A085946 -id:A085946 - cp's OEIS frontend

A085947 a(1) = 1, a(2) = 2 and a(n) = smallest number not included earlier that divides the sum of the two previous terms.

1, 2, 3, 5, 4, 9, 13, 11, 6, 17, 23, 8, 31, 39, 7, 46, 53, 33, 43, 19, 62, 27, 89, 29, 59, 22, 81, 103, 92, 15, 107, 61, 12, 73, 85, 79, 41, 10, 51

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 14 2003

The next term would have to divide 61, but 1 and 61 are both already used. - Franklin T. Adams-Watters, Oct 23 2015

			After 46 and 53 the term is 33 and not 1,3,9 or 11 as they have already been included. 33 divides 46+53 = 99.

Cf. A085946.

a[1]:= 1:
a[2]:= 2:
for n from 3 do
  r:= min(numtheory:-divisors(a[n-1]+a[n-2]) minus {seq(a[i],i=1..n-1)});
  if r = infinity then break fi;
  a[n]:= r
od:
seq(a[i],i=1..n-1); # Robert Israel, Oct 25 2015

More terms from David Wasserman, Feb 14 2005

Name corrected by Franklin T. Adams-Watters, Oct 23 2015

A351629 a(1) = 1, a(2) = 2 and a(n) is the smallest integer not included earlier whose first digit divides the concatenation a(n-2), a(n-1).

Original entry on oeis.org

1, 2, 3, 10, 5, 7, 11, 9, 12, 4, 13, 14, 6, 15, 16, 17, 18, 19, 100, 20, 21, 101, 102, 22, 23, 30, 24, 8, 25, 31, 103, 104, 26, 27, 105, 32, 28, 29, 33, 70, 50, 34, 35, 36, 40, 41, 37, 38, 39, 106, 107, 108, 42, 60, 43, 109, 110, 51, 111, 90, 52, 44, 45, 53, 112, 46, 113, 114, 115, 54, 116, 47

Offset: 1

Eric Angelini and Carole Dubois, Feb 20 2022

The sequence is a permutation of the positive integers.

			a(3) = 3 as 3 is the smallest unused integer whose first digit divides 12;
a(4) = 10 as 10 is the smallest unused integer whose first digit divides 23;
a(5) = 5 as 5 is the smallest unused integer whose first digit divides 310;
a(6) = 7 as 7 is the smallest unused integer whose first digit divides 105; etc.

Carole Dubois, Table of n, a(n) for n = 1..400
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^14.

Cf. A085946.

c[] = 0; u = 1; a[1] = c[1] = 1; a[2] = c[2] = 2; nn = 72; Do[While[c[u] > 0, u++]; k = u; m = a[n - 2]*10^IntegerLength[a[n - 1]] + a[n - 1]; While[Nand[Mod[m, First@ IntegerDigits[k]] == 0, c[k] == 0], k++]; a[n] = k; c[k] = n, {n, 3, nn}]; Array[a[#] &, nn] (* _Michael De Vlieger, Feb 20 2022 *)

from sympy import divisors
def aupton(terms):
    alst, aset, minunused = [1, 2], {1, 2}, 3
    while len(alst) < terms:
        concat = int(str(alst[-2])+str(alst[-1]))
        k = minunused
        while k in aset or concat%int(str(k)[0]): k += 1
        an = k; alst.append(an); aset.add(an)
        while minunused in aset: minunused += 1
    return alst
print(aupton(72)) # Michael S. Branicky, Feb 20 2022

A351820 a(1) = 1, a(2) = 2, a(3) = 3 and a(n) is the smallest number not included earlier that divides the concatenation a(n-3), a(n-2), a(n-1).

Original entry on oeis.org

1, 2, 3, 41, 2341, 9, 1374473, 7, 123, 13, 49, 53, 91, 689, 5391689, 167, 8429, 17, 11, 21, 211, 37, 89, 2113789, 47, 89211378947, 1336372981, 43, 169, 7213, 47966357, 121, 13681, 29863, 9848521381, 173, 23, 2997821, 29, 39, 19, 97973, 130665991, 727, 251, 817

Offset: 1

Carole Dubois and Eric Angelini, Feb 20 2022

			a(4) = 41 is the smallest unused integer that divides 123;
a(5) = 2341 is the smallest unused integer that divides 2341;
a(6) = 9 is the smallest unused integer that divides 3412341;
a(7) = 1374473 is the smallest unused integer that divides 4123419; etc.

Jon E. Schoenfield, Table of n, a(n) for n = 1..5000

Cf. A085946, A351629.

nn = 46; s = Range[3]; c[] = 0; Array[Set[{a[#1], c[#2]}, {#2, #1}] & @@ {#, s[[#1]]} &, Length[s]]; Do[(Set[{a[n], c[#]}, {#, n}] &@ SelectFirst[Divisors[FromDigits@ Flatten@ Map[IntegerDigits, Reverse@ Array[a[n - #] &, 3]]], c[#] == 0 &]), {n, 1 + Length[s], nn}]; Array[a[#] &, nn] (* _Michael De Vlieger, Feb 20 2022 *)

from sympy import divisors
def aupton(terms):
    alst, aset = [1, 2, 3], {1, 2, 3}
    while len(alst) < terms:
        concat = int("".join(map(str, alst[-3:])))
        an = min(d for d in divisors(concat) if d not in aset)
        alst.append(an); aset.add(an)
    return alst
print(aupton(46)) # Michael S. Branicky, Feb 20 2022

a(26) and beyond from Michael S. Branicky, Feb 20 2022

A333905 Lexicographically earliest sequence of distinct positive integers such that a(n) divides the concatenation of a(n+1) to a(n+2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 8, 20, 16, 40, 32, 80, 64, 160, 128, 320, 256, 640, 512, 1280, 1024, 2560, 2048, 5120, 4096, 10240, 8192, 20480, 16384, 40960, 32768, 81920, 65536, 163840, 131072, 327680, 262144, 655360, 524288, 1310720, 1048576, 2621440, 2097152, 5242880, 4194304, 10485760, 8388608, 20971520, 16777216, 41943040

Offset: 1

Eric Angelini, Apr 09 2020

			a(1) = 1 divides 23 (and 23 is a(2) = 2 concatenated to a(3) = 3);
a(2) = 2 divides 34 (and 34 is a(3) = 3 concatenated to a(4) = 4);
a(3) = 3 divides 45 (and 45 is a(4) = 4 concatenated to a(5) = 5);
a(4) = 4 divides 56 (and 56 is a(5) = 5 concatenated to a(6) = 6);
a(5) = 5 divides 610 (and 610 is a(6) = 6 concatenated to a(7) = 10);
a(6) = 6 divides 108 (and 108 is a(7) = 10 concatenated to a(8) = 8);
From a(7) = 10 on, the pattern of the sequence is regular.

Cf. A085946 (a(1) = 1, a(2) = 2 and a(n) = smallest number not included earlier that divides the concatenation a(n-2), a(n-1)).

Conjectures from Colin Barker, Apr 09 2020: (Start)
G.f.: x*(1 + 2*x + x^2 - x^4 - 2*x^5 - 4*x^7) / (1 - 2*x^2).
a(n) = 2*a(n-2) for n>6.
(End)
Conjecture: a(n) = 2^((n-7)/2)*(5 + 2*sqrt(2) + (2*sqrt(2) - 5)*(-1)^n) for n > 6. - Stefano Spezia, Oct 23 2021

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A085947 a(1) = 1, a(2) = 2 and a(n) = smallest number not included earlier that divides the sum of the two previous terms.

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A351629 a(1) = 1, a(2) = 2 and a(n) is the smallest integer not included earlier whose first digit divides the concatenation a(n-2), a(n-1).

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A351820 a(1) = 1, a(2) = 2, a(3) = 3 and a(n) is the smallest number not included earlier that divides the concatenation a(n-3), a(n-2), a(n-1).

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A333905 Lexicographically earliest sequence of distinct positive integers such that a(n) divides the concatenation of a(n+1) to a(n+2).

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