A237589 -id:A237589 - cp's OEIS frontend

A343809 Divide the positive integers into subsets of lengths given by successive primes, then reverse the order of terms in each subset.

2, 1, 5, 4, 3, 10, 9, 8, 7, 6, 17, 16, 15, 14, 13, 12, 11, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59

Offset: 1

Paolo Xausa, Apr 30 2021

From Omar E. Pol, Apr 30 2021: (Start)
Irregular triangle read by rows T(n,k) in which row n lists the next p positive integers in decreasing order, where p is the n-th prime, with n >= 1.
The triangle has the following properties:
Column 1 gives the nonzero terms of A007504.
Column 2 gives A237589.
Column 3 gives A071148.
Column 4 gives the terms > 2 of A343859.
Column 5 gives the absolute values of the terms < -1 of A282329.
Column 6 gives the terms > 7 of A082548.
Column 7 gives the terms > 6 of A115030.
Records are in the column 1.
Indices of records are in the right border.
Right border gives A014284.
Row lengths give A000040.
Row products give A078423.
Row sums give A034956. (End)

			From _Omar E. Pol_, Apr 30 2021: (Start)
Written as an irregular triangle in which row lengths give A000040 the sequence begins:
   2,  1;
   5,  4,  3;
  10,  9,  8,  7,  6;
  17, 16, 15, 14, 13, 12, 11;
  28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18;
  41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29;
  58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42;
  77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59;
  ...
(End)

Paolo Xausa, Table of n, a(n) for n = 1..10887 (rows n = 1..70 of triangle, flattened)
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A000040, A007504, A014284, A034956, A038722, A071148, A073612 (fixed points), A078423, A082548, A115030, A237589, A282329, A343859, A344891.

R:= NULL: t:= 1:
for i from 1 to 20 do
  p:= ithprime(i);
  R:= R, seq(i,i=t+p-1..t,-1);
  t:= t+p;
od:
R; # Robert Israel, Apr 30 2021

With[{nn=10},Reverse/@TakeList[Range[Total[Prime[Range[nn]]]],Prime[Range[nn]]]]//Flatten (* Harvey P. Dale, Apr 27 2022 *)

T(n,k) = A007504(n) - k + 1, with n >= 1 and 1 <= k <= A000040(n). - Omar E. Pol, May 01 2021

A110979 Squares equal to the sum of the first k primes minus 1.

Original entry on oeis.org

1, 4, 9, 16, 196, 839056, 7796654478001

Offset: 1

Giovanni Teofilatto, Sep 30 2005

No more terms < 7472966967498, sum of first 10^6 primes minus 1. - Ray Chandler, Oct 07 2005

Cf. A007504, A110997.

Intersection of A000290 and A237589.

Select[Accumulate[Prime[Range[1000]]]-1,IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Jun 09 2022 *)

lista(nn) = my(s=-1); forprime(p=2, nn, s+=p; if(issquare(s), print1(s, ", "))) \\ Jinyuan Wang, Aug 10 2023

a(6) corrected by Ray Chandler, Oct 07 2005

a(7) from Jinyuan Wang, Aug 10 2023

A110997 Powers equal to (sum of first k primes) minus 1.

Original entry on oeis.org

1, 4, 9, 16, 27, 128, 196, 839056, 7796654478001

Offset: 1

Walter Kehowski, Sep 30 2005

I have checked out to the first 250000 primes but the last entry is at the 504th prime, 3607.

			128 is a term because 128 = -1 + Sum_{i=1..10} prime(i) = 2^7.

Cf. A007504, A110979, A110996.

Intersection of A001597 and A237589.

with(numtheory); egcd := proc(n) local L; L:=map(proc(z) z[2] end, ifactors(n)[2]); igcd(op(L)) end: s := proc(n) option remember; local p; if n=1 then [1,2] else p:=ithprime(n); [n,s(n-1)[2]+p] fi end; t := proc(n) option remember; [n,s(n)[2]-1] end; PW:=[]; for z to 1 do for j from 1 to 250000 do if egcd(t(j)[2])>1 or t(j)[2]=1 then PW:=[op(PW),t(j)] fi od od; PW;

s = 0; Do[s = s + Prime[n]; If[s == 2 || GCD @@ Transpose[ FactorInteger[s - 1]][[2]] > 1, Print[s - 1]], {n, 10^6}] (* Robert G. Wilson v, Oct 02 2005 *)

Initial 1 and a(9) added by Jinyuan Wang, Aug 10 2023

cp's OEIS Frontend

A343809 Divide the positive integers into subsets of lengths given by successive primes, then reverse the order of terms in each subset.

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A110979 Squares equal to the sum of the first k primes minus 1.

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A110997 Powers equal to (sum of first k primes) minus 1.

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