Eric Desbiaux - cp's OEIS frontend

A342621 Sum of the partition number of the prime factors of n with multiplicity.

0, 2, 3, 4, 7, 5, 15, 6, 6, 9, 56, 7, 101, 17, 10, 8, 297, 8, 490, 11, 18, 58, 1255, 9, 14, 103, 9, 19, 4565, 12, 6842, 10, 59, 299, 22, 10, 21637, 492, 104, 13, 44583, 20, 63261, 60, 13, 1257, 124754, 11, 30, 16, 300, 105, 329931, 11, 63, 21, 493, 4567, 831820

Offset: 1

Eric Desbiaux, Mar 16 2021

nonn

			For n = 408 = 2^3*3*17, a(408) = 3 * A000041(2) + A000041(3) + A000041(17) = 3*2 + 3 + 297 = 306.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 3000 terms from Eric Desbiaux)

Cf. A000041, A027746, A058698, A001222.

a:= n-> add(combinat[numbpart](i[1])*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 17 2021

{0}~Join~Array[Total@ Map[#2 PartitionsP[#1] & @@ # &, FactorInteger[#]] &, 58, 2] (* Michael De Vlieger, Mar 17 2021 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,2]*numbpart(f[k,1])); \\ Michel Marcus, Mar 17 2021

def a(n):
    return sum([Partitions(primefactor).cardinality() for (primefactor,exponent) in factor(n) for _ in range(exponent)])
[a(n) for n in (1..100)]

a(A003586(n)) - A001414(A003586(n)) = 0.
a(A006899(n)) * A008480(A006899(n)) - A001414(A006899(n)) = 0.
a(n) = Sum_{k=1..A001222(n)} A000041(A027746(n,k)). - Alois P. Heinz, Apr 09 2021

A276781 a(n) = 1+n-(nearest power of prime <= n); for n > 1, a(n) = minimal b such that the numbers binomial(n,k) for b <= k <= n-b have a common divisor greater than 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 29 2016, following a suggestion from Eric Desbiaux

The definition in the video has "b < k < n-b" rather than "b <= k <= n-b", but that appears to be a typographical error.
From Antti Karttunen, Jan 21 2020: (Start)
a(n) = 1 if n is a power of prime (term of A000961), otherwise a(n) is one more than the distance to the nearest preceding prime power.
For n > 1, a(n) indicates the maximum region on the row n of Pascal's triangle (A007318) such that binomial terms C(n,a(n)) .. C(n,n-a(n)) all share a common prime factor. Because for all prime powers, p^k, the binomial terms C(p^k,1) .. C(p^k,p^k-1) have p as their prime factor, we have a(A000961(n)) = 1 for all n, while for each successive n that is not a prime power, the region of shared prime factor shrinks one step more towards the center of the triangle. From this follows that this is the ordinal transform of A025528 (equally, of A065515, or of A003418(n) from n >= 1 onward), equivalent to the simple definition given above.
(End)

			Row 6 of Pascal's triangle is 1,6,15,20,15,6,1 and [15,20,15] have a common divisor of 5. Since 15 = binomial(6,2), a(6)=2.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms for n = 2..1685 from R. J. Mathar, terms 1686..10000 from Chai Wah Wu).
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Christophe Soulé, Le triangle de Pascal et ses propriétés, Lecture, Soc. Math. de France, Feb 13 2008; SMF link (in French).

Cf. A007318, A010055, A276782 (positions of records), A000961 (positions of ones), A024619 (positions of terms > 1).

Cf. A002944, A003418, A025528, A065515, A175851.

mygcd:=proc(lis) local i,g,m;
m:=nops(lis); g:=lis[1];
for i from 2 to m do g:=gcd(g,lis[i]); od:
g; end;
f:=proc(n) local b,lis; global mygcd;
for b from floor(n/2) by -1 to 1 do
lis:=[seq(binomial(n,i),i=b..n-b)];
if mygcd(lis)=1 then break; fi; od:
b+1;
end;
[seq(f(n),n=2..120)];

Table[b = 1; While[GCD @@ Map[Binomial[n, #] &, Range[b, n - b]] == 1, b++]; b, {n, 92}] (* Michael De Vlieger, Oct 03 2016 *)

A276781(n) = if(1==n,1,forstep(k=n,1,-1,if(isprimepower(k),return(1+n-k)))); \\ Antti Karttunen, Jan 21 2020

from sympy import factorint
def A276781(n): return 1+n-next(filter(lambda m:len(factorint(m))<=1, range(n,0,-1))) # Chai Wah Wu, Oct 25 2024

If A010055(n) == 1, a(n) = 1, otherwise a(n) = 1 + a(n-1). - Antti Karttunen, Jan 21 2020

Term a(1) = 1 prepended and alternative simpler definition added to the name by Antti Karttunen, Jan 20 2020

A188060 Numerator of 8^(2n-1) |B(2n)| / (2n)!, where B() are the Bernoulli numbers.

Original entry on oeis.org

8, 128, 4096, 32768, 1048576, 11593056256, 536870912, 7767448354816, 3014517285978112, 191986824837595136, 2733227576976736256, 66530577009460375453696, 5926115612870995607552, 488951148984934932554973184, 7946710949908368748447692488704, 71105936114697022329949662478336

Offset: 1

Eric Desbiaux, Apr 15 2011

nonn

Start with zeta(2n) = (2Pi)^(2n) |B(2n)| /(2 (2n)!) and replace Pi by 4 arctan(1) and take the numerator of the rational part. The denominator is given by A036278.

			8/3,128/45,4096/945,32768/4725,...

Cf. A002432, A046988, A036278.

 f:=n->8^(2*n)*abs(B(2*n))/(2*(2*n)!); [seq(numer(f(n)),n=1..60)];

Entry revised by N. J. A. Sloane, Apr 17 2011

A177225 Pairs [x.y],[0.2],[2.0],[2.2]... such that 3*x/2+y is prime.

Original entry on oeis.org

0, 2, 2, 0, 2, 2, 2, 4, 4, 5, 4, 7, 6, 8, 6, 10, 8, 11, 10, 14, 10, 16, 12, 19, 14, 20, 14, 22, 16, 23, 18, 26, 20, 29, 20, 31, 22, 34, 24, 35, 24, 37, 26, 40, 28, 41, 30, 44, 32, 49, 34, 50, 34, 52

Offset: 1

Eric Desbiaux, May 05 2010

nonn

Cf. A067076, A173177, A173178

A169631 Recamán's sequence (A005132) written in binary.

Original entry on oeis.org

0, 1, 11, 110, 10, 111, 1101, 10100, 1100, 10101, 1011, 10110, 1010, 10111, 1001, 11000, 1000, 11001, 101011, 111110, 101010, 111111, 101001, 10010, 101010, 10001, 101011, 10000, 101100, 1111, 101101, 1110, 101110, 1001111, 1110001

Offset: 0

Eric Desbiaux, Mar 15 2010

nonn

A169633 The odd terms in Recamán's sequence (A005132).

Original entry on oeis.org

1, 3, 7, 13, 21, 11, 23, 9, 25, 43, 63, 41, 17, 43, 15, 45, 79, 113, 77, 39, 79, 37, 81, 35, 83, 33, 85, 31, 87, 29, 89, 27, 91, 157, 225, 155, 227, 153, 75, 153, 73, 155, 71, 157, 69, 159, 67, 161, 65, 163, 265, 367, 263, 369, 261, 151, 265, 379, 495, 377, 259, 137, 261

Offset: 1

Eric Desbiaux, Mar 15 2010

nonn

More terms from R. J. Mathar, Oct 09 2010

A158942 Nonsquares coprime to 10.

Original entry on oeis.org

3, 7, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 147, 149, 151, 153, 157, 159, 161, 163

Offset: 1

Eric Desbiaux, Mar 31 2009

Odd primes + odd nonprime integers that have an odd numbers of proper divisors A082686, are the result of a suppression of integers satisfying: 2n (A005843); n^2 (A000290); n^2+n (A002378). Of these, we can suppress the multiples of 5 (A008587).
Decimal expansion of 1/10^(n^2+n) + 1/10^(n^2) + 1/10^(5*n) + 1/10^(2*n) gives a 0 for these integers.
2n + n(n+1) + n^2 = 2n^2 + 3n = A014106.
2n^2 + 3n + 5n = 2n^2 + 8n = 2n(n+4) = A067728 8(8+n) is a perfect square.

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

Intersection of A000037 and A045572.

Cf. A082868, A002378, A000290, A005843, A008587, A014106, A067728.

Select[Range@ 163, ! IntegerQ@ Sqrt@ # && CoprimeQ[#, 10] &] (* Michael De Vlieger, Dec 11 2015 *)

isok(n) = (n % 2) && (n % 5) && (isprime(n) || (numdiv(n) % 2 == 0)); \\ Michel Marcus, Aug 27 2013

is(n)=gcd(n,10)==1 && !issquare(n) \\ Charles R Greathouse IV, Sep 05 2013

New name from Charles R Greathouse IV, Sep 05 2013

A169632 The even terms in Recamán's sequence (A005132).

Original entry on oeis.org

0, 6, 2, 20, 12, 22, 10, 24, 8, 62, 42, 18, 42, 16, 44, 14, 46, 78, 114, 78, 38, 80, 36, 82, 34, 84, 32, 86, 30, 88, 28, 90, 26, 224, 156, 226, 154, 228, 152, 74, 154, 72, 156, 70, 158, 68, 160, 66, 162, 64, 164, 264, 368, 262, 370, 40, 152, 494, 378, 258, 138, 260

Offset: 1

Eric Desbiaux, Mar 15 2010

nonn

Index entries for sequences related to Recamán's sequence

Cf. A005132, A169633 (odd).

a(2*n+b) = A005132(4*n+3+b), for b = {0,1}. - Thomas Scheuerle, Oct 06 2022

More terms from R. J. Mathar, Oct 09 2010

A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.

Original entry on oeis.org

1, 4, 7, 10, 13, 19, 22, 25, 28, 34, 40, 43, 49, 52, 55, 64, 67, 73, 82, 85, 88, 94, 97, 112, 115, 118, 124, 127, 130, 133, 139, 145, 154, 157, 172, 175, 178, 190, 193, 199, 208, 214, 220, 223, 229, 232, 238, 244, 250, 253, 259, 277, 280, 283, 292, 295, 298, 307, 319

Offset: 1

Eric Desbiaux, Feb 11 2010

With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 1 + 3 OR 3* 1 + 2 =  5;
  2* 4 + 3 OR 3* 3 + 2 = 11;
  2* 7 + 3 OR 3* 5 + 2 = 17;
  2*10 + 3 OR 3* 7 + 2 = 23;
  2*13 + 3 OR 3* 9 + 2 = 29;
  2*19 + 3 OR 3*13 + 2 = 41;
  2*22 + 3 OR 3*15 + 2 = 47;
  2*25 + 3 OR 3*17 + 2 = 53;
  2*28 + 3 OR 3*19 + 2 = 59.
A024893 Numbers k such that 3k+2 is prime.
A007528 Primes of the form 6k-1.
A024898 Positive integers k such that 6k-1 is prime.
1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?, Frequently asked questions about primes.

Cf. A067076, A024893, A007528, A024898, A059325.

Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* Amiram Eldar, Jul 30 2024 *)

a(n) = 3*A059325(n) + 1. - Amiram Eldar, Jul 30 2024

Data corrected and extended by Amiram Eldar, Jul 30 2024

A159200 Decimal expansion of Sum_{k >= 1} (1/(10^(4k + 2) - 1)) - (1/(10^(2k + 1) - 1)), negated.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 3, 0, 3, 0, 1, 0, 3, 0, 1, 0, 1, 0, 5, 0, 1, 0, 2, 0, 3, 0, 1, 0, 3, 0, 3, 0, 1, 0, 1, 0, 5, 0, 3, 0, 1, 0, 3, 0, 1, 0, 1, 0, 5, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 3, 0, 1, 0, 3, 0, 3, 0, 3, 0, 1, 0, 5, 0

Offset: 0

Eric Desbiaux, Apr 06 2009

It equals Sum_{k >= 1} 1/((2^(4*k + 2)*5^(4*k + 2)) - 1) - 1/((2^(2*k + 1)*5^(2*k + 1)) - 1).
Note that Sum_{k >= 1} (1/(10^k - 1)) / Sum_{k >= 1} ((1/(10^(4*k + 2) - 1)) -(1/(10^(2*k + 1) - 1))) = A073668 / Sum_{k >= 1} ((1/(10^(4*k + 2) - 1)) - (1/(10^(2*k + 1) - 1))) = -121.100.
My idea for this decimal expansion came from the Engel expansion of e - 1, i.e., A000027(n) = n, and the Engel expansion of e^(-1), i.e., A059193(n) = 2*(2*n + 1)*(n - 1), which I have transformed into (2*n + 1)^2 - (6*n + 3) (since 2*(2*n + 1)*(n - 1) = (2*n + 1)^2 - (6*n + 3)). It appears that the Engel expansion of 1/e works like a Sundaram sieve.
Decimal expansion of Sum_{n>=0} (d(2*n+1) - 1)/(10^(2*n+1) - 1), where d = A000005. - Jianing Song, Apr 12 2021

			-0.00101010201010301010301020301010303010301010501020301030301...

Eric Weisstein's World of Mathematics, Engel expansion.
English Wikipedia, Sieve of Sundaram.
French Wikipedia, Crible de Sundaram.

suminf(k=1, 1/(10^(4*k + 2) - 1) - 1/(10^(2*k + 1) - 1)) \\ Michel Marcus, Jun 25 2019

Comments edited by Petros Hadjicostas, Jun 19 2019

cp's OEIS Frontend

User: Eric Desbiaux

A342621 Sum of the partition number of the prime factors of n with multiplicity.

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A276781 a(n) = 1+n-(nearest power of prime <= n); for n > 1, a(n) = minimal b such that the numbers binomial(n,k) for b <= k <= n-b have a common divisor greater than 1.

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A188060 Numerator of 8^(2n-1) |B(2n)| / (2n)!, where B() are the Bernoulli numbers.

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A177225 Pairs [x.y],[0.2],[2.0],[2.2]... such that 3*x/2+y is prime.

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A169631 Recamán's sequence (A005132) written in binary.

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A169633 The odd terms in Recamán's sequence (A005132).

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Extensions

A158942 Nonsquares coprime to 10.

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Mathematica

PARI

PARI

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A169632 The even terms in Recamán's sequence (A005132).

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A173178 Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.

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A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.

A159200 Decimal expansion of Sum_{k >= 1} (1/(10^(4k + 2) - 1)) - (1/(10^(2k + 1) - 1)), negated.