Eric Fox - cp's OEIS frontend

A355851 The difference between the indices of the two most recent instances of a(n-1), not including a(n-1) itself, or 0 if not enough such instances exist.

0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 2, 2, 1, 2, 1, 9, 0, 2, 2, 4, 0, 8, 0, 4, 0, 2, 1, 2, 7, 0, 2, 2, 3, 0, 5, 0, 4, 4, 13, 0, 2, 1, 12, 0, 4, 1, 15, 0, 4, 7, 0, 4, 4, 3, 0, 3, 21, 0, 4, 1, 4, 6, 0, 3, 2, 9, 0, 5, 0, 4, 2, 24, 0, 2, 6, 0, 4, 9, 50, 0, 3, 8, 0, 4, 7, 21, 0, 3, 17

Offset: 1

Eric Fox, Jul 19 2022

nonn

This is similar to Van Eck's sequence: A181391.

			Say you have 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 2, ?
The previous term is a 2, and the distance between the pair of most recent 2s before that is 2, hence ? = 2.

Cf. A181391.

A354856 a(1) = 1, a(n) = the number of times a(n-1) appears among the first n-2 terms.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 0, 2, 1, 3, 0, 3, 1, 4, 0, 4, 1, 5, 0, 5, 1, 6, 0, 6, 1, 7, 0, 7, 1, 8, 0, 8, 1, 9, 0, 9, 1, 10, 0, 10, 1, 11, 0, 11, 1, 12, 0, 12, 1, 13, 0, 13, 1, 14, 0, 14, 1, 15, 0, 15, 1, 16, 0, 16, 1, 17, 0, 17, 1, 18, 0, 18, 1, 19, 0, 19, 1, 20, 0, 20

Offset: 1

Eric Fox, Jun 09 2022

nonn

			a(10) = 3 because a(9) = 1 and 3 other 1s appear before that.

Cf. A342585, A347317.

lista(nn) = my(va=vector(nn)); va[1] = 1; for (n=2, nn, my(vb = vector(n-2, k, va[k])); va[n] = #select(x->(x==va[n-1]), vb);); va; \\ Michel Marcus, Jun 13 2022

from collections import Counter
from itertools import count, islice
def agen(): # generator of terms
    anprev, an, inventory = None, 1, Counter()
    for n in count(2):
        yield an
        anprev, an = an, inventory[an]
        inventory[anprev] += 1
print(list(islice(agen(), 80))) # Michael S. Branicky, Jun 09 2022

a(4n+1..4n+4) = 1, n+1, 0, n+1 for n >= 1. - Michael S. Branicky, Jun 12 2022

a(54) and beyond from Michael S. Branicky, Jun 12 2022

A353219 Positive integers which cannot be expressed as the sum of three or fewer squares, no more than two of which are greater than 1.

Original entry on oeis.org

7, 12, 15, 22, 23, 24, 28, 31, 39, 43, 44, 47, 48, 55, 56, 57, 60, 63, 67, 70, 71, 76, 77, 78, 79, 84, 87, 88, 92, 93, 94, 95, 96, 103, 108, 111, 112, 115, 119, 120, 124, 127, 132, 133, 134, 135, 139, 140, 141, 142, 143, 151, 152, 155, 156, 159, 166, 167, 168, 172, 175

Offset: 1

Eric Fox, Apr 30 2022

nonn

These are the numbers which cannot be a clue in a Tasquare puzzle.

Tasquare also known as Tasukuea.

			A 9 clue can be satisfied in at least one way:
  OOO  OO OO
  OOO  OO9OO
  OOO9   O
So, 9 is not a term in this sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Cross+A, Tasquare rules
Nikoli, Tasukuea rules [broken link]

Complement of A353202.

Cf. A022544.

q:= proc(n) local i; for i to isqrt(n) do if ormap(issqr,
      [n-i^2, n-i^2-1]) then return false fi od: true
    end:
select(q, [$1..175])[];  # Alois P. Heinz, Apr 30 2022

q[n_] := Module[{i}, For[i = 1, i <= Sqrt[n], i++, If[AnyTrue[ {n-i^2, n-i^2-1}, IntegerQ@Sqrt[#]&], Return[False]]]; True];
Select[Range[175], q] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz *)

A353202 Positive integers which can be expressed as the sum of three or fewer squares, no more than two of which are greater than 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 45, 46, 49, 50, 51, 52, 53, 54, 58, 59, 61, 62, 64, 65, 66, 68, 69, 72, 73, 74, 75, 80, 81, 82, 83, 85, 86, 89, 90, 91, 97, 98, 99, 100

Offset: 1

Eric Fox, Apr 30 2022

nonn

These are the numbers which can be a clue in a Tasquare puzzle.

Tasquare also known as Tasukuea.

			A 9 clue can be satisfied in multiple ways:
  OOO  OO OO
  OOO  OO9OO
  OOO9   O

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Cross+A, Tasquare rules
Nikoli, Tasukuea rules [broken link]

Complement of A353219.

Cf. A001481.

q:= proc(n) local i; for i to isqrt(n) do if ormap(issqr,
      [n-i^2, n-i^2-1]) then return true fi od: false
    end:
select(q, [$1..100])[];  # Alois P. Heinz, Apr 30 2022

q[n_] := Module[{i}, For[i = 1, i <= Sqrt[n], i++, If[AnyTrue[ {n-i^2, n-i^2-1}, IntegerQ@Sqrt[#]&], Return[True]]]; False];
Select[Range[100], q] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz *)

A353199 a(1)=a(2)=1. a(n) = the sum of the indices of the terms, from among the first (n-2) terms of the sequence, which divide a(n-1).

Original entry on oeis.org

1, 1, 1, 3, 6, 10, 6, 15, 10, 12, 22, 6, 22, 17, 6, 34, 20, 21, 10, 21, 28, 6, 49, 6, 71, 6, 95, 6, 121, 6, 149, 6, 179, 6, 211, 6, 245, 29, 6, 281, 6, 320, 57, 10, 40, 101, 6, 361, 6, 408, 497, 31, 6, 457, 6, 510, 681, 10, 84, 634, 6, 565, 6, 626, 6, 689, 6, 754, 44, 30, 965

Offset: 1

Eric Fox, Apr 30 2022

nonn

			a(10) = 12. Of the terms before that, a(1), a(2), a(3), a(4), a(5), and a(7) divide 12. Hence, a(11) = 1+2+3+4+5+7 = 22.

Cf. A124063.

a[1] = a[2] = 1; a[n_] := a[n] = Total[Select[Range[n - 2], Divisible[a[n - 1], a[#]] &]]; Array[a, 100] (* Amiram Eldar, Apr 30 2022 *)

first(n) = { n = max(n, 2); res = List([1,1]); for(i = 3, n, v = Vec(select(x->res[#res]%x == 0, res, 1))[^-1]; listput(res, vecsum(v)) ); res } \\ David A. Corneth, Apr 30 2022

A333133 7-Kaprekar numbers.

Original entry on oeis.org

1, 627615, 4444444, 4927941, 5072059, 5555556, 9372385, 9999999

Offset: 1

Eric Fox, Mar 09 2020

No n-Kaprekar number k can have more than n digits because then the number to the left of the plus sign would have more digits than k itself, meaning the sum will always be greater than k.

			627615 is in this sequence because inserting a + before the 7th digit from the right of 627615^2 = 393900588225 yields 39390 + 0588225, which equals 627615 (the starting number).

Cf. A006886, A037042, A053394, A053395, A053396, A053397.

A332261 Numbers that yield a prime whenever a '4' is inserted between any two digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 21, 37, 39, 43, 49, 51, 57, 61, 63, 67, 73, 91, 97, 109, 129, 147, 153, 159, 171, 187, 199, 211, 223, 237, 241, 247, 259, 267, 333, 349, 357, 363, 409, 421, 423, 441, 447, 457, 463, 493, 517, 537, 541, 543, 549, 571, 579, 583, 627, 649, 681

Offset: 1

Eric Fox, Feb 08 2020

For single-digit terms, the condition is voidly satisfied: nothing can be inserted.

			10281 is in this sequence because 1(4)0281, 10(4)281, 102(4)81, and 1028(4)1 are all prime.

Cf. A164329, A304246, A304247, A304248, A050714.

a:=[]; for k in [1..700] do s:=0; v:=Reverse(Intseq(k)); for i in [1..#v-1] do vv:=v[1..i] cat [4] cat v[i+1..#v]; p:=Seqint(Reverse(vv)); if not IsPrime(p) then break; else s:=s+1; end if; end for;  if s eq #v-1 then Append(~a,k); end if; end for; [0] cat a; // Marius A. Burtea, Feb 09 2020

A332240 Palindromes that are the sum of a number and the sum of its digits.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 131, 141, 151, 161, 171, 181, 191, 202, 212, 232, 242, 252, 262, 272, 282, 292, 303, 313, 333, 343, 353, 363, 373, 383, 393, 404, 414, 434, 444, 454, 464, 474, 484, 494, 505, 515, 535, 545, 555, 565, 575

Offset: 1

Eric Fox, Feb 07 2020

			196 + 1 + 9 + 6 = 212, so 212 is in this sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Intersection of A002113 and A176995.

Cf. A062028, A229545.

pal:=func; [k:k in [0..600]| pal(k) and exists(m){s:s in [0..k]| s+&+Intseq(s) eq k}]; // Marius A. Burtea, Feb 08 2020

palQ[n_] := Block[{d = IntegerDigits[n]}, Reverse[d] == d]; Select[ Union[(# + Plus @@ IntegerDigits@#) & /@ Range[0, 600]], # <= 600 && palQ[#] &] (* Giovanni Resta, Feb 07 2020 *)

a(n) in { A062028(A229545(i)) : i >= 1 }. - Amiram Eldar, Feb 12 2020

More terms from Giovanni Resta, Feb 07 2020

A330729 Numbers k that are divisible by all integers less than or equal to the cube root of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 36, 42, 48, 54, 60, 72, 84, 96, 108, 120, 180, 240, 300, 420

Offset: 1

Eric Fox, Dec 28 2019

			42 appears because the cube root of 42 is approximately 3.48 and 42 is divisible by 3, 2, and 1 (all positive integers less than 3.48).

Cf. A003102.

[k: k in [1..500]|forall{m:m in [1..Floor(k^(1/3))]|k mod m eq 0}]; // Marius A. Burtea, Dec 31 2019

Select[Range[1000], Divisible[#, LCM @@ Range @ Floor @ Surd[#, 3]] &] (* Amiram Eldar, Dec 28 2019 *)

is(k) = {my(m=sqrtnint(k, 3)); sum(i=1, m, Mod(k, i)==0) == m; } \\ Jinyuan Wang, Dec 31 2019

A323593 Position on a Dvorak typewriter (or computer) keyboard where the n-th letter of the alphabet can be found.

Original entry on oeis.org

8, 22, 5, 13, 10, 3, 4, 14, 12, 19, 20, 7, 23, 16, 9, 1, 18, 6, 17, 15, 11, 25, 14, 21, 2, 26

Offset: 1

Eric Fox, Aug 30 2019

			a(8)=14 because the 8th letter of the alphabet resides on the 14th key of a Dvorak keyboard.

Cf. A087622.

cp's OEIS Frontend

User: Eric Fox

A355851 The difference between the indices of the two most recent instances of a(n-1), not including a(n-1) itself, or 0 if not enough such instances exist.

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A354856 a(1) = 1, a(n) = the number of times a(n-1) appears among the first n-2 terms.

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PARI

Python

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A353219 Positive integers which cannot be expressed as the sum of three or fewer squares, no more than two of which are greater than 1.

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Maple

Mathematica

A353202 Positive integers which can be expressed as the sum of three or fewer squares, no more than two of which are greater than 1.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

A353199 a(1)=a(2)=1. a(n) = the sum of the indices of the terms, from among the first (n-2) terms of the sequence, which divide a(n-1).

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Examples

Crossrefs

Programs

Mathematica

PARI

A333133 7-Kaprekar numbers.

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A332261 Numbers that yield a prime whenever a '4' is inserted between any two digits.

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Magma

A332240 Palindromes that are the sum of a number and the sum of its digits.

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Author

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Examples

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Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A330729 Numbers k that are divisible by all integers less than or equal to the cube root of k.