun 04 2013

A070560 a(0) = 1; for n > 0, a(n) = (fecundity of n) + 2.

Original entry on oeis.org

1, 12, 11, 11, 10, 3, 10, 9, 9, 8, 2, 10, 9, 9, 8, 3, 8, 8, 7, 5, 2, 7, 7, 6, 7, 4, 6, 7, 4, 5, 2, 5, 6, 4, 4, 3, 5, 5, 5, 4, 2, 6, 3, 4, 3, 5, 3, 4, 3, 6, 2, 7, 5, 10, 4, 3, 6, 4, 4, 3, 2, 4, 4, 7, 7, 4, 3, 3, 9, 7, 2, 6, 6, 4, 3, 3, 8, 7, 5, 4, 2, 6, 4, 3, 9, 5, 5, 5, 6, 5, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2

Offset: 0

N. J. A. Sloane, May 07 2002

Start with n, repeatedly replace x by x + product of digits of x until reach 0; fecundity = number of steps - 1.

			1 -> 2 -> 4 -> 8 -> 16 -> 22 -> 26 -> 38 -> 62 -> 74 -> 102 -> 0 has fecundity 10.

Cf. A070561, A070562, A031346, A003001.

f[n_] := Length@ FixedPointList[ # + Times @@ IntegerDigits@# &, n]; f[0] = 1; Array[f, 105, 0] (* Robert G. Wilson v, Jun 27 2010 *)

More terms from Robert G. Wilson v, Jun 27 2010

A070561 a(0) = 0; for n > 0, a(n) = (fecundity of n) + 1.

Original entry on oeis.org

0, 11, 10, 10, 9, 2, 9, 8, 8, 7, 1, 9, 8, 8, 7, 2, 7, 7, 6, 4, 1, 6, 6, 5, 6, 3, 5, 6, 3, 4, 1, 4, 5, 3, 3, 2, 4, 4, 4, 3, 1, 5, 2, 3, 2, 4, 2, 3, 2, 5, 1, 6, 4, 9, 3, 2, 5, 3, 3, 2, 1, 3, 3, 6, 6, 3, 2, 2, 8, 6, 1, 5, 5, 3, 2, 2, 7, 6, 4, 3, 1, 5, 3, 2, 8, 4, 4, 4, 5, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, May 07 2002

Start with n, repeatedly replace x by x + product of digits of x until the product of digits reaches 0; fecundity = number of steps - 1.

Equivalently, with A230099 = f, a(n) is the number k of distinct values that are obtained with iterations: n, f(n), f(f(n)), f(f(f(n))), ... until a term of this sequence contains a 0. - Bernard Schott, Jul 31 2023

			1 -> 2 -> 4 -> 8 -> 16 -> 22 -> 26 -> 38 -> 62 -> 74 ->102 -> 102 -> ... has fecundity 10.

Diophante, A306, Les nombres féconds (in French).

Cf. A011540, A070560, A070562, A031346, A003001, A230099.

f[n_] := Length@FixedPointList[ # + Times @@ IntegerDigits@# &, n] - 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Jun 27 2010 *)

a(n) = 1 iff n positive is in A011540. - Bernard Schott, Jul 31 2023

More terms from Robert G. Wilson v, Jun 27 2010

A145280 Fecundity of n-th prime.

Original entry on oeis.org

9, 9, 1, 7, 8, 7, 6, 3, 4, 3, 3, 3, 4, 2, 2, 8, 1, 2, 1, 4, 2, 2, 1, 3, 1, 0, 0, 0, 0, 7, 5, 3, 3, 2, 8, 6, 2, 6, 1, 2, 4, 5, 1, 1, 1, 1, 5, 5, 2, 4, 5, 9, 9, 4, 7, 7, 9, 5, 2, 6, 2, 8, 0, 11, 3, 2, 1, 1, 7, 4, 8, 7, 2, 2, 2, 3, 1, 22, 0, 0, 3, 2, 6, 4, 3, 5, 3, 3, 5, 2, 4, 5, 4, 4, 4, 0, 0, 5, 4, 6, 2, 3, 7, 2, 1

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Oct 06 2008

			7 -> 7+7=14 -> 14+1*4=18 -> 18+1*8=26 -> 26+2*6=38 -> 38+3*8=62 -> 62+6*2=74 -> 74+7*4=102 -> 7 steps to reach a zero digit.

Cf. A070562, A145279.

P:=proc(i) local a,b,c,ok,k,w,n; for n from 1 by 1 to i do a:=ithprime(n); b:=1; c:=0; ok:=1; while ok=1 do k:=a; w:=1; while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w=0 then ok:=0; else c:=c+1; a:=a+w; fi; od; print(c); od; end: P(100);

f[n_] := Length@ FixedPointList[ # + Times @@ IntegerDigits@# &, n] - 2; Array[f@Prime@# &, 105] (* Robert G. Wilson v, Jun 27 2010 *)

a(n) = A070562(prime(n)). - Michel Marcus, Aug 01 2015

A145279 Fecundity of n-th Fibonacci number.

Original entry on oeis.org

0, 10, 10, 9, 9, 1, 7, 7, 5, 2, 1, 3, 1, 5, 8, 0, 5, 2, 1, 3, 1, 0, 1, 1, 7, 0, 2, 3, 3, 5, 0, 1, 0, 5, 0, 1, 0, 5, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 1, 3, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Oct 06 2008

Subset of A070562. After the 184th Fibonacci number 127127879743834334146972278486287885163, the fecundity is equal to zero.

The indices of Fibonacci numbers whose fecundity is not zero are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 31, 33, 35, 37, 39, 42, 43, 53, 54, 55, 56, 57, 58, 78, 80, 85, 87, 97, 125, 184}. - Robert G. Wilson v, Jun 27 2010

			Fib(6)=8 -> 8+8=16 -> 16+1*6=22 -> 22+2*2=26 -> 26+2*6=38 -> 38+3*8=62 -> 62+6*2=74 -> 74+7*4=102 -> 7 steps to reach a zero digit.

Cf. A070562, A145280.

P:=proc(i) local a,b,c,d,f,g,ok,k,w,n; d:=0; f:=1; print(d); print(10); for n from 0 by 1 to i do a:=d+f; g:=f; f:=a; d:=g; b:=1; c:=0; ok:=1; while ok=1 do k:=a; w:=1; while k>0 do w:=w*(k-(trunc(k/10)*10)); k:=trunc(k/10); od; if w=0 then ok:=0; else c:=c+1; a:=a+w; fi; od; print(c); od; end: P(200);

f[n_] := Length@ FixedPointList[ # + Times @@ IntegerDigits@# &, n] - 2; Array[f@ Fibonacci@# &, 105, 0] (* Robert G. Wilson v, Jun 27 2010 *)

cp's OEIS Frontend

A070061 Least number of fecundity n (A070562).

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A337789 Numbers k such that trajectory of k under repeated calculation of fecundity (x -> A070562(x)) eventually reaches 0.

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A070257 Fecundity of n sets a new record.

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A070560 a(0) = 1; for n > 0, a(n) = (fecundity of n) + 2.

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A070561 a(0) = 0; for n > 0, a(n) = (fecundity of n) + 1.

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A145280 Fecundity of n-th prime.

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A145279 Fecundity of n-th Fibonacci number.

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