E.J.P. Vening - cp's OEIS frontend

A147781 1 - A147850(n).

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

Offset: 1

E.J.P. Vening, Nov 12 2008

Old name was: Odd/even binary expansion of summed, partitioned subsequent prime numbers. The sum of 8 subsequent prime numbers is multiplied, forming a new sum. The digits of each sum are added per digit, forming a new sum. The odd or even result is visualized by 0/1.

			2+3+5+7+11+13+17+19 = 1384614, 1+3+8+4+6+1+4 = 27 (odd/0).
23+29+31+37+41+43+47+53 = 5466528, 5+4+6+6+5+2+8 = 36 (even/1).
461+463+467+479+487+491+499+503 = 69230700, 6+9+2+3+7 = 27 (odd/0).
509+521+523+541+547+557+563+569 = 77862060 , 7+7+8+6+2+6 = 36 (even/1).

a(n) = 1 - A147850(n).

More terms and better name by Joerg Arndt, Aug 28 2013

A147873 A sequence based on the mechanics of A147781: b(n) = Apply[Plus, IntegerDigits( 17982*Sum_{m=0..7} Prime(n+m) )]; a(n) = 1-(b(n) mod 2).

Original entry on oeis.org

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

Offset: 1

E.J.P. Vening and Roger L. Bagula, Nov 16 2008

a[n_] := Apply[Plus, IntegerDigits[27*666*Sum[Prime[n + m], {m, 0, 7}]]]; Table[1 - Mod[a[n], 2], {n, 1, 100}]

A147850 Parity of the digits sum of Sum_{j = 8n-7..8n} prime(j).

Original entry on oeis.org

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

Offset: 1

E.J.P. Vening, Nov 15 2008

			2+3+5+7+11+13+17+19 = 1384614 (1+3+8+4+6+1+4) = 27 (1).
23+29+31+37+41+43+47+53 = 5466528 (5+4+6+6+5+2+8) = 36 (0).
461+463+467+479+487+491+499+503 = 69230700 (6+9+2+3+7) = 27 (1).
509+521+523+541+547+557+563+569 = 77862060 (7+7+8+6+2+6) = 36 (0).

Cf. A147781.

A127335 := proc(n) add(ithprime(i),i=n..n+7) ; end:
A007953 := proc(n) add(i,i=convert(n,base,10)) ; end:
A147850 := proc(n) A007953(17982*A127335(8*n-7)) mod 2 ; end:
for n from 1 to 200 do printf("%a,",A147850(n)) ; od: # R. J. Mathar, Jan 06 2009

a(n) = 1 - A147781(n).

a(n) = A007953(17982*A127335(8*n-7)) mod 2. - R. J. Mathar, Jan 06 2009

More terms from R. J. Mathar, Jan 06 2009

A152010 Sum of digits of A127335(n).

Original entry on oeis.org

14, 17, 7, 6, 9, 3, 6, 9, 7, 7, 12, 12, 10, 15, 6, 15, 15, 8, 12, 21, 12, 21, 10, 18, 24, 19, 21, 4, 12, 6, 11, 15, 12, 18, 6, 12, 9, 13, 13, 12, 17, 11, 14, 11, 21, 11, 18, 10, 14, 20, 8, 16, 4, 10, 16, 12, 15, 14, 15, 15, 17, 18, 11, 21, 15, 15, 17, 20, 12, 18, 3, 15, 20, 9, 21, 10, 6

Offset: 1

E.J.P. Vening and Roger L. Bagula, Nov 19 2008

Table[Apply[Plus, IntegerDigits[Sum[Prime[n + m], {m, 0, 7}]]], {n, 1, 100}]

a(n) = A007953( A127335(n) ). [Joerg Arndt, Aug 28 2013]

A117722 a(n) = A000045(A003622(n)).

Original entry on oeis.org

1, 3, 8, 34, 144, 377, 1597, 4181, 17711, 75025, 196418, 832040, 3524578, 9227465, 39088169, 102334155, 433494437, 1836311903, 4807526976, 20365011074, 53316291173, 225851433717, 956722026041, 2504730781961, 10610209857723

Offset: 0

E.J.P. Vening, Apr 13 2006

Old name was: Fibonacci numbers from sequence A000045 aligned with instances of '1' in the Fibonacci word sequence A003849.

Cf. A000045, A003622, A003849.

a(n)=fibonacci((n+1)*(3+sqrt(5))\2-1) \\ Charles R Greathouse IV, Apr 19 2015; shifted by Georg Fischer, Jun 22 2022

a(n) = A000045( A003622(n) ). - Joerg Arndt, Aug 28 2013

More terms from R. J. Mathar, Jan 21 2008

Better name by Joerg Arndt, Aug 28 2013

A109755 Row and column sums of A001223 ( Differences between consecutive primes - see example ).

Original entry on oeis.org

1, 3, 9, 21, 51, 117, 251, 523, 1077, 2207, 4521, 9283

Offset: 0

E.J.P. Vening, Aug 11 2005

Resembles the first 6 entries of sequence A005254 (number of weighted voting procedures).

			Column A - Primes (for reference)
Column B - Distance between consecutive primes
A B
2 1
3 2 3
5 2
7 4 6 9
11 2
13 4 6
17 2
19 4 6 12 21
23 6
29 2 8
31 6
37 4 10 18 30 51
41 2
43 4 6
47 6
53 6 12 18 36 66 117
59 2
61 6 8
67 4
71 2 6 14 32 68 134 251

Cf. A001223, A037254, A005254.

A109635 Sum of prime(n) and n-th digit of Pi after the decimal point.

Original entry on oeis.org

3, 7, 6, 12, 20, 15, 23, 24, 26, 34, 39, 46, 48, 52, 50, 55, 62, 69, 71, 77, 75, 85, 87, 92, 100, 109, 106, 109, 116, 122, 132, 131, 139, 147, 157, 155, 158, 172, 174, 174, 185, 190, 194, 202, 206, 202, 218, 228, 228, 229, 238, 247, 243, 251, 266, 270, 273, 280

Offset: 1

E.J.P. Vening, Aug 03 2005

In some cases n resembles the prime number in the subsequent prime row. Some slight intervals can be observed: for example 599(593+6) to 641(641+0) to 691(683+8) to 743(743+0) to 809(811+2), each 8 rows apart and 3 instances 10 rows apart.

			The first column gives the primes, the second column the digits of Pi and the third column their sum.
2 1 3
3 4 7
5 1 6
7 5 12
...

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Total/@With[{nn=60},Thread[{Prime[Range[nn]],Rest[RealDigits[Pi,10,nn+1][[1]]]}]] (* Harvey P. Dale, Apr 25 2020 *)

a(n) ~ n log n.

A109652 a(n) = prime(A000201(n)).

Original entry on oeis.org

2, 5, 7, 13, 19, 23, 31, 37, 43, 53, 59, 67, 73, 79, 89, 97, 103, 109, 113, 131, 137, 149, 157, 163, 173, 181, 191, 197, 199, 223, 229, 233, 241, 257, 263, 271, 277, 283, 307, 311, 317, 331, 347, 353, 359, 373, 383, 389, 401, 409, 421, 433, 439, 449, 457, 463

Offset: 1

E.J.P. Vening, Aug 05 2005

Cf. A003849, A000201, A000040.

a(n) = prime(n*(1+sqrt(5))\2); \\ Michel Marcus, Aug 28 2013

from math import isqrt
from sympy import prime
def A109652(n): return prime(n+isqrt(5*n**2)>>1) # Chai Wah Wu, Aug 10 2022

a(n) ~ phi*n*log n, where phi = (1+sqrt(5))/2. - Charles R Greathouse IV, Apr 19 2015

Better description from Joseph Biberstine and Graeme McRae, Aug 05 2006

A109654 Primes A000040(i) such that A003849(i-1) = 1.

Original entry on oeis.org

3, 11, 17, 29, 41, 47, 61, 71, 83, 101, 107, 127, 139, 151, 167, 179, 193, 211, 227, 239, 251, 269, 281, 293, 313, 337, 349, 367, 379, 397, 419, 431, 443, 461, 467, 491, 503, 523, 557, 569, 587, 599, 613, 631, 643, 659, 677, 691, 719, 733, 751, 769, 787, 811, 823, 839, 859, 877, 887

Offset: 1

E.J.P. Vening, Aug 05 2005

Original sequence name: Primes aligned with instances of '1' in the infinite Fibonacci word sequence A003849.

Cf. A003849.

a(n) = prime(floor(n*((sqrt(5)+1)/2)^2)) /* Georg Fischer, Sep 21 2024 */

a(n) = A000040(A003622(n) + 1) = prime(floor(n*phi^2)), where phi = (1+sqrt(5))/2 is the golden ration. - Charles R Greathouse IV and Danny Rorabaugh, Apr 21 2015

Name clarified by Danny Rorabaugh, Apr 19 2015

a(21)=251 inserted and more terms from Georg Fischer, Sep 21 2024

A102022 First differences of A109652.

Original entry on oeis.org

3, 2, 6, 6, 4, 8, 6, 6, 10, 6, 8, 6, 6, 10, 8, 6, 6, 4, 18, 6, 12, 8, 6, 10, 8, 10, 6, 2, 24, 6, 4, 8, 16, 6, 8, 6, 6, 24, 4, 6, 14, 16, 6, 6, 14, 10, 6, 12, 8, 12, 12, 6, 10, 8, 6, 16, 8, 12, 10, 12, 20, 6, 16, 8, 6, 16, 8, 6, 10, 2, 22, 6, 6, 8, 12, 10, 18, 8, 18, 12, 4, 14, 4, 12, 24, 12, 12, 6, 2, 24, 4

Offset: 1

E.J.P. Vening, Jun 19 2007

Cf. A109652.

Differences[Table[Prime[Floor[n*GoldenRatio]], {n, 1, 92}]] (* James C. McMahon, Jan 06 2024 *)

Terms corrected and more terms added, Joerg Arndt, Aug 28 2013

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User: E.J.P. Vening

A147781 1 - A147850(n).

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A147873 A sequence based on the mechanics of A147781: b(n) = Apply[Plus, IntegerDigits( 17982*Sum_{m=0..7} Prime(n+m) )]; a(n) = 1-(b(n) mod 2).

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A147850 Parity of the digits sum of Sum_{j = 8*n-7..8*n} prime(j).

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Maple

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A152010 Sum of digits of A127335(n).

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A117722 a(n) = A000045(A003622(n)).

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PARI

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A109755 Row and column sums of A001223 ( Differences between consecutive primes - see example ).

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A109635 Sum of prime(n) and n-th digit of Pi after the decimal point.

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A109652 a(n) = prime(A000201(n)).

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A109654 Primes A000040(i) such that A003849(i-1) = 1.

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A147850 Parity of the digits sum of Sum_{j = 8n-7..8n} prime(j).