A007629 -id:A007629 - cp's OEIS frontend

A269310 Consider a number x > 1. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach the Euler totient function of x.

22, 44, 55, 88, 98, 136, 162, 166, 241, 462, 1020, 2040, 2416, 2899, 3060, 4080, 5110, 7942, 10738, 10996, 15006, 24822, 57040, 67054, 70625, 75588, 96888, 261524, 301834, 507471, 735840, 816584, 2893877, 6081064, 8155616, 16513570, 18772258, 40833543

Offset: 1

Paolo P. Lava, Feb 24 2016

			phi(22) = 10: 2 + 2 = 4; 2 + 4 = 6; 4 + 6 = 10.

Lars Blomberg, Table of n, a(n) for n = 1..47

Cf. A000010, A007629, A269307, A269308, A269309, A269311, A269312.

with(numtheory): P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
for n from 2 to q do a:=n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]

Select[Range[2,10^5], (t = EulerPhi[#]; d = IntegerDigits[#]; While[Total[d] < t, d = Join[Rest[d], {Total[d]}]]; Total[d] == t) &] (* Robert Price, May 17 2019 *)

a(38) from Lars Blomberg, Jan 18 2018

A269311 Consider the arithmetic derivative of a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach x.

Original entry on oeis.org

8, 18, 21, 160, 1590, 2420, 18620, 69229, 83790, 279964, 494520, 598810, 676450, 1183147, 4233720, 5600348, 14217074, 20025836, 64278677, 425208387, 604048830, 750851470, 1981942354

Offset: 1

Paolo P. Lava, Feb 24 2016

			8’ = 12 : 1 + 2 = 3; 2 + 3 = 5; 3 + 5 = 8.
18’ = 21 : 2 + 1 = 3; 1 + 3 = 4; 3 + 4 = 7; 4 + 7 = 11; 7 + 11 = 18.

Cf. A003415, A007629, A269307, A269308, A269309, A269310, A269312.

with(numtheory): P:=proc(q,h) local a,b,k,p,t,v; global n; v:=array(1..h);
for n from 1 to q do a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]

dn[n_] := If[Abs@n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@n]]]; (* from Michael Somos, Apr 12 2011 *)
Select[Range[10^5], dn[#] >= 10 && (d = IntegerDigits[dn[#]]; While[Total[d] < #, d = Join[Rest[d], {Total[d]}]]; Total[d] == #) &] (* Robert Price, May 22 2019 *)

a(19)-a(23) from Lars Blomberg, Jan 18 2018

A269308 Consider a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach the sum of the divisors of x.

Original entry on oeis.org

20, 25, 43, 44, 49, 59, 122, 206, 2485, 11899, 17608, 24141, 56207, 195236, 2424613, 2842925, 6241233, 59087970, 111205290, 124735931, 224269761, 1086241193

Offset: 1

Paolo P. Lava, Feb 24 2016

44 works in both directions: n -> sigma(n) and sigma(n) -> n. See A269307.

			sigma(20) = 42 :  2 + 0 = 2; 0 + 2 = 2; 2 + 2 = 4; 2 + 4 = 6; 4 + 6 = 10; 6 + 10 = 16; 10 + 16 = 26; 16 +26 = 42.

Cf. A000203, A007629, A269307, A269309, A269310, A269311, A269312.

with(numtheory): P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
for n from 2 to q do a:=n; b:=ilog10(a)+1; if b>1 then
for k from 1 to b do v[b-k+1]:=(a mod 10); a:=trunc(a/10); od; t:=b+1; v[t]:=add(v[k], k=1..b);
while v[t]

Select[Range[10,10^5], (s = DivisorSigma[1, #]; d = IntegerDigits[#]; While[Total[d] < s, d = Join[Rest[d], {Total[d]}]]; Total[d] == s) &] (* Robert Price, May 21 2019 *)

a(16)-a(22) from Lars Blomberg, Jan 18 2018

A130792 Numbers k whose representation can be split in two parts which can be used as seeds for a Fibonacci-like sequence containing k itself.

Original entry on oeis.org

14, 19, 28, 47, 61, 75, 122, 149, 183, 199, 244, 298, 305, 323, 366, 427, 488, 497, 549, 646, 795, 911, 969, 1292, 1301, 1499, 1822, 1999, 2087, 2602, 2733, 2998, 3089, 3248, 3379, 3644, 3903, 4555, 4997, 5204, 5466, 6178, 6377, 6496, 6505, 7288, 7806, 7995

Offset: 1

Giovanni Resta, Aug 20 2007

The 6 terms with two digits are also Keith numbers. There are 233 numbers below 10^6 in this sequence.
Contribution from Paolo P. Lava, Apr 18 2025: (Start)
If the number k is rewritten as the concatenation of a and b, the problem is to find an integer x such that k = a*F(x) + b*F(x+1), where F(x) is a Fibonacci number (see in Links file with values of k, a, b, x, for k<10^6).
All the listed numbers admit only one concatenation that, through the addition process, leads to themselves. Is there any number that admits more than one single concatenation?
Sequence is infinite. Let us consider the numbers 19, 199, 1999, 19...9 and let us divide them as (1, 9), (1, 99), (1, 999), (1, 9...9). In two steps we have the initial numbers back: 1 + 9 = 10 and 9 + 10 = 19; 1 + 99 = 100 and 99 + 100 = 199, etc. (End)

			122 can be split into 12 and 2 and the Fibonacci-like sequence: 12, 2, 14, 16, 30, 46, 76, 122, ... contains 122 itself.

Michel Marcus, Table of n, a(n) for n = 1..406 (first 200 terms from Paolo Lava)
Paolo P. Lava, List of k, a, b, x

Cf. A007629.

testQ[n_]:= Block[{x, y, z, p = 10, r = False}, While[p < n, x = Floor[n/p]; y = Mod[n, p]; While[y < n, z = x + y; x = y; y = z]; If[y == n, r = True; Break[]]; p *= 10]; r]; Select[Range[10^4],testQ]

isok(n) = {nb = #Str(n); for (i=1, nb-1, x = n\10^i; y = n - 10^i*x; ok = 0; while(!ok, z = x + y; if (z > n, ok = 1); if (z == n, return (1)); x = y; y = z;));} \\ Michel Marcus, Oct 08 2014

A248134 Consider a number x as a concatenation of two integers, a and b: x = concat(a,b). Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Original entry on oeis.org

14, 19, 21, 28, 42, 47, 63, 84, 105, 126, 147, 149, 168, 189, 199, 298, 323, 497, 646, 795, 911, 969, 1292, 1499, 1822, 1999, 2087, 2733, 2998, 3089, 3248, 3379, 3644, 4555, 4997, 5411, 5466, 6178, 6377, 6496, 7288, 7995, 8199, 9161, 9267, 9744, 10822, 12356

Offset: 1

Paolo P. Lava, Oct 02 2014

If the number x is rewritten as concat(a,b), the problem is to find a value of y such that x = a*F(y) + b*F(y+1), if a < b, or x = b*F(y) + a*F(y+1), if a > b, where F(y) is a Fibonacci number (see values of x, a, b, y, for 1

Similar to A130792 but here the minimum number is deleted since the beginning.

All the listed numbers admit only one concatenation, concat(a,b), that, through the addition process, leads to themselves. Is there any number that admit more than one single concatenation?

Sequence is infinite. Let us consider the numbers 19, 199, 1999, 19...9 and let us divide them as concat(1,9), concat(1,99), concat(1,999), concat(1,9...9). In two steps we have the initial numbers back: 1 + 9 = 10 and 9 + 10 = 19; 1 + 99 = 100 and 99 + 100 = 199, etc.

			Let us rewrite 5411 as 54 U 11. Then:
11 + 54 = 65;
54 + 65 =  119;
65 + 119 = 184;
119 + 184 = 303;
184 + 303 = 487;
303 + 487 = 790;
487 + 790 = 1277;
790 + 1277 = 2067;
1277 + 2067 = 3344;
2067 + 3344 = 5411, that is 11*F(10) + 54*F(11) = 11*55 + 54*89 = 605 + 4806 = 5411.

Paolo P. Lava, Table of n, a(n) for n = 1..410
Paolo P. Lava, List of [x,a,b,y] in the equation x = a*F(y) + b*F(y+1), for 1< x <10^6

Cf. A000045, A007629, A130792, A246544, A247012, A247013.

P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
for n from 1 to q do for k from 1 to ilog10(n) do
a:=n mod 10^k; b:=trunc(n/10^k); if a

A097060 Revrepfigits (reverse replicating Fibonacci-like digits): Numbers k whose reversal occurs in a sequence generated by starting with the k digits of a number and then continuing the sequence with a number that is the sum of the previous k terms.

Original entry on oeis.org

12, 24, 36, 48, 52, 71, 341, 682, 1285, 5532, 8166, 17593, 28421, 74733, 90711, 759664, 901921, 1593583, 4808691, 6615651, 6738984, 8366363, 8422611, 26435142, 54734431, 57133931, 79112422, 89681171, 351247542, 428899438, 489044741, 578989902

Offset: 1

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Author

Jason Earls, Sep 15 2004

Keywords

base
nonn

Comments

Numbers ending in zero are not permitted since the zeros are dropped upon reversal. However, terms with internal zeros such as 90711 are permitted. Conjectures: 1. Sequence is infinite. 2. Revrepfigits are more rare than repfigits.

There are no 12-digit revrepfigits.

Examples

8166 is in the sequence since the sequence 8,1,6,6,21,34,67,128,250, 479,924,1781,3434,6618,..., contains the reversal of 8166.

References

J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 11-13. ASIN: B002ACVZ6O [From Jason Earls, Nov 21 2009]

Links

Bernardo Boncompagni and Anton Vrba, Table of n, a(n) for n = 1..59

Carlos Rivera, Primepuzzles.net Puzzle 384

Eric Weisstein's World of Mathematics, Keith Number

Crossrefs

Cf. A007629.

Cf. A128546 (reverse of these numbers).

Programs

Mathematica

rKeithQ[n_Integer] := Module[{b = IntegerDigits[n], r, s, k = 0}, If[Mod[n, 10] == 0, False, r = FromDigits[Reverse[b]]; s = Total[b]; While[s < r, AppendTo[b, s]; k++; s = 2*s - b[[k]]]; s == r]]; Select[Range[10, 100000], rKeithQ] (* T. D. Noe, Mar 15 2011 *)

Extensions

More terms from Bernardo Boncompagni and Anton Vrba (antonvrba(AT)yahoo.com), Jan 05 2007

A098598 Number of primes in sequences formed from the t digits of n where the latter terms are given by rule b(i)=sum of t previous terms; primes are counted from initial t digits up to the largest term < n^2.

Original entry on oeis.org

5, 5, 5, 5, 5, 5, 4, 3, 4, 2, 3, 7, 2, 6, 1, 6, 1, 5, 1, 4, 3, 6, 7, 2, 7, 5, 1, 5, 5, 1, 0, 6, 1, 5, 0, 6, 0, 7, 0, 5, 3, 6, 7, 6, 6, 2, 6, 7, 5, 6, 0, 5, 1, 1, 0, 6, 0, 5, 0, 0, 3, 5, 6, 5, 5, 5, 6, 2, 5, 5, 0, 3, 1, 4, 0, 4, 0, 5, 0, 4, 0, 4, 8, 1, 3, 4, 0, 6, 6, 0, 4, 4, 6, 3, 4, 3, 3, 3, 2, 4, 6, 7, 4, 7, 2

Offset: 10

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Author

Jason Earls, Sep 17 2004

Keywords

base
easy
nonn

Examples

a(21)=7 because there are 7 primes in the sequence 2,1,3,4,7,11,18,29,47,76,123,199,322.

Crossrefs

Cf. A007629, A073882.

A128546 Inrepfigit (INverse REPetitive FIbonacci-like diGIT) numbers (or Htiek numbers).

Original entry on oeis.org

17, 21, 25, 42, 63, 84, 143, 286, 2355, 5821, 6618, 11709, 12482, 33747, 39571, 129109, 466957, 1162248, 1565166, 1968084, 3636638, 3853951, 4898376, 6065280, 13443745, 13933175, 17118698, 22421197, 24153462377

Offset: 1

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Author

Pierre Karpman (pierre.karpman(AT)laposte.net), Oct 23 2007

Keywords

base
nonn

Comments

This sequence is similar to A007629 (Keith numbers). It consists of the numbers n>9 with the following property: n is a term of the sequence S whose first k terms are the k digits of n (with the first term equal to the units digit) and with S(n+1)=sum of the k previous terms.

Examples

42 is in the sequence because the terms of the sequence it creates are 2, 4, 6, 10, 16, 26, 42, ...

Crossrefs

Cf. A007629.

Cf. A097060 (reverse of these numbers).

Programs

Mathematica

iKeithQ[n_Integer] := Module[{b = Reverse[IntegerDigits[n]], s, k = 0}, s = Total[b]; While[s < n, AppendTo[b, s]; k++; s = 2*s - b[[k]]]; s == n]; Select[Range[10, 100000], iKeithQ] (* T. D. Noe, Mar 15 2011 *)

A212875 Primonacci numbers: composite numbers that appear in the Fibonacci-like sequence generated by their own prime factors.

Original entry on oeis.org

4, 9, 12, 25, 27, 169, 1102, 7921, 22287, 54289, 103823, 777627, 876897, 2550409, 20854593, 34652571, 144237401, 144342653, 167901581, 267911895, 792504416, 821223649, 1103528482, 2040412557, 2852002829, 3493254541, 6033671841, 15658859018, 116085000401

Offset: 1

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Author

Herman Beeksma, May 29 2012

Keywords

nonn

Comments

Given n, form a sequence that starts with the k prime factors of n in ascending order. After that, each term is the sum of the preceding k terms. If n eventually appears in the sequence, it is a primonacci number. Primes possess this property trivially and are therefore excluded.

Similar to A007629 (repfigit or Keith numbers), but base-independent. If n is in A005478 (Fibonacci primes), then n^2 is a primonacci number.

The only entries that are semiprimes (A001358) are the squares of A005478. - Robert Israel, Mar 08 2016

Examples

Fibonacci-like sequences for selected values of n: n=12: 2, 2, 3, 7, 12, ... n=25: 5, 5, 10, 15, 25, ... n=1102: 2, 19, 29, 50, 98, 177, 325, 600, 1102, ...

Links

Herman Beeksma, Table of n, a(n) for n = 1..42

Crossrefs

Cf. A000045, A001358, A005478, A007629.

Programs

Maple

with(numtheory): P:=proc(q,h) local a,b,j,k,n,t,v; v:=array(1..h); for n from 2 to q do if not isprime(n) then b:=ifactors(n)[2]; a:=[]; for k from 1 to nops(b) do for j from 1 to b[k][2] do a:=[op(a),b[k][1]]; od; od; a:=sort([op(a)]); b:=nops(a); for k from 1 to b do v[k]:=a[k]; od; t:=b+1; v[t]:=add(v[k], k=1..b); while v[t]Paolo P. Lava, Mar 08 2016

Mathematica

PrimonacciQ[n_]:=Module[{k,seq}, seq=FactorInteger[n]; seq=Map[Table[#[[1]],{#[[2]]}]&, seq]; seq=Flatten[seq]; k=Length[seq]; If[k==1,Return[False]]; seq=Append[seq,Apply[Plus,seq]]; While[seq[[-1]]Michael De Vlieger, Mar 08 2016 *)

Python

from sympy import isprime, factorint from itertools import chain A212875_list = [] for n in range(2,10**6): if not isprime(n): x = sorted(chain.from_iterable([p]*e for p,e in factorint(n).items())) y = sum(x) while y < n: x, y = x[1:]+[y], 2*y-x[0] if y == n: A212875_list.append(n) # Chai Wah Wu, Sep 12 2014

A007603 Power-sum numbers: let n = a_1 a_2 ... a_k be a k-digit number; n is a power-sum number if there are exponents e_1 ... e_m such that n = Sum_{i=1..m} Sum_{j=1..k} a_j^e_i.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 23, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 104, 108, 110, 111, 112, 113, 114, 115, 116, 117, 120, 122, 126, 130, 131, 132, 133, 134, 135, 136, 140, 144, 150, 151, 152, 153, 154, 156, 160, 162, 170, 171, 172, 173, 174, 178, 180, 182

Offset: 1

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Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Keywords

nonn
easy
nice
base

Examples

21 = (2+1)+(2^3+1^3)+(2^3+1^3), with e_1, e_2, e_3 = 1, 3, 3.

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Mike Keith, Power-sum numbers, J. Recreational Mathematics, Vol. 18, No. 4 (1986), pp. 275-278. (Annotated scanned copy)

Crossrefs

Subsequences: A005349, A034087, A034088, A169665, A169666.

Programs

Mathematica

q[n_] := Module[{d = IntegerDigits[n], v = {}, k = 1, s, ans = False}, If[Max[d] == 1, ans = Divisible[n, Total[d]], While[(s = Total[d^k]) <= n, AppendTo[v, s]; If[Length[IntegerPartitions[n, All, v]] > 0, ans = True; Break[]]; k++]]; ans]; Select[Range[200], q] (* Amiram Eldar, Sep 04 2021 *)

Extensions

Corrected and extended by Naohiro Nomoto, Mar 11 2001

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A269310 Consider a number x > 1. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach the Euler totient function of x.

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Extensions

A269311 Consider the arithmetic derivative of a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach x.

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Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A269308 Consider a number x. Take the sum of its digits. Repeat the process deleting the first addendum and adding the previous sum. The sequence lists the numbers that after some iterations reach the sum of the divisors of x.

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Author

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Examples

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Maple

Mathematica

Extensions

A130792 Numbers k whose representation can be split in two parts which can be used as seeds for a Fibonacci-like sequence containing k itself.

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Mathematica

PARI

A248134 Consider a number x as a concatenation of two integers, a and b: x = concat(a,b). Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A097060 Revrepfigits (reverse replicating Fibonacci-like digits): Numbers k whose reversal occurs in a sequence generated by starting with the k digits of a number and then continuing the sequence with a number that is the sum of the previous k terms.

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Mathematica

Extensions

A098598 Number of primes in sequences formed from the t digits of n where the latter terms are given by rule b(i)=sum of t previous terms; primes are counted from initial t digits up to the largest term < n^2.

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A128546 Inrepfigit (INverse REPetitive FIbonacci-like diGIT) numbers (or Htiek numbers).

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