Indranil Ghosh - cp's OEIS frontend

A285098 Row sums of irregular triangle A070168.

1, 3, 23, 7, 20, 29, 124, 15, 147, 30, 117, 41, 63, 138, 296, 31, 106, 165, 231, 50, 84, 139, 281, 65, 294, 89, 40616, 166, 212, 326, 40486, 63, 377, 140, 258, 201, 259, 269, 986, 90, 40589, 126, 588, 183, 253, 327, 40455, 113, 382, 344, 514, 141, 223, 40670, 41000, 222

Offset: 1

Indranil Ghosh, Apr 17 2017

nonn

a(n) is the sum of numbers in trajectory of Terras-modified Collatz problem with first number n.

			The 5th row of irregular triangle A070168 is [5, 8, 4, 2, 1] whose sum is 20. a(5) = 5 + 8 + 4 + 2 + 1 = 20.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A033493, A070168.

a[n_]:=If[n<2, 1, If[OddQ[n], (3n + 1)/2, n/2]]; Table[-1 + Plus @@ FixedPointList[a, n], {n, 100}]

def a(n):
    if n==1: return 1
    l=[n]
    while True:
        if n%2==0: n//=2
        else: n = (3*n + 1)//2
        l.append(n)
        if n<2: break
    return sum(l)
print([a(n) for n in range(1, 101)])

A281553 Write n in binary reflected Gray code, rotate one binary place to the right and convert the code back to decimal.

Original entry on oeis.org

0, 1, 3, 1, 3, 7, 6, 2, 6, 14, 15, 7, 5, 13, 12, 4, 12, 28, 29, 13, 15, 31, 30, 14, 10, 26, 27, 11, 9, 25, 24, 8, 24, 56, 57, 25, 27, 59, 58, 26, 30, 62, 63, 31, 29, 61, 60, 28, 20, 52, 53, 21, 23, 55, 54, 22, 18, 50, 51, 19, 17, 49, 48, 16, 48, 112, 113, 49, 51, 115, 114, 50, 54, 118

Offset: 0

Indranil Ghosh, Jan 24 2017

a(n) = A003188(n), iff the Elias delta code for n contains all 1's without any zeros (see the example section).

			For n = 5, the binary reflected Gray code for n is '111'. Rotating one binary place to the right , '111' gives back '111' again. 111_2 = 7_10. So, a(5) = 7. (For n = 5, A003188(n) = 7).
For n = 15, the binary reflected Gray code for n is '1000'. Rotating one binary place to the right, '1000' gives '0100'. 100_2 = 4_10. So, a(15) = 4.

Indranil Ghosh, Table of n, a(n) for n = 0..10000

Cf. A003188, A014550, A038572.

def rotation(n):
    x=bin(n^(n/2))[2:]
    return int(x[-1]+x[:-1],2)

A281552 Write n in the Elias gamma code and sum the positions where there is a '1' followed immediately to the right by a '0', counting the leftmost digit as position 1.

Original entry on oeis.org

0, 1, 1, 2, 2, 6, 2, 3, 3, 9, 3, 8, 8, 9, 3, 4, 4, 12, 4, 11, 11, 12, 4, 10, 10, 18, 10, 11, 11, 12, 4, 5, 5, 15, 5, 14, 14, 15, 5, 13, 13, 23, 13, 14, 14, 15, 5, 12, 12, 22, 12, 21, 21, 22, 12, 13, 13, 23, 13, 14, 14, 15, 5, 6, 6, 18, 6, 17, 17, 18, 6, 16, 16, 28, 16, 17, 17, 18, 6

Offset: 1

Indranil Ghosh, Jan 24 2017

			For n = 6 , the Elias gamma code for n is '11010'. In '11010', the positions of '1' followed immediately to the right by '0' counting from left are 2 and 4. So, a(6) = 2 + 4 = 6.
For n = 10, the Elias gamma code for n is '1110010'. In '1110010', the positions of '1' followed immediately to the right by '0' counting from left are 3 and 6. So, a(10) = 3 + 6 = 9.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A049501, A171885, A281149, A281388, A281497.

def a(n):
    x= A281149(n)
    s=0
    for i in range(1,len(x)):
        if x[i-1]=="1" and x[i]=="0":
            s+=i
    return s

a(n) = A049501(A171885(n)) for n > = 1.

A281551 Prime numbers p such that the decimal representation of its Elias gamma code is also a prime.

Original entry on oeis.org

3, 23, 41, 47, 59, 89, 101, 149, 179, 227, 317, 347, 353, 383, 389, 479, 503, 599, 821, 887, 929, 977, 1019, 1109, 1229, 1283, 1319, 1511, 1571, 1619, 1667, 1709, 1733, 1787, 1847, 1889, 1907, 1913, 1931, 2207, 2309, 2333, 2357, 2399, 2417, 2459, 2609, 2753, 2789, 2909, 2963, 2999, 3203, 3257, 3299

Offset: 1

Indranil Ghosh, Jan 24 2017

			59 is in the sequence because the decimal representation of its Elias gamma code is 2011 and both 59 and 2011 are prime numbers.

Indranil Ghosh, Table of n, a(n) for n = 1..2014

Cf. A000040, A171885 (decimal representation of Elias gamma code), A281149, A281316.

import math
from sympy import isprime
def unary(n):
    return "1"*(n-1)+"0"
def elias_gamma(n):
    if n ==1:
        return "1"
    k=int(math.log(n,2))
    fp=unary(1+k)    #fp is the first part
    sp=n-2**(k)      #sp is the second part
    nb=k             #nb is the number of bits used to store sp in binary
    sp=bin(sp)[2:]
    if len(sp)

A281550 Number of 2 X 2 matrices with all elements in 0..n such that the sum of the elements is prime.

Original entry on oeis.org

0, 10, 46, 114, 234, 458, 826, 1370, 2090, 3010, 4174, 5658, 7534, 9930, 12954, 16662, 21074, 26242, 32246, 39182, 47186, 56386, 66874, 78798, 92290, 107434, 124282, 142942, 163550, 186266, 211250, 238626, 268526, 301134, 336610, 375086, 416678, 461454, 509434, 560662, 615182, 673106

Offset: 0

Indranil Ghosh, Jan 23 2017

nonn

			For n = 4, a few of the possible matrices are [0,4;2,1], [0,4;3,0], [0,4;3,4], [0,4;4,3], [1,0;0,1], [1,0;0,2], [1,0;0,4], [1,0;1,0], [1,0;1,1], [1,0;1,3], [2,2;3,0], [2,2;3,4], [2,2;4,3], [2,3;0,0], [2,3;0,2], [3,4;3,3], [3,4;4,0], [3,4;4,2], [4,0;0,1], [4,0;0,3], [4,0;1,0], ... There are 234 possibilities.
Here each of the matrices M is defined as M = [a,b;c,d] where a = M[1][1], b = M[1][2], c = M[2][1], d = M[2][2]. So, a(4) = 234.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms 0..200 from Indranil Ghosh, terms 201..3000 from Chai Wah Wu)

Cf. A210000, A281090, A281315.

a(n)=my(X=Pol(vector(n+1,i,1))+O('x^(4*n)),Y=X^4,s); forprime(p=2,4*n, s+=polcoeff(Y,p)); s \\ Charles R Greathouse IV, Feb 15 2017

from sympy import isprime
def t(n):
    s=0
    for a in range(0, n+1):
        for b in range(0, n+1):
            for c in range(0, n+1):
                for d in range(0, n+1):
                    if isprime(a+b+c+d)==True:
                        s+=1
    return s
for i in range(0, 201):
    print(str(i)+" "+str(t(i)))

a(n) = Sum_{p prime} Sum_{k=0..4} (-1)^k * binomial(4, k) * binomial(p+3-k*(n+1), 3). - David Radcliffe, Jun 13 2025

A281496 Numbers n such that the decimal representation of the Elias delta code of n is a palindromic prime.

Original entry on oeis.org

133, 18279, 19279, 24279, 25279, 30379, 263213, 323213, 333213, 374213, 384213, 404213, 33907213, 35117213, 37037213, 37237213, 37537213, 39757213, 41377213, 42088213, 42188213, 43498213, 43598213, 45518213, 46328213, 47138213, 48958213, 49258213, 49658213, 51478213, 51989213, 52289213, 54109213, 55019213, 58049213

Offset: 1

Indranil Ghosh, Jan 23 2017

Numbers n such that A003188(n) is in A002385.

a(n) is always odd.

			133 is in the sequence because the decimal representation of its Elias delta code is 14341 which is both prime and palindromic.
18279 is in the sequence because the decimal representation of its Elias delta code is 1951591 and 1951591 is both prime and palindromic.

Indranil Ghosh, Table of n, a(n) for n = 1..100
Indranil Ghosh, Python program to generate the sequence

Cf. A002385, A281225, A281382.

A281497 Write n in binary reflected Gray code and sum the positions where there is a '1' followed immediately to the left by a '0', counting the rightmost digit as position 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 2, 1, 0, 0, 1, 2, 2, 0, 0, 1, 0, 3, 4, 3, 3, 2, 2, 1, 0, 0, 1, 2, 2, 3, 3, 4, 3, 0, 1, 0, 0, 2, 2, 1, 0, 4, 5, 6, 6, 4, 4, 5, 4, 3, 4, 3, 3, 2, 2, 1, 0, 0, 1, 2, 2, 3, 3, 4, 3, 4, 5, 4, 4, 6, 6, 5, 4, 0, 1, 2, 2, 0, 0, 1, 0, 3, 4, 3, 3, 2, 2, 1, 0, 5, 6, 7, 7, 8, 8, 9, 8, 5, 6, 5, 5, 7, 7, 6

Offset: 1

Indranil Ghosh, Jan 23 2017

			For n = 12, the binary reflected Gray code for 12 is '1010'. In '1010', the position of '1' followed immediately to the left by a '0' counting from right is 2. So, a(12) = 2.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A003188, A014550, A049502, A281388.

Table[If[Length@ # == 0, 0, Total[#[[All, 1]]]] &@ SequencePosition[ Reverse@ IntegerDigits[#, 2] &@ BitXor[n, Floor[n/2]], {1, 0}], {n, 120}] (* Michael De Vlieger, Jan 23 2017, Version 10.1, after Robert G. Wilson v at A003188 *)

def G(n):
    return bin(n^(n/2))[2:]
def a(n):
    x=G(n)[::-1]
    s=0
    for i in range(1,len(x)):
        if x[i-1]=="1" and x[i]=="0":
            s+=i
    return s

a(n) = A049502(A003188(n)).

A281499 Write n in binary reflected Gray code, interchange the 1's and 0's, reverse the code and convert it back to decimal.

Original entry on oeis.org

1, 0, 0, 2, 4, 0, 2, 6, 12, 4, 0, 8, 10, 2, 6, 14, 28, 12, 4, 20, 16, 0, 8, 24, 26, 10, 2, 18, 22, 6, 14, 30, 60, 28, 12, 44, 36, 4, 20, 52, 48, 16, 0, 32, 40, 8, 24, 56, 58, 26, 10, 42, 34, 2, 18, 50, 54, 22, 6, 38, 46, 14, 30, 62, 124, 60, 28, 92, 76, 12, 44, 108, 100, 36, 4, 68, 84, 20, 52, 116, 112, 48, 16, 80, 64, 0, 32, 96, 104, 40, 8, 72, 88, 24, 56, 120

Offset: 0

Indranil Ghosh, Jan 23 2017

			For n = 11, the binary reflected Gray code for 11 is '1110' which after interchanging the 1's and 0's becomes '0001', which on reversing further gives '1000'. Now, 1000_2 = 8_10. So, a(11) = 8.

Indranil Ghosh, Table of n, a(n) for n = 0..10000

Cf. A003188, A014550, A036044.

Table[FromDigits[Reverse@ IntegerDigits[#, 2] &@ BitXor[n, Floor[n/2]] /. { 0 -> 1, 1 -> 0}, 2], {n, 0, 120}] (* Michael De Vlieger, Jan 23 2017 *)

def G(n):
    return bin(n^(n/2))[2:]
def a(n):
    s=""
    x=G(n)
    for i in x:
        if i=="1":s+="0"
        else:s+="1"
    s=s[::-1]
    return int(s,2)

a(n) = A036044(A003188(n)).

A281388 Write n in binary reflected Gray code and sum the positions where there is a '1' followed immediately to the right by a '0', counting the leftmost digit as position 1.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 1, 2, 2, 0, 3, 4, 1, 1, 1, 2, 2, 2, 6, 4, 0, 3, 3, 4, 4, 1, 5, 5, 1, 1, 1, 2, 2, 2, 7, 7, 2, 6, 6, 4, 4, 0, 5, 8, 3, 3, 3, 4, 4, 4, 9, 6, 1, 5, 5, 5, 5, 1, 6, 6, 1, 1, 1, 2, 2, 2, 8, 8, 2, 7, 7, 7, 7, 2, 8, 12, 6, 6, 6, 4, 4, 4, 10, 6, 0, 5, 5, 8, 8, 3, 9, 9, 3, 3, 3

Offset: 1

Indranil Ghosh, Jan 21 2017

			For n = 11, the binary reflected Gray code for 11 is '1110'. In '1110', the position of '1' followed immediately to the right by '0' counting from left is 3. So, a(11) = 3.
For n = 12, the binary reflected Gray code for 12 is '1010'. In '1010', the positions of '1' followed immediately to the right by '0' counting from left are 1 and 3. So, a(12) = 1 + 3 = 4.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A003188, A014550, A049501.

def g(n):
    return bin(n^(n//2))[2:]
def a(n):
    x=g(n)
    s=0
    for i in range(1, len(x)):
        if x[i-1]=="1" and x[i]=="0":
            s+=i
    return s

a(n) = A049501(A003188(n)).

A281382 Numbers n such that the decimal equivalent of the binary reflected Gray code representation of n is a palindromic prime.

Original entry on oeis.org

2, 3, 5, 6, 13, 70, 213, 217, 229, 253, 422, 426, 446, 465, 534, 541, 705, 741, 857, 869, 8441, 8481, 9190, 9221, 9293, 10210, 10349, 10453, 10929, 11049, 12006, 12281, 12329, 12721, 12793, 14109, 14282, 20578, 20934, 21009, 21629, 21701, 22810, 22866, 23221, 23421, 28705, 29397

Offset: 1

Indranil Ghosh, Jan 21 2017

Numbers n such that A003188(n) is both prime and palindromic.

			213 is in the sequence because A003188(213) = 191 and 191 is a palindromic prime.

Indranil Ghosh and Chai Wah Wu, Table of n, a(n) for n = 1..13824, first 1281 terms from Indranil Ghosh.

Cf. A002385, A003188.

Select[Range@ 30000, And[PrimeQ@ #, Reverse@ # == # &@ IntegerDigits@ #] &@ BitXor[#, Floor[#/2]] &] (* Michael De Vlieger, Mar 30 2017 *)

from sympy import isprime
def G(n):
    return int(bin(n^(n//2))[2:],2)
i=0
j=1
while j<=1281:
    if G(i)==int(str(G(i))[::-1]) and isprime(G(i))==True:
        print(j, i)
        j+=1
    i+=1

cp's OEIS Frontend

User: Indranil Ghosh

A285098 Row sums of irregular triangle A070168.

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A281553 Write n in binary reflected Gray code, rotate one binary place to the right and convert the code back to decimal.

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A281552 Write n in the Elias gamma code and sum the positions where there is a '1' followed immediately to the right by a '0', counting the leftmost digit as position 1.

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A281551 Prime numbers p such that the decimal representation of its Elias gamma code is also a prime.

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Python

A281550 Number of 2 X 2 matrices with all elements in 0..n such that the sum of the elements is prime.

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PARI

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A281496 Numbers n such that the decimal representation of the Elias delta code of n is a palindromic prime.

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A281497 Write n in binary reflected Gray code and sum the positions where there is a '1' followed immediately to the left by a '0', counting the rightmost digit as position 1.

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A281499 Write n in binary reflected Gray code, interchange the 1's and 0's, reverse the code and convert it back to decimal.

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