A008578 -id:A008578 - cp's OEIS frontend

A154734 Define k(0) = 2 and k(m) = m^2-k(m-1) for m >= 1. This is a list of those terms k(m) for which k(m)+1 and k(m)-1 are both in A008578 (primes including 1).

2, 4, 12, 138, 822, 2082, 3918, 21738, 39342, 62130, 70878, 106032, 117372, 129288, 135462, 182712, 512580, 524802, 575130, 682698, 769422, 799482, 893118, 1008912, 1026030, 1043292, 1828830, 2368578, 2447580, 3247428, 3278082, 3465030, 4022868, 4056978

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 14 2009

nonn

Cf. A008578, A154736.

a := proc(n) local k; k := n*(n - 1)/2 - 2*(-1)^n:
if k = 2 or isprime(k - 1) and isprime(k + 1) then k else NULL fi end:
seq(a(n), n = 1..1000);  # Peter Luschny, Jul 14 2022

k=2;lst={k};Do[k=n^2-k;If[PrimeQ[k-1]&&PrimeQ[k+1],AppendTo[lst,k]],{n,8!}];lst

a154734(upto,k0=2) = {my(k=k0); print1(k,", "); for(n=1, oo, my(kk=n^2-k);if(isprime(k-1) && isprime(k+1), print1(k,", ")); k=kk; if(k>upto, break))};
a154734(5000000) \\ Hugo Pfoertner, Jul 14 2022

k(n) = n*(n+1)/2+2*(-1)^n. - Peter Luschny, Jul 14 2022

Better name from Pontus von Brömssen, Jul 14 2022

A164653 a(1) = 1, for n>=2: a(n) = sum of two consecutive noncomposite numbers A008578.

Original entry on oeis.org

1, 3, 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, 138, 144, 152, 162, 172, 186, 198, 204, 210, 216, 222, 240, 258, 268, 276, 288, 300, 308, 320, 330, 340, 352, 360, 372, 384, 390, 396, 410, 434, 450, 456, 462, 472, 480, 492, 508

Offset: 1

Jaroslav Krizek, Aug 19 2009

nonn

Basically these are the sums of two successive primes. - N. J. A. Sloane, Nov 16 2018

Essentially the same as A001043, A011974 and A069102.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A008578, A011974, A069102, A076273, A158611.

ListConvolve[{1,1},Join[{0,1},Prime[Range[100]]]] (* Paolo Xausa, Nov 02 2023 *)

a(n) = A158611(n) + A158611(n+1).
a(n) = A008578(n-1) + A008578(n) for n >= 2.
a(n) = A076273(n-1) + 1 for n >= 2.
a(n) = A000040(n-1) + A008578(n-1) for n >= 2. - Jaroslav Krizek, Dec 13 2009

Edited by R. J. Mathar, Aug 21 2009
Correction for change of offset in A158611 and A008578 in Aug 2009 by Jaroslav Krizek, Jan 27 2010
Formulas edited by Paolo Xausa, Nov 04 2023

A175247 Primes (A000040) with noncomposite (A008578) subscripts.

Original entry on oeis.org

2, 3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, 1031, 1063, 1087, 1153, 1171, 1201, 1217, 1297, 1409, 1433, 1447

Offset: 1

Jaroslav Krizek, Mar 13 2010

nonn

A006450 is the main entry for these numbers. [Arkadiusz Wesolowski, Mar 12 2011]

			a(5) = 17 because a(5) = p(q(5)) = p(7) = 17, p = prime, q = noncomposite.

Cf. A006450, A000040.

a(n) U A007821(n+1) = A000040(n) for n >= 1. a(n) = prime(noncomposite(n)) = A000040(A008578(n)). a(1) = 2, a(n) = A006450(n-1) for n >= 2.

A175250 Nonprimes (A018252) with noncomposite (A008578) subscripts.

Original entry on oeis.org

1, 4, 6, 9, 12, 18, 21, 26, 28, 34, 42, 45, 52, 57, 60, 65, 74, 81, 84, 91, 95, 98, 106, 112, 119, 128, 133, 135, 141, 143

Offset: 1

Jaroslav Krizek, Mar 13 2010

nonn

a(n) = nonprime(noncomposite(n)) = A018252(A008578(n)). a(n) U A102615(n+1) = A018252(n) for n >= 1. a(1) = 1, a(n) = A078782(n-1) = nonprimes (A008578) with prime (A000040) subscripts for n >=2.

			a(5) = 12 because a(5) = b(q(5)) = b(7) = 12, q = noncomposite, b = nonprime.

A255325 Half of the difference between the n-th Ludic number and the n-th noncomposite: a(n) = (A003309(n) - A008578(n))/2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 3, 2, 1, 2, 3, 4, 4, 5, 5, 6, 8, 6, 7, 9, 9, 9, 9, 10, 11, 15, 11, 13, 12, 17, 13, 14, 12, 15, 21, 19, 21, 21, 18, 20, 19, 20, 18, 17, 19, 24, 25, 24, 30, 28, 28, 33, 31, 33, 32, 36, 38, 34, 35, 36, 38, 40, 38, 39, 36, 36, 39, 37, 35, 36, 42, 42, 43, 42, 46, 44, 42, 44, 43, 47, 48, 49, 52

Offset: 1

Antti Karttunen, Feb 23 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003309, A008578.

Cf. A255324 (the same terms multipled by 2).

t = Range[2, 1000];
A003309 = {1};
While[Length[t] > 0, k = First[t];
     AppendTo[A003309, k]; t = Drop[t, {1, -1, k}]];
a[n_] := If[n == 1, 0, A003309[[n]] - Prime[n - 1]]/2;
Array[a, Length[A003309]] (* Jean-François Alcover, Jan 08 2022, after Ray Chandler in A003309 *)

Scheme
```
(define (A255325 n) (/ (A255324 n) 2))
```

a(n) = A255324(n)/2 = (A003309(n) - A008578(n))/2.

A256430 The least positive integer in A055744 divisible by A008578(n).

Original entry on oeis.org

1, 4, 18, 50, 294, 1210, 1014, 578, 2166, 58190, 35322, 28830, 8214, 16810, 77658, 5588770, 219102, 4239858, 111630, 1481370, 1058610, 31974, 486798, 2824490, 871310, 56454, 102010, 1082118, 47330166, 71286, 536298, 677418, 6692790, 638146, 146646390, 4928622

Offset: 1

David A. Corneth, Mar 28 2015

nonn

Subset of A256431. Elements from this sequence can be used to find elements from A256431. For example, 18 and 50 are the least number in this sequence divisible by 3 and 5 respectively. These numbers can be used to find the least number in A055744 divisible by both 3 and 5 as follows: 18 = 2^1 * 3^2 and 50 = 2^1 * 5^2. 'Order' these factors together: 2^1|2^1|3^2|5^2. For two consecutive factors, if they have the same base, remove the one with the highest exponent. Leaves 2^1|3^2|5^2. Multiply these factors together. Gives 2 * 3^2 * 5^2 = 450. So 450 is in A256431. This method can be applied recursively to find the least n in A055744 divisible by 3, 5 and 7, for example; applying this to 294 and 450 gives 7350 which is the least element in A055744 divisible by primes 3, 5 and 7.

Andrew Howroyd, Table of n, a(n) for n = 1..72

With[{s = Select[Range[10^6], SameQ @@ Map[FactorInteger[#][[All, 1]] &, {#, EulerPhi@ #}] &]}, TakeWhile[#, IntegerQ] &@ Table[SelectFirst[s, Divisible[#, p] &], {p, {1}~Join~Prime@ Range@ 30}]] (* Michael De Vlieger, Feb 22 2018 *)

a(n)={my(m=0, p=if(n==1,1,prime(n-1))); until(my(f=factor(m)); f[, 1]==factor(eulerphi(f))[, 1], m+=p); m} \\ Andrew Howroyd, Mar 01 2018

a(16), a(18) and other terms corrected by Andrew Howroyd, Mar 01 2018

A289928 p-INVERT of (1,2,3,5,7,11,13,...); i.e., 1 and the primes (A008578), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 14, 48, 162, 547, 1842, 6206, 20906, 70438, 237326, 799629, 2694199, 9077599, 30585239, 103051135, 347211149, 1169861760, 3941626163, 13280557904, 44746308037, 150764154490, 507971076799, 1711511703373, 5766612400708, 19429501132982, 65464000013233

Offset: 0

Clark Kimberling, Aug 15 2017

Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A289780 for a guide to related sequences.

Cf. A008578, A289847.

z = 60; s = x + Sum[Prime[k] x^(k + 1), {k, 1, z}]; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A008578 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x] , 1] (* A289928 *)

A023862 a(n) = 1t(n) + 2t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t = A008578 ({1} U primes).

Original entry on oeis.org

1, 2, 7, 11, 26, 40, 76, 104, 171, 219, 331, 397, 566, 672, 912, 1046, 1369, 1561, 1985, 2223, 2762, 3026, 3688, 4030, 4833, 5215, 6179, 6631, 7764, 8318, 9630, 10270, 11793, 12499, 14245, 15019, 17006, 17930, 20176, 21182, 23701, 24837, 27645, 28965, 32084, 33504

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

CF. A000040, A008578.

[1] cat [(&+[j*NthPrime(n-j): j in [1..Floor((n+1)/2)]]): n in [2..50]]; // G. C. Greubel, Jun 12 2019

Join[{1}, Table[Sum[j*Prime[n-j], {j,1,Floor[(n+1)/2]}], {n, 2, 50}]] (* G. C. Greubel, Jun 12 2019 *)

a(n) = sum(j=1, floor((n+1)/2), j*prime(n-j));
vector(50, n, if(n==1, 1, a(n))) \\ G. C. Greubel, Jun 12 2019

[1]+[sum(j*nth_prime(n-j) for j in (1..floor((n+1)/2))) for n in (2..50)] # G. C. Greubel, Jun 12 2019

Title simplified by Sean A. Irvine, Jun 12 2019

A142074 Prime number superposition a(n) = 10*A008578(2n-1) + A008578(2n).

Original entry on oeis.org

12, 35, 81, 147, 213, 321, 411, 477, 589, 677, 783, 873, 987, 1113, 1179, 1257, 1447, 1539, 1667, 1797, 1909, 2001, 2127, 2201, 2457, 2523, 2631, 2767, 2899, 2987, 3093, 3237, 3423, 3501, 3717, 3843, 3957, 4109, 4219, 4371

Offset: 1

Paul Curtz, Sep 15 2008

The two factors 10 and 1 of this linear combination could be replaced by any other pair of integers.

A008578 := proc(n) if n = 1 then 1; else ithprime(n-1) ; end if; end proc:
A142074 := proc(n) 10*A008578(2*n-1)+A008578(2*n) ; end proc: # R. J. Mathar, Jul 07 2011

Join[{12},10#[[1]]+#[[2]]&/@Partition[Prime[Range[2,80]],2]] (* Harvey P. Dale, Aug 11 2017 *)

a(n) = 10*prime(2n-2)+prime(2n-1), n>1.

A146985 Triangle T(n,m) = f(n-m)*f(n), where f(n) = A008578(n+1).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 6, 6, 5, 7, 10, 9, 10, 7, 11, 14, 15, 15, 14, 11, 13, 22, 21, 25, 21, 22, 13, 17, 26, 33, 35, 35, 33, 26, 17, 19, 34, 39, 55, 49, 55, 39, 34, 19, 23, 38, 51, 65, 77, 77, 65, 51, 38, 23, 29, 46, 57, 85, 91, 121, 91, 85, 57, 46, 29

Offset: 0

Roger L. Bagula, Nov 04 2008

I call this sequence "symmetrical spooky primes" as two prime combinations are used in cryptography.

Row sums are:{1, 4, 10, 22, 43, 80, 137, 222, 343, 508, 737}. The sequence to Floor[n/2] is a way to get all the combinations of primes with one less than the other.

			Triangle T(n,m), n, m >= 0 begins:
   1
   2,  2
   3,  4,  3
   5,  6,  6,  5
   7, 10,  9, 10,  7
  11, 14, 15, 15, 14,  11
  13, 22, 21, 25, 21,  22, 13
  17, 26, 33, 35, 35,  33, 26, 17
  19, 34, 39, 55, 49,  55, 39, 34, 19
  23, 38, 51, 65, 77,  77, 65, 51, 38, 23
  29, 46, 57, 85, 91, 121, 91, 85, 57, 46, 29

Clear[f, t, n, m]; f[n_] := If[n == 0, 1, Prime[n]]; t[n_, m_] = f[n - m]*f[m]; Table[t[n, m], {n, 0, 10}, {m, 0, n}]; Flatten[%]

Edited by Peter Munn, Jun 29 2023

cp's OEIS Frontend

A154734 Define k(0) = 2 and k(m) = m^2-k(m-1) for m >= 1. This is a list of those terms k(m) for which k(m)+1 and k(m)-1 are both in A008578 (primes including 1).

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A164653 a(1) = 1, for n>=2: a(n) = sum of two consecutive noncomposite numbers A008578.

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A175247 Primes (A000040) with noncomposite (A008578) subscripts.

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A175250 Nonprimes (A018252) with noncomposite (A008578) subscripts.

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A255325 Half of the difference between the n-th Ludic number and the n-th noncomposite: a(n) = (A003309(n) - A008578(n))/2.

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A256430 The least positive integer in A055744 divisible by A008578(n).

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A289928 p-INVERT of (1,2,3,5,7,11,13,...); i.e., 1 and the primes (A008578), where p(S) = 1 - S - S^2.

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A023862 a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t = A008578 ({1} U primes).

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Magma

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A142074 Prime number superposition a(n) = 10*A008578(2n-1) + A008578(2n).

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A023862 a(n) = 1t(n) + 2t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t = A008578 ({1} U primes).