Michael Turniansky - cp's OEIS frontend

A339412 a(n) = floor(x(n)) where x(n) = (frac(x(n-1))+1)*floor(x(n-1)) and x(1) = Pi.

3, 3, 4, 5, 5, 7, 10, 10, 13, 17, 31, 35, 67, 123, 223, 305, 414, 822, 1550, 2224, 3273, 4560, 7804, 14372, 15493, 20080, 40039, 44226, 71916, 130773, 183760, 316165, 613602, 1066559, 1138668, 1202427, 2022144, 2251837, 2477524, 4479491, 7192184, 11256849

Offset: 1

Michael Turniansky, Dec 03 2020

nonn

Inspired by A249270.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
James Grime and Brady Haran, 2.920050977316, Numberphile video, Nov 26 2020.

Cf. A000796, A249270.

b:= proc(n) option remember; `if`(n=1, Pi,
      (f-> (frac(f)+1)*floor(f))(b(n-1)))
    end:
a:= n-> floor(b(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 03 2020

Block[{a = {Pi}, $MaxExtraPrecision = 10^3}, Do[AppendTo[a, (FractionalPart[#] + 1) Floor[#]] &@ a[[-1]], 41]; Floor /@ a] (* Michael De Vlieger, Dec 04 2020 *)

{(⌊{(⌊⍵)×1+1|⍵}⍣⍵)○1x}¨0,⍳100

lista(nn) = {localprec(500); my(vx = vector(nn)); vx[1] = Pi; for (n=2, nn, vx[n] = (frac(vx[n-1])+1)*floor(vx[n-1]);); apply(floor, vx);} \\ Michel Marcus, Dec 03 2020

A304011 Number of same-sized pairs of subsets of set of n numbers that might have the same sum.

Original entry on oeis.org

0, 0, 0, 1, 5, 20, 70, 231, 735, 2289, 7029, 21384, 64636, 194480, 583232, 1744847, 5210687, 15540023, 46299143, 137837666, 410127806, 1219804541, 3626853647, 10781440394, 32045015650, 95236794600, 283027305300, 841096898745, 2499595030581, 7428627412260

Offset: 1

Michael Turniansky, Jul 03 2018

nonn

Given a set with n different numbers, you only need to check a(n) pairs of subsets of the same cardinality to prove that no pair of same-cardinality subsets have the same total sum. The others can be eliminated by noting the dominance of members of one totally-ordered subset over the corresponding elements of the other totally-ordered subset.

Michael De Vlieger, Table of n, a(n) for n = 1..2100
Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, Pattern distributions in Dyck paths with a first return decomposition constrained by height, arXiv:1911.03119 [math.CO], 2019.
Project Euler, Problem 106: Special subset sums: meta-testing

Cf. A002054.

Table[1/2 + Hypergeometric2F1[(1 - n)/2, -n/2, 1, 4]/2 - Hypergeometric2F1[(1 - n)/2, -n/2, 2, 4], {n, 1, 30}] (* Vaclav Kotesovec, Aug 04 2018 *)
Join[{0,0,0,1},RecurrenceTable[{(n-4)*(n+2)*a[n]==(3*n^2-7*n-5)*a[n-1]+ (n-3)*(n-1)*a[n-2]-3*(n-2)*(n-1)*a[n-3],a[2]==0,a[3]==0,a[4]==1},a,{n,5,25}]] (* Georg Fischer, Dec 06 2019 *)

a(n) = sum(i=1, n\2, binomial(n, 2*i)*binomial(2*i-1, i-2)); \\ Michel Marcus, Jul 04 2018

a(n) = Sum_{i=1..floor(n/2)} binomial(n, 2*i)*A002054(i-1).
From Vaclav Kotesovec, Aug 04 2018: (Start)
D-finite with recurrence: (n-4)*(n+2)*a(n) = (3*n^2 - 7*n - 5)*a(n-1) + (n-3)*(n-1)*a(n-2) - 3*(n-2)*(n-1)*a(n-3) for n >= 5.
a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi*n)). (End)

a(23) corrected by Georg Fischer, Dec 06 2019

A270004 Run-length encoding of iterated Scrabble function.

Original entry on oeis.org

12, 1, 4, 2, 1, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 3, 4, 1, 1, 1, 3, 1, 2, 3, 3, 3, 1, 1, 1, 5, 6, 3, 1, 2, 4, 3, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 5, 2, 1, 1, 3, 2, 7, 1, 4, 1, 4, 1, 4, 1, 1, 1, 1, 1, 3, 1, 2, 1, 4, 2, 1, 1, 4, 1, 1, 8, 2, 2, 3, 3, 2, 2, 3, 1, 3, 3, 2, 1, 1, 2, 2, 2, 3, 1, 2, 3, 7, 1, 1, 5, 3, 1, 1, 6

Offset: 0

Michael Turniansky, Jul 24 2017

Number of identical consecutive integers in A290205.

Michael Turniansky, Table of n, a(n) for n = 0..455
Wikipedia, Scrabble

Cf. A113172, A290205.

from num2words import num2words
tp = {"aeilnorstu": 1, "dg": 2, "bcmp":3, "fhvwy":4, "k":5, "jx":8, "qz":10}
def pts(c): return ([tp[s] for s in tp if c in s]+[0])[0]
def A113172(n): return sum(map(pts, num2words(n).replace(" and", "")))
def A290205(n):
    while n not in {12, 4, 7, 8, 9}: n = A113172(n)
    return 12 if n == 12 else 4
def aupton(terms):
    alst, prev, k, rl = [], A290205(0), 1, 1
    while len(alst) < terms:
        while A290205(k) == prev: k += 1; rl += 1
        alst.append(rl); rl = 0; prev = 12 if prev == 4 else 4
    return alst
print(aupton(105)) # Michael S. Branicky, Dec 01 2021

A290205 Group identifier of iterated Scrabble function.

Original entry on oeis.org

4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 12, 4, 12, 4, 12, 12, 4, 12, 12, 12, 4, 4, 12, 4, 12, 12, 4, 12, 12, 12, 4, 4, 4, 4, 12, 4, 12, 4, 4, 4, 12, 4, 4, 12, 12, 12, 4, 4, 4, 12, 12, 12, 4, 12, 4, 12, 12, 12, 12, 12, 4, 4, 4, 4, 4

Offset: 0

Michael Turniansky, Jul 24 2017

Take the sequence A113172. Map it onto itself repeatedly. You will either end up at the fixed point, 12, or a loop of 4->7->8->9->4 (here represented by 4).

			For example 10, spelled TEN is worth 3 points, THREE is worth 8 points and 8 is in the 4/7/8/9 loop, so a(10)=4

Wikipedia, Scrabble

Cf. A113172.

from num2words import num2words
def A290205(n):
    f = lambda n:sum((1,3,3,2,1,4,2,4,1,8,5,1,3,1,1,3,10,1,1,1,1,4,4,8,4,10)[a] for d in num2words(n).replace(' and ','') if 0<=(a:=ord(d)-97)<=25)
    while True:
        n = f(n)
        if n in {4,7,8,9,4}: return 4
        if n == 12: return 12 # Chai Wah Wu, Apr 21 2023

A283793 Number of elements formable in <= n steps, starting with 4 elements, combining 2 elements into a new element at each step.

Original entry on oeis.org

4, 14, 109, 5999, 17997004, 161946085486514, 13113267302202731189080679359, 85978889669509647874887802052390686151982448025024665124

Offset: 0

Michael Turniansky, Mar 16 2017

nonn

In a game such as Doodle God (see links), you start with Earth, Air, Fire and Water, and combine them two at a time (including combining an element with itself) into new elements. A(n) is the hypothetical maximum number of different possible elements you could reach from clicking at most n times.

This list is akin to A006894, which is the sequence if we started with 1 element instead of 4.

			Starting with {A, B, C, D}, we can make {AA, AB, AC, AD, BB, BC, BD, CC, CD, and DD}.  The union of these two sets has cardinality 14 = a(1).

Michael Turniansky, Table of n, a(n) for n = 0..9
Anton Rybakov as Joybits Ltd., Doodle God game homepage
Cary Kaiming Huang, Elements 3. An online version which is not bounded by predefined elements, so a(n) could theoretically be reached. (No longer works.)

Cf. A006894.

a[0]=4; a[n_] := a[n] = 4 + a[n-1] (a[n-1] + 1)/2; a /@ Range[0, 7] (* Giovanni Resta, Mar 16 2017 *)

a(n) = if(n<1, 4, 4 + a(n - 1) * (a(n - 1) + 1) / 2);
for(n=0, 7, print1(a(n),", ")) \\ Indranil Ghosh, Mar 16 2017

a(n) = 4 + T(a(n-1)) where T(m) is the m-th triangular number.

A261148 Prime-Indexed Primes (PIPs) k such that the sum of all PIPs <= k is a prime.

Original entry on oeis.org

3, 11, 31, 59, 83, 211, 331, 773, 1297, 1433, 1471, 1621, 2027, 2477, 3637, 4153, 4787, 4877, 5623, 7699, 9103, 9619, 11743, 12097, 12959, 13037, 13591, 13709, 14177, 14969, 15299, 16411, 16703, 16921, 19463, 19577, 21379, 22093, 22721, 24107, 24151, 24419, 24509, 24671, 28657

Offset: 1

Michael Turniansky, Aug 10 2015

This is a strict subset of A006450: {k from A006450 | Sum_{j=1..k} A006450(j) is prime}.

It seems from observation that asymptotically a(n)/A006450(n) ~ 7.5*log(n) - e. But might this just be coincidence? I certainly have no proof. - Michael Turniansky, Aug 21 2015

			11 is in the sequence because A006450(1) + A006450(2) + A006450(3) = 3 + 5 + 11 = 19, a prime number.

Michael Turniansky, Table of n, a(n) for n = 1..508

Cf. A006450.

L={}; s=0; p=2; While[Length@L < 100, If[PrimeQ[s+=(q = Prime@p)], AppendTo[L, q]]; p = NextPrime@ p]; L (* Giovanni Resta, Aug 21 2015 *)

lista(nn) = {s = 0; forprime(p=2, nn, q = prime(p); s += q; if (isprime(s), print1(q, ", ")););} \\ Michel Marcus, Aug 20 2015

A239751 Numbers whose number of nonzero binary digits is less than their number of nonzero decimal digits.

Original entry on oeis.org

16, 32, 64, 128, 129, 132, 136, 144, 192, 256, 257, 258, 264, 272, 288, 384, 512, 513, 514, 516, 528, 544, 576, 768, 1024, 1025, 1026, 1028, 1032, 1056, 1088, 1152, 1153, 1154, 1156, 1168, 1184, 1216, 1280, 1281, 1282, 1284, 1288, 1296, 1312, 1344, 1536, 1537, 1538, 1544, 1552, 1568, 1664

Offset: 1

Michael Turniansky, Mar 26 2014

If Hebrew gematria (A051596) were replaced with a "binary gematria" where aleph=1, bet=2, gimel=4, etc., this sequence would be a subset of "numbers whose binary gematria has fewer letters than their original gematria".

Michael Turniansky, Table of n, a(n) for n = 1..200

Cf. A000120, A051596, A055640.

((+⌿(22⍴2)⊤C)<+⌿0≠(5⍴10)⊤C)/C←⍳10000

Select[Range[1664], Less @@ Total /@ Sign /@ IntegerDigits[#, {2, 10}] &] (* Giovanni Resta, Mar 26 2014 *)

n such that A000120(n)<A055640(n).

A235029 Numbers whose English name requires fewer letters than twice the number of decimal digits.

Original entry on oeis.org

10, 1000000, 1000001, 1000002, 1000006, 1000010, 2000000, 2000001, 2000002, 2000006, 2000010, 3000000, 4000000, 5000000, 6000000, 6000001, 6000002, 6000006, 6000010, 7000000, 8000000, 9000000, 10000000, 10000001, 10000002, 10000003

Offset: 1

Michael Turniansky, Jan 02 2014

This is also the list of numbers whose English name is shorter than their name in the artificial language Lojban.

John Woldemar Cowan, 18.2. Lojban numbers, in: The Complete Lojban Language (CLL v1.1), 2016

Numbers n such that 2*A055642(n) > A005589(n).

Comment extended by Georg Fischer, Dec 18 2020

A228158 Numbers n such that the cardinality of (natural numbers <=n with a first digit of 1) = n/2.

Original entry on oeis.org

2, 16, 22, 176, 222, 1776, 2222, 17776, 22222, 177776, 222222, 1777776, 2222222, 17777776, 22222222, 177777776, 222222222, 1777777776, 2222222222, 17777777776, 22222222222, 177777777776, 222222222222, 1777777777776, 2222222222222, 17777777777776

Offset: 1

Michael Turniansky, Aug 14 2013

This is conceptually related to Benford's law.

Index entries for sequences related to Benford's law

Vec(2*(1+8*x)/((1-x)*(1+x)*(1-10*x^2))+O(x^66)) \\ Joerg Arndt, Aug 14 2013

a(n) for odd values of n is simply "2" repeated (n+1)/2 times (sum as i=1->(n/2) of 10^(i-1)). For even values of n, it's a(n-1)*8.

G.f.: 2*(1+8*x)/((1-x)*(1+x)*(1-10*x^2)). [Joerg Arndt, Aug 14 2013]

A213394 The difference between n and the product of the digits of the n-th prime.

Original entry on oeis.org

-1, -1, -2, -3, 4, 3, 0, -1, 3, -8, 8, -9, 9, 2, -13, 1, -28, 12, -23, 13, 0, -41, -1, -48, -38, 26, 27, 28, 29, 27, 17, 29, 12, 7, -1, 31, 2, 20, -3, 19, -22, 34, 34, 17, -18, -35, 45, 36, 21, 14, 33, -2, 45, 44, -15, 20, -51, 44, -39, 44, 13, 8, 63, 61, 56

Offset: 1

Michael Turniansky, Jun 28 2012

The first three zeros occur at terms 7, 21 and 181440.

T. D. Noe, Table of n, a(n) for n = 1..10000

Table[n - Times @@ IntegerDigits[Prime[n]], {n, 100}] (* T. D. Noe, Jun 28 2012 *)

a(n) = A000027(n) - A053666(n).

cp's OEIS Frontend

User: Michael Turniansky

A339412 a(n) = floor(x(n)) where x(n) = (frac(x(n-1))+1)*floor(x(n-1)) and x(1) = Pi.

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A304011 Number of same-sized pairs of subsets of set of n numbers that might have the same sum.

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A270004 Run-length encoding of iterated Scrabble function.

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A290205 Group identifier of iterated Scrabble function.

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A283793 Number of elements formable in <= n steps, starting with 4 elements, combining 2 elements into a new element at each step.

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Mathematica

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A261148 Prime-Indexed Primes (PIPs) k such that the sum of all PIPs <= k is a prime.

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Mathematica

PARI

A239751 Numbers whose number of nonzero binary digits is less than their number of nonzero decimal digits.

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APL

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A235029 Numbers whose English name requires fewer letters than twice the number of decimal digits.