cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A123177 Main diagonal of semiprime power sum array.

Original entry on oeis.org

2, 81, 20494, 1315073, 6115250626, 548619740497, 558551290190815706, 83010387915319808001, 718992177811939511654842, 110011000001100011001010001, 23225155720141324351494556519644062
Offset: 1

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Author

Jonathan Vos Post, Oct 03 2006

Keywords

Comments

Semiprime analog of A123113 Main diagonal of prime power sum array. a(n) is prime for n = 1, 4; what is the next prime value in this sequence?

Examples

			a(1) = 1 + 1^semiprime(1) = 1 + 1^4 = 2.
a(2) = 1 + 2^semiprime(1) + 2^semiprime(2) = 1 + 2^4 + 2^6 = 81.
a(3) = 1 + 3^semiprime(1) + 3^semiprime(2) + 3^semiprime(3) = 1 + 3^4 + 3^6 + 3^9 = 20494.
a(4) = 1 + 4^semiprime(1) + 4^semiprime(2) + 4^semiprime(3) + 4^semiprime(4) = 1 + 4^4 + 4^6 + 4^9 + 4^10 = 1315073 (which is prime).
a(5) = 1 + 5^semiprime(1) + 5^semiprime(2) + 5^semiprime(3) + 5^semiprime(4) + 5^semiprime(5) = 1 + 5^4 + 5^6 + 5^9 + 5^10 + 5^14 = 6115250626.
a(6) = 1 + 6^semiprime(1) + 6^semiprime(2) + 6^semiprime(3) + 6^semiprime(4) + 6^semiprime(5) + 6^semiprime(6) = 1 + 6^4 + 6^6 + 6^9 + 6^10 + 6^14 + 6^15 = 548619740497.
a(7) = 1 + 7^4 + 7^6 + 7^9 + 7^10 + 7^14 + 7^15 + 7^21 = 558551290190815706.
a(8) = 1 + 8^4 + 8^6 + 8^9 + 8^10 + 8^14 + 8^15 + 8^21 + 8^22 = 83010387915319808001.
a(9) = 1 + 9^4 + 9^6 + 9^9 + 9^10 + 9^14 + 9^15 + 9^21 + 9^22 + 9^25 = 718992177811939511654842.
a(10) = 1 + 10^4 + 10^6 + 10^9 + 10^10 + 10^14 + 10^15 + 10^21 + 10^22 + 10^25 + 10^26 = 110011000001100011001010001.
a(11) = 1 + 11^4 + 11^6 + 11^9 + 11^10 + 11^14 + 11^15 + 11^21 + 11^22 + 11^25 + 11^26 + 11^33 = 23225155720141324351494556519644062.
a(12) = 1 + 12^4 + 12^6 + 12^9 + 12^10 + 12^14 + 12^15 + 12^21 + 12^22 + 12^25 + 12^26 + 12^33 + 12^34 = 5332421525600135159678023844770734337.
a(13) = 1 + 13^4 + 13^6 + 13^9 + 13^10 + 13^14 + 13^15 + 13^21 + 13^22 + 13^25 + 13^26 + 13^33 + 13^34 + 13^35 = 1053371868623897220377558669169756037622.

Crossrefs

Cf. A001358, A123113.

Formula

a(n) = 1 + n^4 + n^4 + n^6 + ... + n^semiprime(n) = 1 + SUM[i=1..n]n^semiprime(i) = Main diagonal A(n,n), of the infinite array A(k,n) = 1 + SUM[i=1..k]n^semiprime(i) = 1 + SUM[i=1..k]n^A001358(i). If we deem semiprime(0) = 1, the array is A(k,n) = SUM[i=0..k]n^A001358(i).

A123650 a(n) = 1 + n^2 + n^3 + n^5.

Original entry on oeis.org

4, 45, 280, 1105, 3276, 8029, 17200, 33345, 59860, 101101, 162504, 250705, 373660, 540765, 762976, 1052929, 1425060, 1895725, 2483320, 3208401, 4093804, 5164765, 6449040, 7977025, 9781876, 11899629, 14369320, 17233105, 20536380, 24327901
Offset: 1

Views

Author

Jonathan Vos Post, Oct 04 2006

Keywords

Comments

3rd row, A(3,n), of the infinite array A(k,n) = 1 + Sum_{i=1..k} n^prime(i). If we deem prime(0) = 1, the array is A(k,n) = Sum_{i=0..k} n^prime(i). The first row is A002522 = 1 + n^2. The second row is A098547 = 1 + n^2 + n^3. The 4th row, A(4,n), is A123111 1 + n^2 + n^3 + n^5 + n^7. 10101101 (base n). A(n,n) is A123113 Main diagonal of prime power sum array. The current sequence, A(3,n), can never be prime because of the polynomial factorization a(n) = 1 + n^2 + n^3 + n^5 = +/- (n+1)*(n^2-n+1)*(n^2+1). Its fewest prime factors are 2 for the semiprime a(1) = 4. We similarly have polynomial factorizations for A123651 = A(7,n) = 1+n^2+n^3+n^5+n^7+n^11+n^13+n^17 and A123652 = A(13,n) = 1+n^2+n^3+n^5+...+n^41.

Links

Crossrefs

Cf. A002522, A098547, A123111, A123113, A123651, A123652.

Programs

  • Magma
    [1+n^2+n^3+n^5: n in [1..25]]; // G. C. Greubel, Oct 17 2017
  • Mathematica
    Table[1+n^2+n^3+n^5,{n,30}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{4,45,280,1105,3276,8029},30] (* Harvey P. Dale, Jan 18 2014 *)
  • PARI
    for(n=1,25, print1(1+n^2+n^3+n^5, ", ")) \\ G. C. Greubel, Oct 17 2017

Formula

a(n) = 1 + n^2 + n^3 + n^5 = 101101 (base n) = +/- (n+1)*(n^2-n+1)*(n^2+1).
G.f.: x*(4 +21*x +70*x^2 +20*x^3 +6*x^4 -x^5)/(1-x)^6. - Colin Barker, May 25 2012

A123651 a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17.

Original entry on oeis.org

8, 141485, 130914100, 17251189841, 764209065776, 16940083223773, 232729381165100, 2252358161564225, 16679754951397336, 100010100010101101, 505481836542757988, 2218718842990269265, 8650720586711446400
Offset: 1

Views

Author

Jonathan Vos Post, Oct 04 2006

Keywords

Comments

7th row, A(7,n), of the infinite array A(k,n) = 1 + Sum_{i=1..k} n^prime(i). If we deem prime(0) = 1, the array is A(k,n) = Sum_{i=0..k} n^prime(i). The first row is A002522 = 1 + n^2. The second row is A098547 = 1 + n^2 + n^3. The 3rd row, A(3,n), is A123650. The 4th row, A(4,n), is A123111 1 + n^2 + n^3 + n^5 + n^7. 10101101 (base n). A(n,n) is A123113 Main diagonal of prime power sum array. The current sequence, A(7,n), can never be prime, because of the polynomial factorization a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 = +/- (n^2+1)*(n^15 -n^13 +2n^11 -n^9 +n^7 +n^3 +1). It can be semiprime, as with a(2) and with a(10) = 100010100010101101 = 101 * 990199010001001 and a(14). We similarly have polynomial factorization for A123652 = A(13,n) = 1 +n^2 +n^3 +n^5 +...+ n^41.

Links

Crossrefs

Cf. A002522, A098547, A123111, A123113, A123650, A123652.

Programs

  • Magma
    [1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17: n in [1..25]]; // G. C. Greubel, Oct 17 2017
  • Mathematica
    Table[Total[n^Prime[Range[7]]]+1,{n,20}] (* Harvey P. Dale, Aug 22 2012 *)
  • PARI
    for(n=1,25, print1(1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17, ", ")) \\ G. C. Greubel, Oct 17 2017

Formula

a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 = 100010100010101101 (base n) = +/- (n^2+1)*(n^15-n^13+2n^11-n^9+n^7+n^3+1).

A123652 a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 + n^19 + n^23 + n^29 + n^31 + n^37 + n^41.

Original entry on oeis.org

14, 2339155617965, 36923966682271786990, 4854597644377050732053585, 45547499507677574921923909526, 80266855145143309588022024772829, 44586202603279528645530450127574150
Offset: 1

Views

Author

Jonathan Vos Post, Oct 04 2006

Keywords

Comments

13th row, A(13,n), of the infinite array A(k,n) = 1 + Sum_{i=1..k} n^prime(i). If we deem prime(0) = 1, the array is A(k,n) = Sum_{i=0..k} n^prime(i). The first row is A002522 = 1 + n^2. The second row is A098547 = 1 + n^2 + n^3. The 3rd row, A(3,n), is A123650. The 4th row, A(4,n), is A123111 1 +n^2 +n^3 +n^5 +n^7. 10101101 (base n). A(n,n) is A123113 Main diagonal of prime power sum array. The sequence A(13,n) = a(n) can never be prime because of the polynomial factorization. It can be semiprime, as with a(1) = 14 and a(2) = 2339155617965 = 5 * 467831123593 and a(6) and 100010000010100000100010100010100010101101 = 101 * 990198019901980199010000990199010001001. We similarly have polynomial factorization for the 7th row, A123651 = A(7,n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 = +/- (n^2+1)*(n^15-n^13+2n^11-n^9+n^7+n^3+1).

Links

Crossrefs

Cf. A002522, A098547, A123111, A123113, A123650, A123651.

Programs

  • Magma
    [1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 + n^19 + n^23 + n^29 + n^31 + n^37 + n^41: n in [1..25]]; // G. C. Greubel, Oct 17 2017
  • Mathematica
    Table[1+Total[n^Prime[Range[PrimePi[41]]]],{n,8}] (* Harvey P. Dale, Dec 20 2010 *)
  • PARI
    for(n=1,25, print1(1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 + n^19 + n^23 + n^29 + n^31 + n^37 + n^41, ", ")) \\ G. C. Greubel, Oct 17 2017

Formula

a(n) = 1+n^2+n^3+n^5+n^7+n^11+n^13+n^17+n^19+n^23+n^29+n^31+n^37+n^41 = 100010000010100000100010100010100010101101(base n) = +/-(n^2+1)*(n^39-n^37+2n^35-2n^33+2n^31-n^29+2n^27-2n^25+2n^23-n^21+n^19+n^15-n^13+2n^11-n^9+n^7+n^3+1).
Showing 1-4 of 4 results.

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