Dann Toliver - cp's OEIS frontend

A029910 Start with n; if prime, stop; repeatedly sum prime factors (counted with multiplicity) and add 1, until reach 1, 6 or a prime.

1, 2, 3, 5, 5, 6, 7, 7, 7, 7, 11, 7, 13, 7, 7, 7, 17, 7, 19, 7, 11, 7, 23, 7, 11, 7, 7, 7, 29, 11, 31, 11, 7, 7, 13, 11, 37, 7, 17, 7, 41, 13, 43, 7, 7, 7, 47, 7, 7, 13, 11, 7, 53, 7, 17, 7, 23, 11, 59, 13, 61, 7, 7, 13, 19, 17, 67, 7, 7, 7, 71, 13, 73, 7, 7, 7, 19, 19, 79, 7

Offset: 1

Dann Toliver

nonn

			20 -> 2+2+5+1 = 10 -> 2+5+1 = 8 -> 2+2+2+1 = 7 so a(20)=7.

A029911 Start with n; if prime, skip; repeatedly sum prime factors (counted with multiplicity) and add 1, until reach 1, 6 or a prime.

Original entry on oeis.org

1, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 7, 7, 11, 7, 7, 7, 11, 11, 7, 7, 13, 11, 7, 17, 7, 13, 7, 7, 7, 7, 7, 13, 11, 7, 7, 17, 7, 23, 11, 13, 7, 7, 13, 19, 17, 7, 7, 7, 13, 7, 7, 7, 19, 19, 7, 13, 7, 7, 23, 7, 7, 7, 7, 11, 7, 13, 13, 11, 7, 17, 7, 7, 23, 7, 7, 7, 7, 19, 41, 7

Offset: 1

Dann Toliver

nonn

			20 -> 2+2+5+1 = 10 -> 2+5+1 = 8 -> 2+2+2+1 = 7.

Cf. A029910.

A036430 Number of iterations needed to reach 1 under the map n -> Omega(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 3, 2

Offset: 1

Dann Toliver

			16 -> 4 -> 2 -> 1 and thus a(16) = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A029908, A073855.

a(n)=my(s);while(n>1,n=bigomega(n);s++);s \\ Charles R Greathouse IV, Apr 25 2012

a(n) = a(bigomega(n)) + 1 for n > 1. - Vladeta Jovovic, Jul 10 2004

a(n) = O(log* n). - Charles R Greathouse IV, Apr 25 2012

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003

Formula corrected by Charles R Greathouse IV, Apr 25 2012

A042938 Reverse or triple: if reverse(a(n)) > a(n), a(n+1) = reverse(a(n)), else a(n+1) = 3*a(n).

Original entry on oeis.org

1, 3, 9, 27, 72, 216, 612, 1836, 6381, 19143, 34191, 102573, 375201, 1125603, 3065211, 9195633, 27586899, 99868572, 299605716, 617506992, 1852520976, 6790252581, 20370757743, 34775707302, 104327121906, 609121723401, 1827365170203, 3020715637281, 9062146911843

Offset: 1

Dann Toliver

			Starting with 1, the reverse is 1 and not greater than 1 so the next term is 3. The reverse of 3 is also not greater than 3 and so the next term is 9. The reverse of 9 is not greater than 9 and so the next term is 27. However, the reverse of 27 is 72 which is greater and so the next term in the sequence is 72 and so on.

Robert Israel, Table of n, a(n) for n = 1..1951

reverse:= proc(n)
local L,i,m;
L:= convert(n,base,10);
  m:= nops(L);
add(L[i]*10^(m-i),i=1..m);
end proc:
a[1]:= 1:
for n from 2 to 100 do
r:= reverse(a[n-1]);
if r > a[n-1] then a[n]:= r
else a[n]:= 3*a[n-1]
fi
od:
seq(a[i],i=1..100); # Robert Israel, Jun 24 2015

rd[n_]:=Module[{rev=FromDigits[Reverse[IntegerDigits[n]]]}, If[rev>n, rev, 3 n]]; NestList[rd, 1, 30] (* Vincenzo Librandi, Jun 24 2015 *)
NestList[If[IntegerReverse[#]>#,IntegerReverse[#],3#]&,1,30] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 09 2018 *)

lista(nn) = {print1(a=1, ", "); for (n=2, nn, r = eval(concat(Vecrev(Str(a)))); if (r > a, a = r, a *= 3); print1(a, ", "););} \\ Michel Marcus, Jan 31 2016

Corrected by Robert Israel, Jun 24 2015

A029913 Start with n; if prime, stop; repeatedly sum squares of prime factors (counted with multiplicity), until reach 16 or a prime; set a(n) = 0 if no limit exists.

Original entry on oeis.org

2, 3, 17, 5, 13, 7, 17, 59, 29, 11, 17, 13, 53, 293, 16, 17, 59, 19

Offset: 2

Dann Toliver

			9 -> 3^2 + 3^2 = 18 -> 4 + 9 + 9 = 22 -> 4 + 121 = 125 -> 25 + 25 + 25 = 75 -> 9 + 25 + 25 = 59, so a(9) = 59.

Cf. A029914.

More terms from Michel ten Voorde, Apr 12 2001

Incorrect extension reverted by Sean A. Irvine, Mar 08 2020

A029914 Start with n; repeatedly sum squares of prime factors (counted with multiplicity), until reach a prime p, then set a(n) = p; if reach a fixed point q, set a(n) = q; set a(n) = 0 if no limit exists.

Original entry on oeis.org

17, 59, 17, 31, 13

Offset: 2

Dann Toliver

			2 -> 2^2 = 4 -> 2^2 + 2^2 = 8 -> 2^2 + 2^2 + 2^12 = 12 -> 2^2 + 2^2 + 3^2 = 17, prime, so stop; a(2)=17.

Cf. A029913.

More terms from Michel ten Voorde, Apr 12 2001

Incorrect extension reverted by Sean A. Irvine, Mar 08 2020

A041012 Concatenate the next a(n) integers to get the n+1 term.

Original entry on oeis.org

1, 2, 34, 35363738394041424344454647484950515253545556575859606162636465666768

Offset: 0

Dann Toliver

NestList[FromDigits[Flatten[IntegerDigits/@Range[#+1,2#]]]&,1,3] (* Harvey P. Dale, Aug 13 2022 *)

a(n+1) = (a(n)+1).(a(n)+2). ... .(a(n)+a(n))

A029912 Start with n; repeatedly sum prime factors (counted with multiplicity) and add 1, until reach 1, 6 or a prime.

Original entry on oeis.org

1, 3, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 7, 7, 7, 11, 7, 7, 7, 11, 11, 11, 11, 7, 7, 13, 11, 7, 7, 17, 7, 13, 13, 7, 7, 7, 7, 7, 7, 7, 13, 11, 7, 7, 7, 17, 7, 23, 11, 13, 13, 7, 7, 7, 13, 19, 17, 7, 7, 7, 7, 13, 13, 7, 7, 7, 7, 19, 19, 7, 7, 13, 7, 7, 7, 23, 7

Offset: 1

Dann Toliver

nonn

If p is in A023200 then a(3*p) = p+4. It appears that all n > 35 such that a(n) > n/3 are 3*p for p in A023200. - Robert Israel, Dec 18 2019

			20 -> 2+2+5+1 = 10 -> 2+5+1 = 8 -> 2+2+2+1 = 7 so a(20)=7.

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

f:= proc(n) option remember;
  local v;
  v:= add(t[1]*t[2],t=ifactors(n)[2])+1;
  if v = 1 or v = 6 or isprime(v) then return v fi;
  procname(v)
end proc:
map(f, [$1..100]); # Robert Israel, Dec 18 2019

a[n_] := a[n] = If[n==1, 1, Module[{v}, v = Sum[t[[1]]*t[[2]], {t, FactorInteger[n]}]+1; If[v==1 || v==6 || PrimeQ[v], Return[v]]; a[v]]];
a /@ Range[100] (* Jean-François Alcover, Aug 21 2022, after Robert Israel *)

A041013 Reverse or double: if reverse of a(n) > a(n), then a(n+1) = a(n) reversed, otherwise a(n+1) = 2*a(n).

Original entry on oeis.org

1, 2, 4, 8, 16, 61, 122, 221, 442, 884, 1768, 8671, 17342, 24371, 48742, 97484, 194968, 869491, 1738982, 2898371, 5796742, 11593484, 48439511, 96879022, 193758044, 440857391, 881714782, 1763429564, 4659243671, 9318487342, 18636974684, 48647963681

Offset: 0

Dann Toliver

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A004086.

a041013 n = a041013_list !! n
a041013_list = 1 : f 1 where
   f x | rev <= x  = (2*x) : f (2*x)
       | otherwise = rev : f rev where rev = a004086 x
-- Reinhard Zumkeller, Aug 08 2011

rd[n_]:=Module[{rev=FromDigits[Reverse[IntegerDigits[n]]]},If[ rev>n, rev, 2n]]; NestList[rd,1,40] (* Harvey P. Dale, Jan 25 2013 *)
NestList[If[IntegerReverse[#]>#,IntegerReverse[#],2#]&,1,40] (* Harvey P. Dale, Aug 23 2023 *)

Typo in definition corrected by K. Viswanathan Iyer, Mar 23 2010

A030455 Numbers having the same number of digits as letters in their US English spelling.

Original entry on oeis.org

1000000000, 2000000000, 6000000000, 3000000000000, 7000000000000, 8000000000000, 10000000000001, 10000000000002, 10000000000006, 10000000000010, 11000000000000, 12000000000000, 20000000000000, 30000000000000, 80000000000000, 90000000000000, 3000000000000000

Offset: 1

Dann Toliver, 1999

Or, numbers N such that A005589(N)=A055642(N).

			"One billion" has 10 letters and "1000000000" has 10 digits.
a(7)=10^13+1 (ten trillion one) has 14 digits and also 14 letters in the US English spelling (while the preferred British spelling is "...and one"). The same applies to a(8)=10^13+2, a(9)=10^13+6, a(10)=10^13+10, a(11)=11*10^12, a(12)=12*10^12, a(13)=20*10^12, a(14)=30*10^12, a(15)=80*10^12, a(16)=90*10^12. - _M. F. Hasler_, Feb 13 2012

L. C. Noll, The English name of a number.
Wiktionary, One hundred one.

See A204593 for the French version.

{N=1; while(1, while(0>d=#Str(N*=10)-A005589(N),);
d | for(k=1,3,print1(k!*N", ")) | next;  for(k=1,90, for(u=0,90, A005589(k*N+u)==#Str(k*N) & print1(k*N+u","))))}

Corrected by M. F. Hasler, Feb 13 2012

cp's OEIS Frontend

User: Dann Toliver

A029910 Start with n; if prime, stop; repeatedly sum prime factors (counted with multiplicity) and add 1, until reach 1, 6 or a prime.

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A029911 Start with n; if prime, skip; repeatedly sum prime factors (counted with multiplicity) and add 1, until reach 1, 6 or a prime.

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A036430 Number of iterations needed to reach 1 under the map n -> Omega(n).

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A042938 Reverse or triple: if reverse(a(n)) > a(n), a(n+1) = reverse(a(n)), else a(n+1) = 3*a(n).

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A029913 Start with n; if prime, stop; repeatedly sum squares of prime factors (counted with multiplicity), until reach 16 or a prime; set a(n) = 0 if no limit exists.

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A029914 Start with n; repeatedly sum squares of prime factors (counted with multiplicity), until reach a prime p, then set a(n) = p; if reach a fixed point q, set a(n) = q; set a(n) = 0 if no limit exists.

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A041012 Concatenate the next a(n) integers to get the n+1 term.

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Mathematica

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A029912 Start with n; repeatedly sum prime factors (counted with multiplicity) and add 1, until reach 1, 6 or a prime.

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Maple

Mathematica

A041013 Reverse or double: if reverse of a(n) > a(n), then a(n+1) = a(n) reversed, otherwise a(n+1) = 2*a(n).

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Haskell

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A030455 Numbers having the same number of digits as letters in their US English spelling.

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