A023506 -id:A023506 - cp's OEIS frontend

A087207 A binary representation of the primes that divide a number, shown in decimal.

0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258

Offset: 1

Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017
From Antti Karttunen, Jun 18 & 20 2017: (Start)
A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).
For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.
Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.
If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.
(End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

			a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from _T. D. Noe_]
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

For partial sums see A288566.
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332.
Sequences with related definitions: A007947, A008472, A027748, A048675, A248663, A276379 (same sequence shown in base 2), A288569, A289271, A297404.
Cf. A286608 (numbers n for which a(n) < n), A286609 (n for which a(n) > n), and also A286611, A286612.
A003986, A003961, A059896 are used to express relationship between terms of this sequence.
Related to A267116 via A225546.
Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17).
A048675 gives binary rank of prime indices.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices (listed A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- sum A029931, product A096111
- max A029837 or A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.

a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
-- Reinhard Zumkeller, Jul 16 2013

a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)

a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[,1]), v = concat(v, vecsearch(f[,1], p)!=0);); fromdigits(Vecrev(v), 2));} \\ Michel Marcus, Jun 05 2017

A087207(n)=vecsum(apply(p->1<M. F. Hasler, Jun 23 2017

from sympy import factorint, primepi
def a(n):
    return sum(2**primepi(i - 1) for i in factorint(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

(definec (A087207 n) (if (= 1 n) 0 (+ (A000079 (+ -1 (A055396 n))) (A087207 (A028234 n))))) ;; This uses memoization-macro definec
(define (A087207 n) (A048675 (A007947 n))) ;; Needs code from A007947 and A048675. - Antti Karttunen, Jun 19 2017

Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003
a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003
A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003
G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start)
a(n) = A048675(A007947(n)).
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)).
A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.]
A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.]
a(A293214(n)) = A218403(n).
a(A293442(n)) = A267116(n).
A069010(a(n)) = A287170(n).
A007088(a(n)) = A276379(n).
A038374(a(n)) = A300820(n) for n >= 1.
(End)
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)).
a(A003961(n)) = 2*a(n).
a(n^2) = a(n).
a(n) = A267116(A225546(n)).
a(A225546(n)) = A267116(n).
(End)

More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003

Name clarified by Antti Karttunen, Jun 18 2017

A372429 Sum of binary indices of prime(n). Sum of positions of ones in the reversed binary expansion of prime(n).

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 6, 8, 11, 13, 15, 10, 11, 13, 16, 15, 18, 19, 10, 13, 12, 17, 15, 17, 14, 17, 19, 20, 21, 19, 28, 11, 13, 15, 17, 19, 21, 17, 20, 22, 22, 23, 29, 16, 19, 21, 23, 30, 24, 25, 26, 31, 27, 33, 10, 15, 17, 19, 18, 19, 21, 19, 23, 26, 25, 28, 23

Offset: 1

Gus Wiseman, May 02 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Do 2, 3, 4, 7, 12, 14 appear just once?
Are 1, 5, 9 missing?
The above questions hold true up to n = 10^6. - John Tyler Rascoe, May 21 2024

			The primes together with their binary expansions and binary indices begin:
   2:      10 ~ {2}
   3:      11 ~ {1,2}
   5:     101 ~ {1,3}
   7:     111 ~ {1,2,3}
  11:    1011 ~ {1,2,4}
  13:    1101 ~ {1,3,4}
  17:   10001 ~ {1,5}
  19:   10011 ~ {1,2,5}
  23:   10111 ~ {1,2,3,5}
  29:   11101 ~ {1,3,4,5}
  31:   11111 ~ {1,2,3,4,5}
  37:  100101 ~ {1,3,6}
  41:  101001 ~ {1,4,6}
  43:  101011 ~ {1,2,4,6}
  47:  101111 ~ {1,2,3,4,6}
  53:  110101 ~ {1,3,5,6}
  59:  111011 ~ {1,2,4,5,6}
  61:  111101 ~ {1,3,4,5,6}
  67: 1000011 ~ {1,2,7}
  71: 1000111 ~ {1,2,3,7}
  73: 1001001 ~ {1,4,7}
  79: 1001111 ~ {1,2,3,4,7}

John Tyler Rascoe, Table of n, a(n) for n = 1..9438

The number instead of sum of binary indices is A014499.
Restriction of A029931 (sum of binary indices) to the primes A000040.
The maximum instead of sum of binary indices is A035100, see also A023506.
Row-sums of A372471.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020.
A056239 adds up prime indices.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A372427 lists numbers whose binary and prime indices have the same sum.
Cf. A005940, A030101, A059893, A071814, A230877, A231204, A358134, A359359, A359401, A372430-A372437, A372441.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[bix[Prime[n]]],{n,100}]

a(n) = A029931(prime(n)).

A372471 Irregular triangle read by rows where row n lists the binary indices of the n-th prime number.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 07 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			We have prime(12) = (2^1 + 2^3 + 2^6)/2, so row 12 is (1,3,6).
Each prime followed by its binary indices:
   2: 2
   3: 1 2
   5: 1 3
   7: 1 2 3
  11: 1 2 4
  13: 1 3 4
  17: 1 5
  19: 1 2 5
  23: 1 2 3 5
  29: 1 3 4 5
  31: 1 2 3 4 5
  37: 1 3 6
  41: 1 4 6
  43: 1 2 4 6
  47: 1 2 3 4 6

Row lengths are A014499.
Second column is A023506(n) + 1.
Final column is A035100.
Prime-indexed rows of A048793.
Row-sums are A372429, restriction of A029931 (sum of binary indices).
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020.
A070939 gives length of binary expansion.
Cf. A000040, A005940, A056239, A071814, A096111, A191232, A230877, A231204, A372427-A372442.

Table[Join@@Position[Reverse[IntegerDigits[Prime[n],2]],1],{n,15}]

A057023 Largest odd factor of (n-th prime-1); k when n-th prime is written as k*2^m+1 [with k odd].

Original entry on oeis.org

1, 1, 1, 3, 5, 3, 1, 9, 11, 7, 15, 9, 5, 21, 23, 13, 29, 15, 33, 35, 9, 39, 41, 11, 3, 25, 51, 53, 27, 7, 63, 65, 17, 69, 37, 75, 39, 81, 83, 43, 89, 45, 95, 3, 49, 99, 105, 111, 113, 57, 29, 119, 15, 125, 1, 131, 67, 135, 69, 35, 141, 73, 153, 155, 39, 79, 165, 21, 173, 87

Offset: 1

Henry Bottomley, Jul 24 2000

			a(5)=5 because 5th prime is 11 and 11=5*2^1+1.

Cf. A057024.

Table[p = Prime[n]; ie = IntegerExponent[p - 1, 2]; (p - 1)/2^ie, {n, 100}] (* Zak Seidov, Mar 25 2014 *)

lista(nn) = forprime (p=2, nn, my(m = p-1); print1(m >> valuation(m, 2), ", ")); \\ Michel Marcus, Jan 30 2016

a(n) = {my(m = prime(n) - 1); m >> valuation(m, 2);} \\ Michel Marcus, Jan 30 2016

a(n) = A000265(A000040(n)-1) = A000265(A006093(n)) =(A000040(n)-1)/A007814(A000040(n)-1) = A006093(n)/A023506(n).

A023512 Exponent of 2 in prime factorization of prime(n) + 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

2^a(n) is the largest power of 2 dividing (prime(n)+1).

By Dirichlet's theorem on arithmetic progressions, the asymptotic density of primes p such that p == 2^k-1 (mod 2^k) within all the primes is 1/2^(k-1), for k >= 1. This is also the asymptotic density of terms in this sequence that are >= k. Therefore, the asymptotic density of the occurrences of k in this sequence is d(k) = 1/2^(k-1) - 1/2^k = 1/2^k, and the asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 2. - Amiram Eldar, Mar 14 2025

			a(9) = 3 because the 9th prime is 23 and the largest power of 2 dividing 24 is 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2261 from K. G. Stier)

Cf. A007814, A008864, A023506, A239114.

[Valuation(NthPrime(n)+1, 2): n in [1..110]]; // Bruno Berselli, Aug 05 2013

with(numtheory): a:=proc(n) local div,s,j,c: div:=divisors(1+ithprime(n)): s:=nops(div): for j from 1 to s do if type(simplify(log[2](div[j])), integer)=true then c[j]:=simplify(log[2](div[j])) else c[j]:=0 fi od: max(seq(c[j],j=1..s)) end: seq(a(n),n=1..120); # most probably not the simplest Maple program - Emeric Deutsch, Jul 20 2005

Join[{0}, Table[FactorInteger[Prime[n] + 1][[1]][[2]], {n, 2, 100}]] (* Clark Kimberling, Oct 01 2013 *)
IntegerExponent[Prime[Range[100]] + 1, 2] (* Zak Seidov, Apr 25 2014 *)

a(n)=valuation(prime(n)+1,2);
vector(100,n,a(n)) \\ Joerg Arndt, Mar 11 2014

a(n) = A007814(A008864(n)). - Amiram Eldar, Mar 14 2025

Corrected by Yasutoshi Kohmoto, Feb 25 2005

Edited by N. J. A. Sloane, Dec 23 2006

A280954 Number of integer partitions of n using predecessors of prime numbers.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 11, 11, 17, 17, 26, 26, 37, 37, 53, 53, 74, 74, 101, 101, 137, 137, 183, 183, 240, 240, 314, 314, 406, 406, 520, 520, 662, 662, 837, 837, 1049, 1049, 1311, 1311, 1627, 1627, 2008, 2008, 2469, 2469, 3021, 3021, 3678, 3678, 4466, 4466

Offset: 0

Gus Wiseman, Jan 11 2017

nonn

The predecessors of prime numbers are {1, 2, 4, 6, 10, 12, ...} = A006093.

			The partitions for n=0..7 are:
(),
(1),
(2), (11),
(21),(111),
(4), (22), (211), (1111),
(41),(221),(2111),(11111),
(6), (42), (411), (222), (2211), (21111), (111111),
(61),(421),(4111),(2221),(22111),(211111),(1111111).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A006093, A023506.

Even (and odd) bipartition gives A280962.

b:= proc(n, i) option remember; `if`(n=0 or i=2, 1,
      b(n, prevprime(i))+`if`(i-1>n, 0, b(n-i+1, i)))
    end:
a:= n-> b(n, nextprime(n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 11 2017
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(`if`(
      isprime(d+1), d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nn=60;invser=Series[Product[1-x^(Prime[n]-1),{n,PrimePi[nn+1]}],{x,0,nn}];
CoefficientList[1/invser,x]

A372688 Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.

Original entry on oeis.org

0, 0, 2, 2, 1, 3, 3, 6, 3, 6, 9, 20, 13, 22, 22, 45, 47, 70, 75, 100, 107, 132, 157, 202, 229, 302, 396, 495, 536, 699, 820, 962, 1193, 1507, 1699, 2064, 2455, 2945, 3408, 4026, 4691, 5749, 6670, 7614, 9127, 10930, 12329, 14370, 16955, 19961, 22950, 26574, 30941

Offset: 0

Gus Wiseman, May 16 2024

nonn

Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

			The partition (3,2,1) has rank 2^(3-1) + 2^(2-1) + 2^(1-1) = 7, which is prime, so (3,2,1) is counted under a(6).
The a(2) = 2 through a(10) = 9 partitions:
(2)   (21)   (31)  (221)    (51)    (421)      (431)   (441)     (91)
(11)  (111)        (2111)   (321)   (2221)     (521)   (3321)    (631)
                   (11111)  (3111)  (4111)     (5111)  (4221)    (721)
                                    (22111)            (33111)   (3331)
                                    (211111)           (42111)   (7111)
                                    (1111111)          (411111)  (32221)
                                                                 (322111)
                                                                 (3211111)
                                                                 (31111111)

For all positive integers (not just prime) we get A000041.
For even instead of prime we have A087787, strict A025147, odd A096765.
These partitions have Heinz numbers A277319.
The strict case is A372687, ranks A372851.
The version counting only distinct parts is A372887, ranks A372850.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A048793 and A272020 (reverse) list binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
A058698 counts partitions of prime numbers, strict A064688.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372689.

Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^#]/2]&]],{n,0,30}]

A372689 Positive integers whose binary indices (positions of ones in reversed binary expansion) sum to a prime number.

Original entry on oeis.org

2, 3, 4, 6, 9, 11, 12, 16, 18, 23, 26, 29, 33, 38, 41, 43, 44, 48, 50, 55, 58, 61, 64, 69, 71, 72, 74, 79, 81, 86, 89, 91, 92, 96, 101, 103, 104, 106, 111, 113, 118, 121, 131, 132, 134, 137, 142, 144, 149, 151, 152, 154, 159, 163, 164, 166, 169, 174, 176, 181

Offset: 1

Gus Wiseman, May 18 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Note the function taking a set s to its binary rank Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

			The terms together with their binary expansions and binary indices begin:
   2:      10 ~ {2}
   3:      11 ~ {1,2}
   4:     100 ~ {3}
   6:     110 ~ {2,3}
   9:    1001 ~ {1,4}
  11:    1011 ~ {1,2,4}
  12:    1100 ~ {3,4}
  16:   10000 ~ {5}
  18:   10010 ~ {2,5}
  23:   10111 ~ {1,2,3,5}
  26:   11010 ~ {2,4,5}
  29:   11101 ~ {1,3,4,5}
  33:  100001 ~ {1,6}
  38:  100110 ~ {2,3,6}
  41:  101001 ~ {1,4,6}
  43:  101011 ~ {1,2,4,6}
  44:  101100 ~ {3,4,6}
  48:  110000 ~ {5,6}
  50:  110010 ~ {2,5,6}
  55:  110111 ~ {1,2,3,5,6}
  58:  111010 ~ {2,4,5,6}
  61:  111101 ~ {1,3,4,5,6}

Numbers k such that A029931(k) is prime.
Union of prime-indexed rows of A118462.
For even instead of prime we have A158704, odd A158705.
For prime indices instead of binary indices we have A316091.
The prime case is A372885, indices A372886.
A000040 lists the prime numbers, A014499 their binary indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A058698 counts partitions of prime numbers, strict A064688.
A372471 lists binary indices of primes, row-sums A372429.
A372687 counts strict partitions of prime binary rank, counted by A372851.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A035100, A071814, A096111, A023506, A277319, A372688, A372850, A372887.

Select[Range[100],PrimeQ[Total[First /@ Position[Reverse[IntegerDigits[#,2]],1]]]&]

A334006 Triangle read by rows: T(n,k) = (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)) for nonnegative k < n, n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 7, 1, 3, 1, 3, 1, 1, 4, 1, 5, 1, 5, 1, 5, 1, 9, 1, 3, 1, 3, 1, 7, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 6, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 13, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 7, 1, 3, 1, 3

Offset: 1

Juri-Stepan Gerasimov, Apr 12 2020

If the sum of proper divisors of q in row q <= q, then q are 1, 2, 3, 4, 5, 8, 16, 17, 32, 64, 128, 256, 257, ...(union of Fermat primes and powers of 2).

			Triangle T(n,k) begins:
  n\k| 0   1  2  3  4   5  6  7  8   9 10 11 12  13 14 15 16
  ---+------------------------------------------------------
   1 | 1;
   2 | 1,  1;
   3 | 1,  3, 1;
   4 | 1,  2, 1, 3;
   5 | 1,  5, 1, 1, 1;
   6 | 1,  3, 1, 3, 1,  3;
   7 | 1,  7, 1, 3, 1,  3, 1;
   8 | 1,  4, 1, 5, 1,  5, 1, 5;
   9 | 1,  9, 1, 3, 1,  3, 1, 7, 1;
  10 | 1,  5, 1, 1, 1,  5, 1, 1, 1,  5;
  11 | 1, 11, 1, 3, 1,  3, 1, 3, 1,  3, 1;
  12 | 1,  6, 1, 9, 1,  9, 1, 9, 1,  9, 1, 9;
  13 | 1, 13, 1, 1, 1,  5, 1, 1, 1,  5, 1, 1, 1;
  14 | 1,  7, 1, 3, 1,  3, 1, 7, 1,  3, 1, 3, 1, 7;
  15 | 1, 15, 1, 3, 1, 15, 1, 3, 1, 15, 1, 3, 1, 15, 1;
  16 | 1,  8, 1, 5, 1,  9, 1, 5, 1,  9, 1, 5, 1,  9, 1, 5;
  17 | 1, 17, 1, 1, 1,  1, 1, 1, 1,  1, 1, 1, 1,  1, 1, 1, 1;
  ...
For (n, k) = (7, 3), there are three nonnegative values of m < n such that m^3 == m (mod 7) (namely 0, 1, and 6) and one nonnegative value of m < n such that -m^3 == m (mod 7) (namely 0), so T(7,3) = 3/1 = 3.

Cf. A000012, A000079, A002322, A010684, A019434, A023506, A026741, A182816, A333570, A337454, A337632, A337633, A337820, A337910.

[[#[m: m in [0..n-1] | m^k mod n eq m]/#[m: m in [0..n-1] | -m^k mod n eq m]: k in [0..n-1]]: n in [1..17]];

T(n, k) = sum(m=0, n-1, Mod(m, n)^k == m)/sum(m=0, n-1, -Mod(m, n)^k == m);
matrix(7, 7, n, k, k--; if (k>=n, 0, T(n,k))) \\ to see the triangle \\ Michel Marcus, Apr 17 2020

Name corrected by Peter Kagey, Sep 12 2020

cp's OEIS Frontend

A087207 A binary representation of the primes that divide a number, shown in decimal.

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A372429 Sum of binary indices of prime(n). Sum of positions of ones in the reversed binary expansion of prime(n).

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A372471 Irregular triangle read by rows where row n lists the binary indices of the n-th prime number.

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A057023 Largest odd factor of (n-th prime-1); k when n-th prime is written as k*2^m+1 [with k odd].

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A023512 Exponent of 2 in prime factorization of prime(n) + 1.

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A280954 Number of integer partitions of n using predecessors of prime numbers.

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A372688 Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.

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