Maghraoui Abdelkader - cp's OEIS frontend

A262254 a(1) = 11; for n>1, a(n) is the smallest prime that starts with the least significant digit of the previous term and has not occurred earlier.

11, 13, 31, 17, 71, 19, 97, 73, 37, 79, 907, 701, 101, 103, 307, 709, 911, 107, 719, 919, 929, 937, 727, 733, 311, 109, 941, 113, 313, 317, 739, 947, 743, 331, 127, 751, 131, 137, 757, 761, 139, 953, 337, 769, 967, 773, 347, 787, 797, 7001, 149, 971, 151, 157, 7013, 349, 977, 7019, 983

Offset: 1

Maghraoui Abdelkader, Sep 16 2015

This sequence is different from A089755, as this one does not include single-digit primes.
This sequence is different from A089755, for which a(n+1) uses all the digit of a(n) except the most-significant digit of a(n).
In this sequence, a(n+1) uses only the least-significant digit of a(n).

			a(3) = 31 where the most significant digit of 31 is 3, which is the least significant digit of a(2) = 13.

Maghraoui Abdelkader, Table of n, a(n) for n = 1..200

Cf. A076653, A082238, A089755, A107809, A180022.

f[s_List] := Block[{lsd = Mod[s[[-1]], 10], p = 3}, While[ IntegerDigits[p][[1]] != lsd || MemberQ[s, p], p = NextPrime@ p]; Append[s, p]]; s = {11}; s = Nest[f, s, 70] (* Robert G. Wilson v, Sep 16 2015 *)

msd(n)=(n\10^(#Str(n)-1)); l1=1;l3=1;l7=1;l9=1; q=1; lsd(n)=n%10;
t1 = vector(200); i=1;forprime(n=11,10000, if( digits(n)[1]==1,t1[i]=n; i=i+1 ; )) ;
t3 = vector(130); i=1;forprime(n=31,3900 ,  if( digits(n)[1]==3,t3[i]=n; i=i+1 ; ));
t7 = vector(130); i=1;forprime(n=71,70000, if( digits(n)[1]==7,t7[i]=n; i=i+1 ; )) ;
t9 = vector(130); i=1;forprime(n=97,90000, if( digits(n)[1]==9,t9[i]=n; i=i+1 ; )) ;lsd(n)=n%10;
t = vector(200);
findnextnumber(m) =
{if(lsd(m)==1, l1=t1[i1] ; i1=i1+1;return(l1);); if(lsd(m)==3, l3=t3[i3];  i3=i3+1 ; return(l3););
if(lsd(m)==7, l7=t7[i7] ; i7=i7+1;return(l7);); if(lsd(m)==9, l9=t9[i9];  i9=i9+1;return(l9);); }
i1=1;  i3=1;  i7=1;  i9=1;  j=1;s=11;   for(n=1,200,   p=findnextnumber(s) ; t[j]=p; s=p;j++);for(i=1,200, print1(t[i],", ") )

A261507 Fibonacci-numbered rows of Pascal's triangle. Triangle read by rows: T(n,k)= binomial(Fibonacci(n), k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 10, 5, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1, 1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1

Offset: 0

Maghraoui Abdelkader, Aug 22 2015

Subsequence of A007318.

			1,
1,  1,
1,  1,
1,  2,  1,
1,  3,  3,   1,
1,  5, 10,  10,   5,    1,
1,  8, 28,  56,  70,   56,   28,    8,    1,
1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1

Jean-François Alcover, Table of n, a(n) for n = 0..388 (corrected by _N. J. A. Sloane_, Jan 18 2019)

Cf. A000045, A007318.

Cf. A036355, A037027, A228074, A162741, A034871, A034870, A191797.

Table[Binomial[Fibonacci[n], k], {n, 0, 8}, {k, 0, Fibonacci[n]}]//Flatten (* Jean-François Alcover, Nov 12 2015*)

v = vector(101,j,fibonacci(j)); i=0; n=0; while(n<100, for(k=0, n, print1(binomial(n, k), ", ","")); print(); i=i+1; n=v[i] ;)

T(n, k) = binomial(fibonacci(n), k).
T(n, 1) = fibonacci(n) = A000045(n).
T(n, 2) = A191797(n) for n>3.

A262463 If n is prime a(n) = n else a(n) = nextprime(reverse(n)), where "next prime" is the smallest prime >= n, see A007918.

Original entry on oeis.org

2, 2, 3, 5, 5, 7, 7, 11, 11, 2, 11, 23, 13, 41, 53, 61, 17, 83, 19, 2, 13, 23, 23, 43, 53, 67, 73, 83, 29, 3, 31, 23, 37, 43, 53, 67, 37, 83, 97, 5, 41, 29, 43, 47, 59, 67, 47, 89, 97, 5, 17, 29, 53, 47, 59, 67, 79, 89, 59, 7, 61, 29, 37, 47, 59, 67, 67, 89, 97, 7, 71, 29, 73, 47, 59, 67, 79, 89, 79, 11, 19, 29

Offset: 1

Maghraoui Abdelkader, Sep 23 2015

			For n=7 a(7)=7.
For n=12, reverse(12)=21; a(12)=nextprime(21)=23.

Maghraoui Abdelkader, Table of n, a(n) for n = 1..200

Cf. A000040, A077018.

Table[Which[PrimeQ[n],n,PrimeQ[IntegerReverse[n]],IntegerReverse[n], True, NextPrime[ IntegerReverse[ n]]],{n,100}] (* Harvey P. Dale, May 12 2018 *)

rev(n)={d=digits(n); p=""; for(i=1, #d, p=concat(Str(d[i]), p)); return(eval(p))}
i=0; t=vector(200);
findn(n)={if(isprime(n),t[i++]=n, a=rev(n); b=nextprime(a); t[i++]=b); }
for(n=1,200,findn(n)); t

a(n)=n if n is prime;
       else b=reverse(n);
       if b is prime a(n)=b else a(n)=nextprime(b);
(using "next prime" function as "smallest prime >= n"; see A007918. )

A261198 Start with n and repeat the map x -> x+sumdigits(x) until reaching a prime, which is a(n), or 0 if no prime is reached.

Original entry on oeis.org

2, 23, 0, 23, 11, 0, 19, 23, 0, 11, 13, 0, 17, 19, 0, 23, 37, 0, 29, 41, 0, 41, 101, 0, 37, 41, 0, 101, 59, 0, 43, 37, 0, 41, 43, 0, 47, 101, 0, 59, 67, 0, 89, 59, 0, 67, 71, 0, 101, 89, 0, 59, 61, 0, 89, 67, 0, 71, 73, 0, 103, 101, 0, 127, 89, 0, 109, 103, 0, 101, 79, 0, 83, 127, 0, 89, 101, 0, 109, 109, 0, 103, 107, 0, 127, 101, 0, 109, 113, 0, 101, 103, 0, 107, 109, 0, 113, 127, 0, 101

Offset: 1

Maghraoui Abdelkader, Sep 30 2015

Multiples of 3 never reach a prime because (3*x + sumdigits(3*x)) is always a multiple of 3.

			a(3)=0; a(6)=0; a(9)=0  as 3,6,9 are multiples of 3.
n=2; a0=2; a1=2+sumdigits(2)=4; a2=4+sumdigits(4)=8; a3=8+sumdigits(8)=16;
a4=16+sumdigits(16)=16+7=23; a4 is prime, so a(2)=23;
a(14)=14+(1+4=19); 19 is prime.
a(16)=16+(1+6)=23; 23 is prime.

Maghraoui Abdelkader, Table of n, a(n) for n = 1..100

Cf. A000040, A001651, A008585, A154561.

verif(n)={if((n%3)==0, print1(0,", ");return(););
b=1; a=n;
while(b<10, a=a+sumdigits(a) ;if(isprime(a),print1(a,", "); b=20))}
for(n=1, 100, verif(n);)

A261365 Prime-numbered rows of Pascal's triangle.

Original entry on oeis.org

1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 10, 5, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1, 1, 17, 136, 680, 2380, 6188, 12376, 19448, 24310, 24310, 19448, 12376, 6188, 2380, 680, 136, 17, 1, 1, 19, 171, 969, 3876, 11628, 27132, 50388, 75582, 92378, 92378, 75582, 50388, 27132, 11628, 3876, 969, 171, 19, 1

Offset: 1

Maghraoui Abdelkader, Aug 16 2015

			1,2,1;
1,3,3,1;
1,5,10,10,5,1;
1,7,21,35,35,21,7,1;
1,11,55,165,330,462,462,330,165,55,11,1;

Maghraoui Abdelkader, Table of n, a(n) for n = 1..4273

Cf. A007318, A034870, A034871.

Cf. A000040 (2nd column), A008837 (3rd column).

Table[Binomial[Prime@ n, k], {n, 8}, {k, 0, Prime@ n}] // Flatten (* Michael De Vlieger, Aug 20 2015 *)

forprime(n=2, 20, for(k=0,n,print1(binomial(n,k),", ")))

T(n,k) = binomial(prime(n), k).

A030657 Parity of digits of Pi.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Maghraoui Abdelkader

Zak Seidov, Table of n, a(n) for n = 1..10000 [a(10000)=1 corrected by _Georg Fischer_, Jun 22 2020]

Cf. A173999, A174000.

s = First[RealDigits[N[Pi, 1000]]]; aa = {}; Do[If[OddQ[s[[n]]], AppendTo[aa, 1], AppendTo[aa, 0]], {n, 1, Length[s]}]; aa (* Artur Jasinski, Mar 05 2010 *)
s= RealDigits[N[Pi, 10000]][[1]];Table[{n,Mod[s[[n]],2]},{n,10000}](* this gives b-file *)(* Zak Seidov, Oct 05 2011*)

a(n) = A000035(A000796(n)). - Omar E. Pol, Oct 26 2013

More terms from Simon Plouffe.

A030656 Pair up the numbers.

Original entry on oeis.org

1, 23, 45, 67, 89, 1011, 1213, 1415, 1617, 1819, 2021, 2223, 2425, 2627, 2829, 3031, 3233, 3435, 3637, 3839, 4041, 4243, 4445, 4647, 4849, 5051, 5253, 5455, 5657, 5859, 6061, 6263, 6465, 6667, 6869, 7071, 7273, 7475, 7677, 7879, 8081, 8283, 8485, 8687, 8889

Offset: 0

Maghraoui Abdelkader

Michel Marcus, Table of n, a(n) for n = 0..5000

Cf. A030655.

Subsequence of A005408.

[Seqint(Intseq(n+1) cat Intseq(n)): n in [0..86 by 2]];  // Bruno Berselli, Jun 18 2011

seq[n_] := FromDigits[Flatten[IntegerDigits /@ #]] & /@ Partition[Range[0, (n + 1)*2], 2];
seq[44] (* Harvey P. Dale, Jan 24 2012, edited by Robert P. P. McKone, Jan 23 2021 *)

a(n)=eval(Str(2*n,2*n+1)) \\ Charles R Greathouse IV, Jul 21 2016

def a(n): return int(str(2*n)+str(2*n+1))
print([a(n) for n in range(45)]) # Michael S. Branicky, Jan 23 2021

More terms from Erich Friedman

A030658 1 iff n-th digit of Pi is >= (n+1)st digit.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Maghraoui Abdelkader

a(n) = if A095916(n) <= 0 then 1 else 0. - Reinhard Zumkeller, Mar 12 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences related to the number Pi

Cf. A095916, A193496, A000796.

a030658 = fromEnum . (<= 0) . a095916  -- Reinhard Zumkeller, Mar 12 2015

If[First[#]>=Last[#],1,0]&/@Partition[RealDigits[Pi,10,130][[1]],2,1] (* Harvey P. Dale, Jul 25 2011 *)

import sympy as sp
length = 100
[int(x>=y) for l in [str(sp.N(sp.pi/10,length))[2:]] for x,y in zip(l[:-1],l[1:])]
# Nicholas Stefan Georgescu, Feb 27 2023

More terms from David Radcliffe

A027746 Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition.

Original entry on oeis.org

1, 2, 3, 2, 2, 5, 2, 3, 7, 2, 2, 2, 3, 3, 2, 5, 11, 2, 2, 3, 13, 2, 7, 3, 5, 2, 2, 2, 2, 17, 2, 3, 3, 19, 2, 2, 5, 3, 7, 2, 11, 23, 2, 2, 2, 3, 5, 5, 2, 13, 3, 3, 3, 2, 2, 7, 29, 2, 3, 5, 31, 2, 2, 2, 2, 2, 3, 11, 2, 17, 5, 7, 2, 2, 3, 3, 37, 2, 19, 3, 13, 2, 2, 2, 5, 41, 2, 3, 7, 43, 2, 2, 11, 3, 3, 5

Offset: 1

Maghraoui Abdelkader

n-th row has length A001222(n) (n>1).

			Triangle begins
  1;
  2;
  3;
  2, 2;
  5;
  2, 3;
  7;
  2, 2, 2;
  3, 3;
  2, 5;
  11;
  2, 2, 3;
  ...

N. J. A. Sloane, First 2048 rows of triangle, flattened
S. von Worley (?), Animated Factorization Diagrams, Oct. 2012.
Brent Yorgey, Factorization diagrams, The Math Less Traveled, Oct 05 2012.

Cf. A000027, A001222, A027748.
a(A022559(A000040(n))+1) = A000040(n).
Column 1 is A020639, columns 2 and 3 correspond to A014673 and A115561.
A281890 measures frequency of each prime in each column, with A281889 giving median values.
Cf. A175943 (partial products), A265110 (partial row products), A265111.

import Data.List (unfoldr)
a027746 n k = a027746_tabl !! (n-1) !! (k-1)
a027746_tabl = map a027746_row [1..]
a027746_row 1 = [1]
a027746_row n = unfoldr fact n where
   fact 1 = Nothing
   fact x = Just (p, x `div` p) where p = a020639 x
-- Reinhard Zumkeller, Aug 27 2011

P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1],k=1..FM[j][2]),j=1..nops(FM)) end: 1; for n from 2 to 45 do P(n) od; # yields sequence in triangular form; Emeric Deutsch, Feb 13 2005

row[n_] := Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]; Flatten[ Table[ row[n], {n, 1, 45}]] (* Jean-François Alcover, Dec 01 2011 *)

A027746_row(n,o=[1])=if(n>1,concat(apply(t->vector(t[2],i,t[1]), Vec(factor(n)~))),o) \\ Use %(n,[]) if you want the more natural [] for the first row. - M. F. Hasler, Jul 29 2015

def factors(n: int) -> list[int]:
    p = n
    L:list[int] = []
    for f in range(2, p + 1):
        if f * f > p: break
        while True:
            q, r = divmod(p, f)
            if r != 0: break
            L.append(f)
            p = q
            if p == 1: return L
    L.append(p)
    return L  # Peter Luschny, Jul 18 2024

v=[1]
for k in [2..45]: v.extend(p for (p, m) in factor(k) for _ in range(m))
print(v) # Giuseppe Coppoletta, Dec 29 2017

Product_{k=1..A001222(n)} T(n,k) = n.
From Reinhard Zumkeller, Aug 27 2011: (Start)
A001414(n) = Sum_{k=1..A001222(n)} T(n,k), n>1;
A006530(n) = T(n,A001222(n)) = Max_{k=1..A001222(n)} T(n,k);
A020639(n) = T(n,1) = Min_{k=1..A001222(n)} T(n,k). (End)

More terms from James Sellers

A036467 a(n) + a(n-1) = n-th prime.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 6, 11, 8, 15, 14, 17, 20, 21, 22, 25, 28, 31, 30, 37, 34, 39, 40, 43, 46, 51, 50, 53, 54, 55, 58, 69, 62, 75, 64, 85, 66, 91, 72, 95, 78, 101, 80, 111, 82, 115, 84, 127, 96, 131, 98, 135, 104, 137, 114, 143, 120, 149, 122, 155, 126, 157, 136, 171, 140, 173

Offset: 0

Maghraoui Abdelkader

After the initial 1,1, this sequence contains no duplicate values: terms thereafter have opposite parity, and a(n+2) > a(n). Do even and odd values trade the lead infinitely often? (We would expect them to if we model their difference as a random walk.) - Franklin T. Adams-Watters, Jan 25 2010

T. D. Noe, Table of n, a(n) for n = 0..2000

Cf. A001223, A008347.

a036467 n = a036467_list !! n
a036467_list = 1 : zipWith (-) a000040_list a036467_list
-- Reinhard Zumkeller, Nov 02 2011

[n lt 2 select 1 else NthPrime(n)-NthPrime(n-1)+Self(n-1): n in [0..65]];  // Bruno Berselli, Jun 18 2011

a[n_] := Abs[1+Sum[(-1)^(k+1)*Prime[k], {k, 2, n}]]; a /@ Range[0, 65] (* Jean-François Alcover, Apr 22 2011 *)
t={1,1};Do[AppendTo[t,NextPrime[t[[-2]]+t[[-1]]]-t[[-1]]],{n,64}];t (* Vladimir Joseph Stephan Orlovsky, Jan 26 2012 *)
Transpose[NestList[{First[#]+1,Prime[First[#]+1]-Last[#]}&,{0,1},70]][[2]] (* Harvey P. Dale, Sep 14 2012 *)

print1(t=1);forprime(p=2,1e3,print1(", ",t=p-t)) \\ Charles R Greathouse IV, Jun 18 2011

More terms from Jud McCranie

cp's OEIS Frontend

User: Maghraoui Abdelkader

A262254 a(1) = 11; for n>1, a(n) is the smallest prime that starts with the least significant digit of the previous term and has not occurred earlier.

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A261507 Fibonacci-numbered rows of Pascal's triangle. Triangle read by rows: T(n,k)= binomial(Fibonacci(n), k).

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A262463 If n is prime a(n) = n else a(n) = nextprime(reverse(n)), where "next prime" is the smallest prime >= n, see A007918.

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A261198 Start with n and repeat the map x -> x+sumdigits(x) until reaching a prime, which is a(n), or 0 if no prime is reached.

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A261365 Prime-numbered rows of Pascal's triangle.

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A030657 Parity of digits of Pi.

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A030656 Pair up the numbers.

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A030658 1 iff n-th digit of Pi is >= (n+1)st digit.