A071148 -id:A071148 - cp's OEIS frontend

A373827 Position of first appearance of n in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

4, 1, 38, 6781, 26100, 23238

Offset: 1

Gus Wiseman, Jun 22 2024

Positions of first appearances in A373820 (run-lengths of A027833 with 1 prepended).

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with antiruns (differing by > 2):
(3), (5), (7,11), (13,17), (19,23,29), (31,37,41), (43,47,53,59), ...
with lengths:
1, 1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, ...
which have runs:
(1,1), (2,2), (3,3), (4), (3), (6), (2), (5), (2), (6), (2,2), (4), ...
with lengths:
2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
with positions of first appearances a(n).

Positions of first appearances in A373820.
For runs instead of antiruns we have A373825, sorted A373824.
The sorted version is A373826.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
Cf. A001359, A006512, A025584, A027833, A037201, A038664, A054265, A067774, A176246, A251092, A373404, A373405, A373406.

t=Length/@Split[Length /@ Split[Select[Range[3,10000],PrimeQ],#1+2!=#2&]//Most]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

A058845 Numbers n such that the sum of the first n odd primes is palindromic.

Original entry on oeis.org

1, 2, 49, 54, 172, 921, 1421, 12485, 78653, 1969457, 5606014, 90638394

Offset: 1

Patrick De Geest, Dec 15 2000

a(13) > 2.8*10^12, if it exists. - Giovanni Resta, Sep 01 2018

			Palindromes are 3 + 5 + 7 + 11 + 13 + 17 + ... + z. For values of z see A058846.

P. De Geest, Palindromic Sums

Cf. A058846, A058847, A038582, A071148.

Position[Accumulate[Prime[Range[2,91*10^6]]],?PalindromeQ]//Flatten (* _Harvey P. Dale, Sep 12 2023 *)

A058846 Numbers k such that the sum of odd primes up to k is palindromic.

Original entry on oeis.org

3, 5, 229, 257, 1031, 7213, 11863, 133853, 1002073, 31924583, 97137589, 1837875227

Offset: 1

Patrick De Geest, Dec 15 2000

Sum is 3 + 5 + 7 + 11 + 13 + 17 + ... + k.

a(13) > 8.79*10^13, if it exists. - Giovanni Resta, Sep 01 2018

Patrick De Geest, Palindromic Sums

Cf. A065091, A058845, A058847, A071148, A038584.

lista(nn) = {s = 0; p = 2; for (n=1, nn, p = nextprime(p+1); s += p; d = digits(s); if (Vecrev(d) == d, print1(p, ", ")););} \\ Michel Marcus, Aug 09 2017

from sympy import primerange
def ispal(n): s = str(n); return s == s[::-1]
def afind(limit):
  s = 0
  for p in primerange(3, limit):
    s += p
    if ispal(s): print(p, end=", ")
afind(2*10**6) # Michael S. Branicky, Mar 05 2021

a(n) = A065091(A058845(n)). - R. J. Mathar, Sep 09 2015

A058847 Palindromes that are the sum of consecutive initial odd primes.

Original entry on oeis.org

3, 8, 5115, 6336, 81218, 3091903, 7843487, 792727297, 37706560773, 30398022089303, 263888373888362, 81120957675902118

Offset: 1

Patrick De Geest, Dec 15 2000

3 + 5 + 7 + 11 + 13 + 17 + ... + z = n. For values of z see A058846.

a(13) > 1.2*10^26, if it exists. - Giovanni Resta, Sep 01 2018

P. De Geest, Palindromic Sums

Cf. A058845, A058846, A038583.

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; Select[ Accumulate[ Prime[Range[2,25000000]]],palQ] (* The program takes a long time to run and only generates the first 11 terms of the sequence *) (* Harvey P. Dale, Aug 24 2014 *)

a(n) = A071148(A058845(n)). - R. J. Mathar, Sep 09 2015

A097961 Numbers k such that the sum of the first k odd primes is divisible by k.

Original entry on oeis.org

1, 2, 3, 60, 73, 357, 690, 970, 1560, 1844, 2016, 2071, 3267, 7034, 22388, 37244, 137166, 808334, 1126996, 3420839, 4971830, 14647946, 15553569, 21957090, 31327140, 90514444, 98576118, 204198604, 210662116, 553825420, 1395717645, 2820805440, 6780317160

Offset: 1

Anne M. Donovan (anned3005(AT)aol.com), Oct 22 2004

			a(1) = 1 since 3 is divisible by 1.
a(2) = 2 since 3 + 5 = 8 is divisible by 2.
a(3) = 3 since 3 + 5 + 7 = 15 is divisible by 3.
a(4) != 4 since 3 + 5 + 7 + 11 = 26 is not divisible by 4.
98576118 * 977748014 = 96382603602329652.

Cf. A071148, A363477.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 2; s = 0; Do[p = NextPrim[p]; s = s + p; If[ Mod[s, n] == 0, Print[n]], {n, 151666666}] (* Robert G. Wilson v, Oct 23 2004 *)

from sympy import sieve
L = sieve.primerange(3, 1.7*10**11); s, k = 0, 0
for p in L:
    s += p;  k += 1
if s%k == 0: print(k, end = ", ")  # Ya-Ping Lu, Jun 16 2023

Numbers k such that A071148(k)/k or (A007504(k+1)-2)/k is an integer.

Sum_{i=1..a(n)} prime(i) = n*A363477(n). - Ya-Ping Lu, Jun 16 2023

More terms from Robert G. Wilson v, Oct 23 2004
a(28)-a(30) from Rémy Sigrist, Sep 25 2016
a(31)-a(33) from Ya-Ping Lu, Jun 16 2023

A286263 The smallest weight possible for a prime vector of order n.

Original entry on oeis.org

2, 8, 19, 26, 43, 56, 79, 104, 127, 166, 223, 258, 307, 348

Offset: 1

Dmitry Kamenetsky, May 05 2017

A prime vector of order n is an array of n distinct primes P = (p_1, p_2, ..., p_n), such that every sum of an odd number of consecutive elements is also prime. The weight of the prime vector is the sum of its elements. For full details see Kamenetsky's paper.
Calculations by Kamenetsky and J. K. Andersen show that a(15-17) are likely to be 443, 522 and 641.
Calculations by J. K. Andersen show that a(18-21) are likely to be 762, 881, 1002 and 1259.
J. K. Andersen found the best upper bounds for a(22-23) as 1716 and 1931.
For odd n, a(n) <= A068873(n) (smallest prime which is a sum of n distinct primes).
For even n, a(n) <= A071148(n) (sum of the first n odd primes).

			The best solution for n=5 is (3,11,5,7,17) with a weight of 43. This is a prime vector because all the generated sums are prime: 3+11+5=19, 11+5+7=23, 5+7+17=29, 3+11+5+7+17=43.

Dmitry Kamenetsky, Prime sums of primes, arXiv:1703.06778 [math.HO], 2017.
Carlos Rivera, Puzzle 875: Vector of primes that generates distinct primes

Cf. A068873, A071148, A286269.

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

Original entry on oeis.org

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

A373817 Positions of terms > 1 in the run-lengths of the first differences of the odd primes.

Original entry on oeis.org

2, 14, 34, 36, 42, 49, 66, 94, 98, 100, 107, 117, 147, 150, 169, 171, 177, 181, 199, 219, 250, 268, 315, 333, 361, 392, 398, 435, 477, 488, 520, 565, 570, 585, 592, 595, 628, 642, 660, 666, 688, 715, 744, 765, 772, 778, 829, 842, 897, 906, 931, 932, 961, 1025

Offset: 1

Gus Wiseman, Jun 23 2024

nonn

Positions of terms > 1 in A333254. In other words, the a(n)-th run of differences of odd primes has length > 1.

			Primes 54 to 57 are {251, 257, 263, 269}, with differences (6,6,6). This is the 49th run, and the first of length > 2.

Positions of adjacent equal prime gaps are A064113.
Positions of adjacent unequal prime gaps are A333214.
Positions of terms > 1 in A333254, run-lengths A373821, firsts A335406.
A000040 lists the primes, differences A001223.
A027833 gives antirun lengths of odd primes, run-lengths A373820.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
Cf. A006560, A006562, A029707, A031217, A037201, A038664, A069010, A073051, A089180, A251092.

Join@@Position[Length /@ Split[Differences[Select[Range[1000],PrimeQ]]] // Most,x_Integer?(#>1&)]

A373823 Half the sum of the n-th maximal run of first differences of odd primes.

Original entry on oeis.org

2, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 6, 1, 3, 2, 1, 3, 2, 3, 4, 2, 1, 2, 1, 2, 7, 2, 3, 1, 5, 1, 6, 2, 6, 1, 5, 1, 2, 1, 12, 2, 1, 2, 3, 1, 5, 9, 1, 3, 2, 1, 5, 7, 2, 1, 2, 7, 3, 5, 1, 2, 3, 4, 6, 2, 3, 4, 2, 4, 5, 1, 5, 1, 3, 2, 3, 4, 2, 1, 2, 6, 4, 2, 4, 2, 3

Offset: 1

Gus Wiseman, Jun 22 2024

nonn

Halved run-sums of A001223.

			The odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with first differences:
2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, ...
with runs:
(2,2), (4), (2), (4), (2), (4), (6), (2), (6), (4), (2), (4), (6,6), ...
with halved sums a(n).

Halved run-sums of A001223.
For run-lengths we have A333254, run-lengths of run-lengths A373821.
Multiplying by two gives A373822.
A000040 lists the primes.
A027833 gives antirun lengths of odd primes (partial sums A029707).
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
Cf. A037201, A038664, A054265, A176246, A251092, A373404, A373825.

Total/@Split[Differences[Select[Range[3,1000],PrimeQ]]]/2

A134182 Difference between the sums of the first 10^n odd primes and the first 10^n odd positive integers > 1.

Original entry on oeis.org

38, 14478, 2688838, 396250152, 52261798440, 6472980453364, 770530574266708, 89262852894258444, 10138479465982004008, 1134379338819040693132, 125436174619351016716668, 13738971133578180130155676, 1493061976858459711065006050, 161191473337955042966337346114

Offset: 1

Enoch Haga, Oct 13 2007

nonn

Original name: 10^n-th difference between cumulative prime and odd sums.

Beginning at 3, compute the sums of the prime and odd sequences at 10^n and take the difference.

			a(1) = (3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31) - (3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21) = 158 - 120 = 38.
a(2) = 14478 because at 10^2, 100 sums of primes and odds, the prime sum is 24678, the odd sum is 10200 and the difference is 14478.

Chai Wah Wu, Table of n, a(n) for n = 1..23 (using terms from A006988 and A099824)

Cf. A006988, A071148, A005563, A099824, A134181.

10 N=1: A=2
20 A=nxtprm(A): B=B+A
30 N=N+2: D=D+N
40 if C=9 then print A;N;B;D;B-D: stop
50 C=C+1: if C<10 then 20

a(n) = A134181(10^n).
a(n) = A099824(n) + prime(10^n+1) - (10^n*(10^n+2)) - 2. - Chai Wah Wu, Mar 30 2020
a(n) = A071148(10^n) - (10^n+1)^2 + 1, where A071148 are the partial sums of odd primes, and N^2 is the sum of the first N odd integers. - M. F. Hasler, Aug 08 2025

Edited by M. F. Hasler, Aug 08 2025

cp's OEIS Frontend

A373827 Position of first appearance of n in the run-lengths (differing by 0) of the antirun-lengths (differing by > 2) of the odd primes.

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A058845 Numbers n such that the sum of the first n odd primes is palindromic.

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A058846 Numbers k such that the sum of odd primes up to k is palindromic.

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PARI

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A058847 Palindromes that are the sum of consecutive initial odd primes.

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A097961 Numbers k such that the sum of the first k odd primes is divisible by k.

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A286263 The smallest weight possible for a prime vector of order n.

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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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Maple

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A373817 Positions of terms > 1 in the run-lengths of the first differences of the odd primes.

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