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 1-10 of 18 results. Next

A134245 Terms in A007468 which are multiples of their indices.

Original entry on oeis.org

2, 8, 88, 384, 1056, 2310, 207936, 417219, 2978610, 6215400, 9216124, 205006774, 255230655, 576178034, 1157525280, 2038109955, 3053762208, 10584038058, 25042362120, 1025402527504, 2304427934330, 11623068703428
Offset: 1

Views

Author

Enoch Haga, Oct 15 2007, Oct 16 2007

Keywords

Comments

If A007468(j) is a multiple of j, we add A007468(j) to this sequence. This sequence is a subsequence of A007468. - R. J. Mathar, Nov 16 2007

Examples

			3+5=8 is a multiple of i=2 and is added to the sequence. 7+11+13 is not a multiple of 3 and is skipped. 17+19+23+29=88 is a multiple of 4 and is added to the sequence. 31+37+41+43+47=199 is not a multiple of 5 and is skipped.

Links

Crossrefs

Cf. A134244, A134246, A007468.

Programs

Formula

a(n) = A007468(A134244(n)).

Extensions

Edited and corrected by R. J. Mathar, Nov 16 2007
Corrected by R. J. Mathar, Nov 16 2007
Edited by Max Alekseyev, Nov 08 2011

A134244 Indices k such that k divides A007468(k).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 38, 47, 86, 108, 122, 322, 345, 446, 556, 665, 756, 1122, 1476, 4838, 6274, 10556, 12383, 38774, 42776, 46239, 210215, 232276, 517144, 1125321, 2561652
Offset: 1

Views

Author

Enoch Haga, Oct 15 2007, Oct 16 2007

Keywords

Crossrefs

Cf. A134245, A134246.

Programs

  • PARI
    { p=2 ; i=1 ; for(j=1,13000, psum=p ; for(k=2,j, p=nextprime(p+1) ; psum += p ; ) ; if(psum % j ==0, print(j) ; ) ; p=nextprime(p+1) ; ) ; } \\ R. J. Mathar, Jan 24 2008

Extensions

Better definition from R. J. Mathar, Nov 16 2007
Entries checked by R. J. Mathar, Jan 24 2008
a(24)-a(26) from Max Alekseyev, Jul 29 2011
a(27)-a(28) from Chai Wah Wu, Jan 14 2020
a(29) from Chai Wah Wu, Jan 16 2020
a(30)-a(31) from Giovanni Resta, Jan 24 2020

A134179 Primes in A007468.

Original entry on oeis.org

2, 31, 199, 659, 1601, 5693, 38917, 51797, 84463, 697373, 8059169, 8510617, 9465251, 28268543, 36254039, 40121999, 48667219, 54973229, 63627491, 90443813, 97626083, 102534899, 110364689, 139100699, 161967391, 183528421, 219566989
Offset: 1

Views

Author

Enoch Haga, Oct 16 2007

Keywords

Links

Crossrefs

Cf. A134180, A134244, A134245, A134246.

Programs

  • Mathematica
    n = 1000; Select[Total /@ TakeList[Prime[Range@(n(n+1)/2)], Range@n], PrimeQ] (* Zhining Yang, May 22 2023 *)
  • UBASIC
    10 K=1
20 A=nxtprm(A):B=B+A:C=C+1: if C<>K then 20:else 30
30 L=B/K
31 print K;B;:Q=prmdiv(B): if Q=B then print B;"-":stop:else 40
40 B=0:K=K+1:C=0:goto 20

Extensions

Edited by Zak Seidov, Feb 05 2010

A134180 Indices of primes in A007468.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 23, 25, 29, 55, 117, 119, 123, 173, 187, 193, 205, 213, 223, 249, 255, 259, 265, 285, 299, 311, 329, 351, 393, 397, 411, 483, 487, 513, 515, 525, 563, 587, 597, 607, 637, 645, 647, 679, 709, 789, 871, 877, 911, 953, 971, 985, 1015, 1051, 1075
Offset: 1

Views

Author

Enoch Haga, Oct 16 2007

Keywords

Comments

Compute sums of k distinct sequential primes (no overlap). If the sums are prime add their indices to the sequence.

Examples

			a(2)=3 because this k value is the index for the next 3 primes in sequence to be summed. k=1 is 2, k=2 is 3+5 and k=3 is 7+11+13=31. The sums at k=1 and k=3 are prime, while k=2 is composite.

Links

Crossrefs

Cf. A007468. Corresponding primes are listed in A134179.
Cf. A134244, A134245, A134246.

Programs

  • UBASIC
    10 K=1
  20 A=nxtprm(A): B=B+A: C=C+1: if C<>K then 20: else 30
  30 L=B/K
  31 print K;B;: Q=prmdiv(B): if Q=B then print B; "-": stop: else 40
  40 B=0: K=K+1: C=0: goto 20

A058871 Sum of next n primes (A007468) is prime for odd n or half the sum is prime for even n.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 18, 20, 23, 25, 29, 48, 55, 98, 117, 119, 123, 130, 173, 187, 193, 198, 205, 213, 218, 223, 248, 249, 255, 259, 265, 268, 285, 292, 299, 311, 329, 346, 351, 386, 390, 393, 397, 411, 412, 448, 483, 487, 513, 515, 525, 554, 563, 574, 587, 597
Offset: 1

Views

Author

Robert G. Wilson v, Jan 06 2001

Keywords

Crossrefs

Cf. A007468.

Programs

  • Mathematica
    Do[ s = Sum[ Prime[i], {i, n(n-1)/2 + 1, n(n+1)/2} ]; If[ PrimeQ[s] || PrimeQ[s/2], Print[n] ], {n, 10^3} ]

A082749 Difference between the sum of next prime(n) natural numbers and the sum of next n primes.

Original entry on oeis.org

1, 4, 9, 10, 54, 71, 191, 236, 446, 1025, 1310, 2259, 3245, 3820, 5048, 7321, 10060, 11473, 15328, 18358, 20381, 25672, 30222, 36561, 46367, 53031, 58108, 65444, 70971, 78391, 104184, 116542, 133095, 142728, 169931, 181324, 203429, 226622
Offset: 1

Views

Author

Amarnath Murthy, Apr 17 2003

Keywords

Comments

Group the natural numbers with prime(n) elements in each group. (1,2),(3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17),... The sum corresponding the groups is 3,12,40,98,... Group the prime numbers such that the n-th group contains n primes. (2),(3,5),(7,11,13),(17,19,23,29),... The sum corresponding the groups is 2,8,31,88,... The required difference is 1,4,9,10,...
The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004

Programs

  • Mathematica
    Module[{nn=80,trms=40,c,nat,pr},c=(nn(nn+1))/2;nat=Total/@TakeList[Range[c],Prime[Range[trms]]];pr=Total/@TakeList[Prime[Range[c]], Range[trms]]; Differences/@ Thread[{pr,nat}]]//Flatten (* Harvey P. Dale, Apr 13 2025 *)

Formula

a(n) = ((A061802(n-1) + 1)*A000040(n))/2 - A007468(n). - Gionata Neri, May 17 2015

Extensions

More terms from Ray Chandler, May 13 2003

A072475 Sum of next n composite numbers.

Original entry on oeis.org

4, 14, 31, 63, 112, 176, 264, 385, 529, 712, 932, 1184, 1503, 1833, 2234, 2689, 3207, 3779, 4408, 5117, 5913, 6747, 7657, 8667, 9766, 10938, 12240, 13612, 15071, 16578, 18266, 20081, 22007, 23989, 26100, 28334, 30695, 33221, 35811, 38569, 41474
Offset: 1

Views

Author

Amarnath Murthy, Jun 20 2002

Keywords

Links

Crossrefs

Cf. A072474, A007468, A006003.

Programs

  • Mathematica
    Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[ Sum[ Composite[i], {i, n(n - 1)/2 + 1, n(n + 1)/2}], {n, 1, 42}]
With[{terms=50},cnos=With[{c=(terms(terms+1)(terms+2))/6}, Complement[ Range[5,c], Prime[Range[PrimePi[c]]]]];Join[{4}, Total/@Table[Take[ cnos,{n (n+1)/2,(n+1) (n+2)/2-1}],{n,terms-1}]]] (* Harvey P. Dale, Oct 10 2011 *)

Extensions

More terms from Jim Nastos and Robert G. Wilson v, Jun 21 2002

A134246 Quotients A134245(n)/A134244(n).

Original entry on oeis.org

2, 4, 22, 64, 132, 231, 5472, 8877, 34635, 57550, 75542, 636667, 739799, 1291879, 2081880, 3064827, 4039368, 9433189, 16966370, 211947608, 367298045, 1101086463, 1541079585, 16915330550, 20775909209, 24450416187, 575276374739, 707965641746, 3731987263894
Offset: 1

Views

Author

Enoch Haga, Oct 15 2007, Oct 16 2007

Keywords

Comments

Alternatively: integers of the form A007468(j)/j sorted w.r.t. increasing j. - R. J. Mathar, Nov 16 2007

Crossrefs

Cf. A134244 A134245.

Programs

Extensions

Better definition from R. J. Mathar, Nov 16 2007
a(24)-a(26) from Max Alekseyev, Jul 29 2011
a(27)-a(28) from Chai Wah Wu, Jan 14 2020
a(29) from Chai Wah Wu, Jan 16 2020

A075673 Sum of next n integer interprimes (cf. A024675).

Original entry on oeis.org

4, 15, 45, 111, 232, 422, 704, 1129, 1667, 2403, 3287, 4470, 5810, 7508, 9414, 11663, 14363, 17454, 20715, 24739, 29214, 33957, 39183, 45540, 52056, 59497, 67181, 75862, 84831, 95697, 106608, 117812, 130356, 143759, 158617, 174312, 190500
Offset: 1

Views

Author

Zak Seidov, Sep 24 2002

Keywords

Comments

Sum of next n primes is A007468. Sum of next n odd interprimes is A075674. Sum of next n even interprimes is A075675.

Examples

			a(1) = (3+5)/2 = 4; a(2) = (5+7)/2+(7+11)/2 = 15; a(3) = (11+13)/2+(13+17)/2 +(17+19)/2 = 45.

Links

Crossrefs

Cf. A024675, A007468, A075673, A075674, A075675.

Programs

  • Mathematica
    (* sum of next n integer interprimes*) i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; Table[Sum[(Prime[i+2]+Prime[i+1])/2, {i, i1, i2}], {n, 1, 40}]
With[{nn=40},Total/@TakeList[Mean/@Partition[Prime[Range[2,(nn(nn+1))/2+2]],2,1],Range[nn]]] (* Harvey P. Dale, Feb 24 2023 *)

A075674 Sum of next n odd interprimes.

Original entry on oeis.org

9, 36, 153, 378, 805, 1576, 2733, 3818, 5857, 8006, 10537, 14812, 19389, 23472, 29757, 36416, 44067, 52274, 61307, 71740, 86195, 104056, 120463, 138128, 158209, 181160, 205973, 229728, 255013, 281608
Offset: 1

Views

Author

Zak Seidov, Sep 24 2002

Keywords

Comments

Sum of next n primes in A007468. Sum of next n integer interprimes in A075673. Sum of next n even interprimes in A075675.

Examples

			a(1) = (7+11)/2 = 9; a(2) = (13+17)/2+(19+23)/2 = 15 + 21 = 36.

Crossrefs

Cf. A007468, A075673, A075674, A075675.

Programs

  • Mathematica
    (* sum of next n odd interprimes*) od=Select[Table[(Prime[i]+Prime[i+1])/2, {i, 2, 2000}], OddQ]; i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; A075674=Table[Sum[od[[i]], {i, i1, i2}], {n, 30}]
Showing 1-10 of 18 results. Next

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