Irina Gerasimova - cp's OEIS frontend

A242667 Number of ways of representing n as the sum of one or more consecutive squarefree numbers.

1, 1, 2, 0, 2, 2, 1, 1, 0, 2, 3, 0, 2, 2, 1, 1, 3, 1, 1, 0, 3, 1, 3, 2, 0, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 1, 0, 2, 1, 2, 0, 4, 0, 3, 2, 3, 0, 3, 2, 1, 1, 2, 3, 2, 0, 3, 3, 3, 3, 1, 1, 1, 1, 2, 3, 2, 2

Offset: 1

Irina Gerasimova, May 20 2014

nonn

			a(6)=2 because n=6 itself is already a squarefree number (sum of 1 term), and 6 can in addition be written as A005117(1)+ A005117(2)+A005117(3), a sum of 3 consecutive squarefree numbers.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005117.

A242667 := proc(n)
    a := 0 ;
    for i from 1 do
        if A005117(i) > n then
            return a;
        end if;
        for k from i do
            su := add(A005117(s),s=i..k) ;
            if su = n then
                a := a+1 ;
            elif su > n then
                break;
            fi ;
        end do:
    end do:
end proc:
seq(A242667(n),n=1..80) ; # R. J. Mathar, Jun 12 2014
# Alternative:
N:= 1000:# to get the first N entries
A005117:= select(numtheory:-issqrfree,[$1..N]):
M:= nops(A005117);
A:= Array(1..N):
t0:= 0:
for n from 1 to M-1 do
  t0:= t0 + A005117[n];
  t:= t0;
  for i from 1 while t <= N do
     A[t] := A[t]+1;
     if n+i > M then break fi;
     t:= t + A005117[n+i]-A005117[i];
  od;
od:
seq(A[i],i=1..N); # Robert Israel, Jun 25 2014

With[{N = 100}, (* to get the first N entries *)
A005117 = Select[Range[N], SquareFreeQ];
M = Length[A005117];
A = Table[0, {N}];
t0 = 0;
For[n = 1, n <= M-1, n++,
   t0 = t0+A005117[[n]];
   t = t0;
   For[i = 1, t <= N, i++,
      A[[t]] = A[[t]]+1;
      If[n+i > M, Break[]];
      t = t + A005117[[n+i]] - A005117[[i]]]
   ]
];
A (* Jean-François Alcover, Feb 07 2023, after Robert Israel *)

A240546 a(n) = prime(n+1)^n mod prime(n).

Original entry on oeis.org

1, 1, 3, 4, 10, 1, 9, 5, 16, 9, 26, 10, 33, 1, 2, 49, 33, 3, 35, 48, 3, 32, 62, 64, 4, 20, 8, 62, 93, 83, 64, 41, 68, 79, 138, 125, 88, 56, 4, 169, 72, 36, 40, 144, 73, 140, 63, 120, 24, 218, 67, 48, 58, 194, 126, 54, 74, 223, 74, 59, 176, 161, 280, 208, 215, 236, 82, 141, 139, 344, 7

Offset: 1

Irina Gerasimova, Apr 07 2014

prime(k+1)^k mod prime(k) = k: 1, 3, 4, 76, 7743, ... .

			a(5) = prime(5+1)^5 mod prime(5) = 13^5 mod 11 = 10.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A038702.

[(NthPrime(n+1)^n mod NthPrime(n)): n in [1..100]]; // Juri-Stepan Gerasimov, Apr 07 2014

a:= n-> ithprime(n+1) &^n mod ithprime(n):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 30 2014

Table[Mod[Prime[n + 1]^n, Prime[n]], {n, 80}] (* Alonso del Arte, Apr 11 2014 *)
Table[PowerMod[Prime[n+1],n,Prime[n]],{n,80}] (* Harvey P. Dale, Jun 11 2019 *)

a(n)=my(p=prime(n)); lift(Mod(nextprime(p+1),p)^n) \\ Charles R Greathouse IV, Apr 08 2014

A237417 Numbers that are the product of an odiousfree number and an evilfree number.

Original entry on oeis.org

3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 23, 24, 27, 29, 30, 33, 34, 35, 36, 39, 40, 42, 43, 45, 46, 48, 51, 53, 54, 55, 57, 58, 60, 63, 65, 66, 68, 70, 71, 72, 78, 80, 83, 84, 85, 86, 89, 90, 92, 93, 95, 96, 99, 101, 102, 105, 106, 108, 110, 111, 113, 114, 116, 117, 119, 120, 123, 126, 129

Offset: 1

Irina Gerasimova, Feb 23 2014, following a suggestion from Juri-Stepan Gerasimov

Odiousfree*evilfree numbers: numbers of the form odiousfree*evilfree.
Subsequence of this sequence (A237417): numbers that are not the products of two odious numbers or the products of two evil numbers: 3, 5, 6, 10, 12, 17, 20, 23, 24, 29, 33, 34, 39, 40, 43, 46, 48, 57, 58, 63, 65, 66, 68, 71, 78, 80, 83, 86, 89, 92, 95, 101, 105, 106, 111, 113, 114, 116, 119,...
Putting the 1 aside in A093688, these could be called odiousfree numbers, and are a subsequence of A001969. A093696 would be the evilfree numbers then, and are a subsequence of A000069.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A093688, A093696.

N:= 200: # to get all terms <= N
Ofree:= {$2..N}: Efree:= {$1..N/3}:
for n from 2 to N do
  t:= convert(convert(n,base,2),`+`) mod 2;
  if t = 0 then Efree:= Efree minus {seq(i,i=n..N/3,n)}
  else Ofree:= Ofree minus {seq(i,i=n..N,n)}
  fi
od:
sort(convert(select(`<=`,{seq(seq(s*t,s=Ofree),t=Efree)},N),list)); # Robert Israel, May 09 2019

odFreeQ[n_] := AllTrue[Rest @ Divisors[n], EvenQ[DigitCount[#, 2, 1]] &]; evFreeQ[n_] := AllTrue[Divisors[n], OddQ[DigitCount[#, 2, 1]] &]; m = 100; o = Select[Range[2, m], odFreeQ]; e = Select[Range[m], evFreeQ]; Union @ Select[Times @@@ Tuples[{o, e}], # <= m &] (* Amiram Eldar, Oct 16 2020 *)

isA093696(n)= fordiv(n, d, if(hammingweight(d)%2==0, return(0))); 1;
isA093688(n)= if (n==1, 0, sumdiv(n, d, hammingweight(d)%2)==1);
lista(nn) = {vn = vector(2*nn, i, i); vof = select(n->isA093696(n), vn); vef = select(n->isA093688(n), vn); vp = []; for (i = 1, #vof, for (j = 1, #vef, vp = Set(concat(vp, vof[i]*vef[j])););); for (i = 1, #vp, if (vp[i] <= nn, print1(vp[i], ", ")););} \\ Michel Marcus, Mar 05 2014

a(n) = A093688(k+1)*A093696(m).

Definition corrected by Jon E. Schoenfield, Feb 26 2014

A236695 The n-th prime with n 0-bits in its binary expansion.

Original entry on oeis.org

2, 43, 41, 139, 269, 773, 1049, 2309, 4357, 8737, 16673, 34819, 66569, 139393, 279553, 589829, 1051649, 2621569, 4260097, 9437189, 17039489, 33817601, 67649537, 167903233, 269484097, 545260033, 1074267137, 2155872769, 4311760897, 12884901893, 17184063521

Offset: 1

Irina Gerasimova, Jan 30 2014

			Primes p(k) such that
A035103(p(k)) = 0: 3, 7, 31, 127, 8191,...
A035103(p(k)) = 1: 2, 5, 11, 13, 23, 29,...
A035103(p(k)) = 2: 19, 43, 53, 79, 103, 107,...
A035103(p(k)) = 3: 17, 37, 41, 71, 83, 89, 101,...
A035103(p(k)) = 4: 67, 73, 97, 139, 149, 163,...
A035103(p(k)) = 5: 131, 137, 193, 263, 269, 277,...

Alois P. Heinz, Table of n, a(n) for n = 1..500 (first 40 terms from Charles R Greathouse IV)

Cf. A066195 (least prime having n zeros in binary), A236513 (the n-th prime with n 1-bits in its binary expansion).

nz(n)=#binary(n)-hammingweight(n)
a(n)=my(k=n);forprime(p=2,,if(nz(p)==n&&k--==0,return(p))) \\ Charles R Greathouse IV, Feb 04 2014

New name from Ralf Stephan and Charles R Greathouse IV, Feb 04 2014
a(14)-a(27) from Charles R Greathouse IV, Feb 04 2014
a(28)-a(31) from Giovanni Resta, Feb 04 2014

A236514 Primes with a binary weight greater than or equal to the binary weight of their squares.

Original entry on oeis.org

2, 3, 7, 23, 31, 47, 79, 127, 157, 191, 223, 317, 367, 379, 383, 479, 727, 751, 887, 1087, 1151, 1277, 1279, 1451, 1471, 1531, 1663, 1783, 1789, 1951, 2297, 2557, 2927, 3067, 3259, 3319, 3581, 3583, 3967, 4253, 4349, 5119, 5231, 5503, 5807, 5821, 6079, 6143, 6271, 6653, 6871, 6911, 7039, 7103, 7151

Offset: 1

Irina Gerasimova, Jan 27 2014

Primes p such that A000120(p) = A000120(p^2): 2, 3, 7, 31, 79, 127, 157, 317, 379, 751, 1087, 1151, 1277, 1279,...

			2 is in this sequence because 2 is 10 in binary representation, and it has as many 1s as its square 4, which is 100 in binary.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A077436, A094694.

bc[n_] := DigitCount[n, 2][[1]]; Select[Range[7151], PrimeQ[#] && bc[#] >= bc[#^2] &] (* Giovanni Resta, Jan 28 2014 *)
Select[Prime[Range[1000]], DigitCount[#, 2, 1] >= DigitCount[#^2, 2, 1] &] (* Alonso del Arte, Jan 28 2014 *)

is(n)=hammingweight(n^2)<=hammingweight(n) && isprime(n) \\ Charles R Greathouse IV, Mar 18 2014

Primes p such that A000120(p) >= A000120(p^2).

A236513 The n-th prime with n 1-bits in its binary expansion.

Original entry on oeis.org

2, 5, 13, 53, 79, 373, 379, 983, 1783, 6007, 7151, 21503, 31231, 98207, 129919, 259967, 507839, 1564159, 1830911, 4193263, 8355583, 25157567, 33288191, 92274671, 134180863, 394264447, 536838139, 1072693243, 2145382399, 6442188791, 8522825599, 17179836413

Offset: 1

Irina Gerasimova, Jan 27 2014

			Primes p such that
A000120(p) = 1: 2;
A000120(p) = 2: 3, 5, 17, 257,...
A000120(p) = 3: 7, 11, 13, 19, 37, 41,...
A000120(p) = 4: 23, 29, 43, 53, 71, 83, 89,...
A000120(p) = 5: 31, 47, 59, 61, 79, 103, 107, 109,...
A000120(p) = 6: 311, 317, 347, 349, 359, 373,...

Chai Wah Wu, Table of n, a(n) for n = 1..1014

Cf. A061712 (least prime having n ones in binary).

nn = 20; t = Table[-n + 1, {n, nn}]; p = 1; While[Min[t] <= 0, p = NextPrime[p]; b = Total[IntegerDigits[p, 2]]; If[b <= nn, If[t[[b]] < 0, t[[b]]++, If[t[[b]] == 0, t[[b]] = p]]]]; t (* T. D. Noe, Jan 27 2014 *)

lista(nn) = {prm = primes(5000000); for (n = 1, nn, ltp = select(p->hammingweight(p)== n, prm); print1(ltp[n], ", "););} \\ Michel Marcus, Jan 27 2014

from itertools import combinations
from sympy import isprime
def A236513(n):
    l, k, c = n-1, 2**n, 0
    while True:
        for d in combinations(range(l-1,-1,-1),l-n+1):
            m = k-1 - sum(2**(e) for e in d)
            if isprime(m):
                c += 1
                if c == n:
                    return m
        l += 1
        k *= 2 # Chai Wah Wu, Sep 02 2021

a(24)-a(32) from Giovanni Resta, Feb 04 2014

A234431 Numbers that are the sum of 2 successive evil numbers (A001969).

Original entry on oeis.org

3, 8, 11, 15, 19, 22, 27, 32, 35, 38, 43, 47, 51, 56, 59, 63, 67, 70, 75, 79, 83, 88, 91, 94, 99, 104, 107, 111, 115, 118, 123, 128, 131, 134, 139, 143, 147, 152, 155, 158, 163, 168, 171, 175, 179, 182, 187, 191, 195, 200, 203, 207, 211, 214, 219, 224, 227, 230, 235, 239, 243, 248, 251

Offset: 1

Irina Gerasimova, Dec 26 2013

First differences are in {3, 4, 5}; 4*n - 2 <= a(n) <= 4*n. - Charles R Greathouse IV, Dec 26 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001969, A003159 (indices of odd numbers in A234431), A131323 (odd numbers in A234431).

Total/@Partition[Select[Range[0,200],EvenQ[DigitCount[#,2,1]]&],2,1] (* Harvey P. Dale, Nov 02 2015 *)

a(n)=4*n+hammingweight(n-1)%2+hammingweight(n)%2-2 \\ Charles R Greathouse IV, Dec 26 2013

a(n) = A001969(n) + A001969(n + 1).

A234218 Primes whose cubes are odious.

Original entry on oeis.org

2, 13, 23, 29, 43, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 131, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 229, 241, 251, 293, 311, 313, 337, 353, 367, 383, 389, 397, 409, 419, 431, 439, 443, 461, 467, 479, 499, 509, 521, 541, 563, 577, 601

Offset: 1

Irina Gerasimova, Dec 21 2013

Primes p with odious p^3.
Note: "odious" means having an odd number of 1-bits in number's binary representation. So, put in another way, primes p such that A010060(A000578(p)) = 1. - Antti Karttunen, Dec 22 2013
Subsequence of the numbers 1, 2, 4, 8, 13, 16, 23, 25, 26, 29, 32, 35, 43, 45, 46, ... which have odious cubes.

			Prime 2 is in this sequence because 2^3 = 8 and 8 is odious number. Prime 13 is in this sequence because 13*3 = 2197 and 2197 is odious number.

Antti Karttunen, Table of n, a(n) for n = 1..1001

Cf. A000069, A000120, A000578, A010060, A125498, A192085, A194991.

Select[Prime[Range[1000]], OddQ[DigitCount[#^3, 2, 1]] &] (* Indranil Ghosh, Apr 02 2017 *)

is(n)=isprime(n) && hammingweight(n^3)%2 \\ Charles R Greathouse IV, Mar 17 2014

Missing terms added by Antti Karttunen, Dec 22 2013

A233866 Ramanujan primes A104272 that are primes p(k) such that (k+1)p(k)>kp(k+1).

Original entry on oeis.org

2, 11, 17, 29, 41, 59, 71, 101, 107, 127, 149, 179, 227, 229, 281, 307, 311, 347, 349, 419, 431, 439, 461, 487, 569, 599, 641, 643, 659, 739, 769, 809, 821, 823, 853, 857, 937, 967, 983, 1009, 1019, 1031, 1049, 1061, 1087, 1091, 1151, 1187, 1217, 1229

Offset: 1

Irina Gerasimova, Dec 17 2013

Non-Ramanujan primes (A174635) that are primes p(k) such that (k+1)*p(k)>k*p(k+1): 137, 163, 191, 197, 223, 277, 379, 397, 457, 499, 521, 613, 617, 673, 757, 859, 877, 907, 1093, 1181, 1213, 1223, 1231,...
Primes p(k) such that (k+1)*p(k)> k*p(k+1): 2, 11, 17, 29, 41, 59, 71,...
primes p(k) such that (k+1)*p(k) - k*p(k+1)=1: 2, 11,...
Primes p(m) such that (m+1)*p(m) < m*p(m+1): 3, 5, 7, 13, 19, 23, 31,...

			Ramanujan prime 2 is in this sequence because 2 = p(1) such that (1 + 1)*p(1) = 2*2 = 4 > 1*p(1 + 1) = 3;
Ramanujan prime 11 is in this sequence because 11 = p(5) such that (5 + 1)*p(5) = 6*11 = 66 > 5*p(5 + 1) = 65.

A233409 Squares with squarefree neighbors.

Original entry on oeis.org

4, 16, 36, 144, 196, 256, 400, 484, 900, 1156, 1296, 1600, 1764, 2704, 2916, 3136, 3364, 3600, 4356, 5184, 6084, 7056, 7396, 7744, 8100, 8464, 8836, 9216, 10404, 10816, 11236, 11664, 12100, 12544, 12996, 16384, 16900, 19044, 19600, 20164, 20736, 22500

Offset: 1

Irina Gerasimova, Dec 09 2013

nonn

All terms are multiples of 4. Whether n is congruent to 1 or 3 mod 4, n^2 is congruent to 1 mod 3 and therefore mu(n^2 - 1) = 0. - Alonso del Arte, Dec 12 2013

			36 is in this sequence because 35 and 37 are both squarefree.
64 is not in this sequence because 63 = 3^2 * 7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000290, A005117, A088068.

Select[Table[n^2, {n, 150}], SquareFreeQ[# - 1] && SquareFreeQ[# + 1] &] (* Vaclav Kotesovec, Dec 11 2013 *)
Select[Range[150]^2, Abs[MoebiusMu[# - 1] MoebiusMu[# + 1]] == 1 &] (* Alonso del Arte, Dec 11 2013 *)
SequencePosition[Table[Which[IntegerQ[Sqrt[n]],1,SquareFreeQ[n],2,True,0],{n,25000}],{2,1,2}][[;;,1]]+1 (* Harvey P. Dale, Jun 27 2023 *)

forstep(n=2,1e3,[2, 2, 6, 2, 2, 2, 2],if(issquarefree(n-1) && issquarefree(n+1) && issquarefree(n^2+1), print1(n^2", "))) \\ Charles R Greathouse IV, Mar 18 2014

cp's OEIS Frontend

User: Irina Gerasimova

A242667 Number of ways of representing n as the sum of one or more consecutive squarefree numbers.

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A240546 a(n) = prime(n+1)^n mod prime(n).

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Magma

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A237417 Numbers that are the product of an odiousfree number and an evilfree number.

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A236695 The n-th prime with n 0-bits in its binary expansion.

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PARI

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A236514 Primes with a binary weight greater than or equal to the binary weight of their squares.

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Mathematica

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A236513 The n-th prime with n 1-bits in its binary expansion.

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A234431 Numbers that are the sum of 2 successive evil numbers (A001969).

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A233866 Ramanujan primes A104272 that are primes p(k) such that (k+1)p(k)>kp(k+1).