A126646 -id:A126646 - cp's OEIS frontend

A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

1, 3, 1, 7, 4, 1, 15, 13, 6, 1, 31, 40, 31, 8, 1, 63, 121, 156, 57, 12, 1, 127, 364, 781, 400, 133, 14, 1, 255, 1093, 3906, 2801, 1464, 183, 18, 1, 511, 3280, 19531, 19608, 16105, 2380, 307, 20, 1, 1023, 9841, 97656, 137257, 177156, 30941, 5220, 381, 24, 1, 2047, 29524, 488281, 960800, 1948717

Offset: 0

Omar E. Pol, Sep 09 2018

T(n,k) is also the sum of the divisors of the n-th nonnegative power of the k-th prime, n >= 0, k >= 1.

			The corner of the square array is as follows:
         A126646 A003462 A003463  A023000    A016123    A091030     A091045
A000012        1,      1,      1,       1,         1,         1,          1, ...
A008864        3,      4,      6,       8,        12,        14,         18, ...
A060800        7,     13,     31,      57,       133,       183,        307, ...
A131991       15,     40,    156,     400,      1464,      2380,       5220, ...
A131992       31,    121,    781,    2801,     16105,     30941,      88741, ...
A131993       63,    364,   3906,   19608,    177156,    402234,    1508598, ...
.......      127,   1093,  19531,  137257,   1948717,   5229043,   25646167, ...
.......      255,   3280,  97656,  960800,  21435888,  67977560,  435984840, ...
.......      511,   9841, 488281, 6725601, 235794769, 883708281, 7411742281, ...
...

Rows 0-5: A000012, A008864, A060800, A131991, A131992, A131993.
Columns 1-15: A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.
Main diagonal gives A319074.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A000040, A000203, A006093, A319075.

T(n, k) = sigma(prime(k)^n); \\ Michel Marcus, Sep 13 2018

T(n,k) = A000203(A000040(k)^n).
T(n,k) = Sum_{j=0..n} A000040(k)^j.
T(n,k) = Sum_{j=0..n} A319075(j,k).
T(n,k) = (A000040(k)^(n+1) - 1)/(A000040(k) - 1).
T(n,k) = (A000040(k)^(n+1) - 1)/A006093(k).

A330725 a(0) = 0; thereafter a(n) = a(n-1) + sigma(n) if sigma(n) > a(n-1), otherwise a(n) = a(n-1) - sigma(n), where sigma is the sum of divisors function A000203.

Original entry on oeis.org

0, 1, 4, 0, 7, 1, 13, 5, 20, 7, 25, 13, 41, 27, 3, 27, 58, 40, 1, 21, 63, 31, 67, 43, 103, 72, 30, 70, 14, 44, 116, 84, 21, 69, 15, 63, 154, 116, 56, 0, 90, 48, 144, 100, 16, 94, 22, 70, 194, 137, 44, 116, 18, 72, 192, 120, 0, 80, 170, 110, 278, 216, 120, 16

Offset: 0

Alois P. Heinz, Jan 11 2020

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000203, A008344, A008345, A008348, A076042, A111466, A126646, A327442, A331165.

a:= proc(n) option remember; `if`(n=0, 0, ((s, t)-> s+
      `if`(s

nxt[{n_,a_}]:={n+1,If[DivisorSigma[1,n+1]>a,a+DivisorSigma[1,n+1],a- DivisorSigma[ 1,n+1]]}; NestList[nxt,{0,0},70][[All,2]] (* Harvey P. Dale, May 14 2022 *)

A331165 a(n) = a(n-1) + p(n) if p(n) > a(n-1), otherwise a(n) = a(n-1) - p(n), where p is the partition function A000041 (assuming a(n) = 0 for n < 0).

Original entry on oeis.org

1, 0, 2, 5, 0, 7, 18, 3, 25, 55, 13, 69, 146, 45, 180, 4, 235, 532, 147, 637, 10, 802, 1804, 549, 2124, 166, 2602, 5612, 1894, 6459, 855, 7697, 16046, 5903, 18213, 3330, 21307, 42944, 16929, 48114, 10776, 55359, 2185, 65446, 140621, 51487, 157045, 32291, 179564

Offset: 0

Alois P. Heinz, Jan 11 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000041, A008344, A008345, A008348, A076042, A111466, A126646, A327442, A330725.

a:= proc(n) option remember; `if`(n<0, 0, ((s, t)-> s+
     `if`(s

a[n_] := a[n] = If[n<0, 0, With[{a1 = a[n-1], p = PartitionsP[n]}, If[p>a1, a1 + p, a1 - p]]];
a /@ Range[0, 70] (* Jean-François Alcover, Jan 05 2021 *)

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(p = numbpart(n-1)); va[n] = va[n-1] - p; if (va[n] < 0, va[n] += 2*p);); va;} \\ Michel Marcus, Jan 06 2021

A334627 T(n,k) is the number of k's in the n-th row of Stern's triangle (A337277); triangle T(n,k), n >= 0, 1 <= k <= A000045(n+1), read by rows.

Original entry on oeis.org

1, 3, 5, 2, 7, 4, 4, 9, 6, 8, 4, 4, 11, 8, 12, 8, 12, 0, 8, 4, 13, 10, 16, 12, 20, 4, 16, 8, 8, 4, 8, 4, 4, 15, 12, 20, 16, 28, 8, 28, 12, 16, 8, 24, 8, 16, 8, 4, 4, 8, 8, 8, 0, 4, 17, 14, 24, 20, 36, 12, 40, 20, 24, 12, 40, 12, 36, 16, 8, 16, 28, 16, 24, 4, 8, 8, 16, 4, 12, 8, 8, 0, 12, 4, 8, 0, 0, 4

Offset: 0

Alois P. Heinz, Sep 09 2020

All terms in the first column are odd, all other terms are even.

			T(0,1) = 1 because Stern's triangle has one 1 in row n=0.
T(2,2) = 2 because Stern's triangle has two 2's in row n=2.
T(4,3) = 8 because Stern's triangle has eight 3's in row n=4.
Triangle T(n,k) begins:
   1;
   3;
   5,  2;
   7,  4,  4;
   9,  6,  8,  4,  4;
  11,  8, 12,  8, 12, 0,  8,  4;
  13, 10, 16, 12, 20, 4, 16,  8,  8, 4,  8, 4,  4;
  15, 12, 20, 16, 28, 8, 28, 12, 16, 8, 24, 8, 16, 8, 4, 4, 8, 8, 8, 0, 4;
  ...

Alois P. Heinz, Rows n = 0..21, flattened

Column k=1 gives A005408.
Row sums give A126646.
Row lengths give A000045(n+1).
Cf. A000244, A337277.

b:= proc(n) option remember; `if`(n=0, [1], (l-> [1, l[1],
      seq([l[i-1]+l[i], l[i]][], i=2..nops(l)), 1])(b(n-1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add(x^j, j=b(n))):
seq(T(n), n=0..8);

Sum_{k=1..A000045(n+1)} k * T(n,k) = A000244(n).

A341746 If the runs in the binary expansion of n are (r_1, ..., r_k), then the runs in the binary expansion of a(n) are (r_1 + ... + r_k, r_1, ..., r_{k-1}).

Original entry on oeis.org

1, 6, 3, 14, 29, 28, 7, 30, 123, 122, 61, 60, 121, 120, 15, 62, 503, 502, 251, 250, 501, 500, 125, 124, 499, 498, 249, 248, 497, 496, 31, 126, 2031, 2030, 1015, 1014, 2029, 2028, 507, 506, 2027, 2026, 1013, 1012, 2025, 2024, 253, 252, 2023, 2022, 1011, 1010

Offset: 1

Rémy Sigrist, Feb 18 2021

This sequence is related to A341694 (see Formula section).
All terms are distinct.
If a(n) > n, then a(n) does not appear in A341699.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   1     1       1          1
   2     6      10        110
   3     3      11         11
   4    14     100       1110
   5    29     101      11101
   6    28     110      11100
   7     7     111        111
   8    30    1000      11110
   9   123    1001    1111011
  10   122    1010    1111010
  11    61    1011     111101
  12    60    1100     111100
  13   121    1101    1111001
  14   120    1110    1111000
  15    15    1111       1111

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, Binary plot of the first 255 terms
Index entries for sequences related to binary expansion of n

Cf. A005811, A070939, A090996, A126646, A341694, A341699.

toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
a(n) = { my (r=toruns(n)); fromruns(concat(vecsum(r), r[1..#r-1])) }

A341694(a(n), k) = A341694(n, k+1).
a(n) = n iff n belongs to A126646.
A090996(a(n)) = A070939(n).
A090996(a(n)) > A070939(a(n)) / 2.
A005811(a(n)) = A005811(n).

A367669 Number of degree 3 number fields unramified outside the first n prime numbers.

Original entry on oeis.org

0, 9, 32, 108, 360, 1168, 3638, 11492, 35638, 111059

Offset: 1

Robin Visser, Nov 26 2023

B. Matschke showed that a(11) = 340618 assuming the Generalized Riemann Hypothesis.

			For n = 1, there are no cubic number fields unramified away from 2, so a(1) = 0.
For n = 2, the a(2) = 9 cubic number fields unramified away from {2,3} can be given by Q(a) where a is a root of x^3 - 3x - 1, x^3 - 2, x^3 + 3x - 2, x^3 - 3, x^3 - 3x - 4, x^3 - 3x - 10, x^3 - 12, x^3 - 6, or x^3 - 9x - 6.

K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), no. 219, 1213-1237.
J. W. Jones and D. P. Roberts, A database of number fields, LMS J. Comput. Math. 17 (2014), no. 1, 595-618.
B. Matschke, Number fields unramified outside S.

Cf. A126646 (degree 2), A368057 (degree 4).

A368057 Number of degree 4 number fields unramified outside the first n prime numbers.

Original entry on oeis.org

7, 62, 379, 2165, 12315, 67887

Offset: 1

Robin Visser, Dec 09 2023

			For n = 1, the a(1) = 7 degree 4 number fields unramified away from {2} can be given by Q(a) where a is a root of x^4 + 1, x^4 - 2x^2 + 2, x^4 - 2x^2 - 1, x^4 - 2, x^4 + 2, x^4 - 4x^2 + 2, or x^4 + 4x^2 + 2.

J. W. Jones and D. P. Roberts, A database of number fields, LMS J. Comput. Math. 17 (2014), no. 1, 595-618.

Cf. A126646 (degree 2), A367669 (degree 3).

A368479 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} 2^j * j^k.

Original entry on oeis.org

1, 0, 3, 0, 2, 7, 0, 2, 10, 15, 0, 2, 18, 34, 31, 0, 2, 34, 90, 98, 63, 0, 2, 66, 250, 346, 258, 127, 0, 2, 130, 714, 1274, 1146, 642, 255, 0, 2, 258, 2074, 4810, 5274, 3450, 1538, 511, 0, 2, 514, 6090, 18458, 24810, 19098, 9722, 3586, 1023

Offset: 0

Seiichi Manyama, Dec 26 2023

			Square array begins:
    1,   0,    0,     0,      0,      0,       0, ...
    3,   2,    2,     2,      2,      2,       2, ...
    7,  10,   18,    34,     66,    130,     258, ...
   15,  34,   90,   250,    714,   2074,    6090, ...
   31,  98,  346,  1274,   4810,  18458,   71626, ...
   63, 258, 1146,  5274,  24810, 118458,  571626, ...
  127, 642, 3450, 19098, 107754, 616122, 3557610, ...

OEIS Wiki, Eulerian polynomials.

Columns k=0..3 give A126646, A036799, A036800, A036827.
Main diagonal gives A368466.
Cf. A103438, A173018, A368486.

PARI
```
T(n, k) = sum(j=0, n, 2^j*j^k);
```

G.f. of column k: 2*x*A_k(2*x)/((1-x) * (1-2*x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.

A369277 Distinct values of A369317, in order of appearance.

Original entry on oeis.org

1, 3, 7, 5, 15, 9, 31, 21, 11, 13, 63, 17, 51, 127, 85, 33, 73, 255, 27, 45, 511, 65, 341, 23, 107, 29, 19, 189, 195, 25, 1023, 273, 69, 81, 455, 129, 585, 79, 93, 819, 207, 121, 243, 2047, 1365, 279, 635, 443, 889, 465, 4095, 257, 1419, 1677, 1057, 313, 1335

Offset: 1

Rémy Sigrist, Jan 20 2024

All terms are even.
This sequence is infinite as it contains A126646.
Will every odd number appear in the sequence?
Empirically, each odd number, say v, appears in A369317, and the first index is of the form v*2^k - 1 for some k > 0 (see Example section).

			The first terms, alongside their index m in A369317, in decimal and in binary, are:
  n   a(n)  m     bin(a(n))  bin(m)
  --  ----  ----  ---------  ------------
   1     1     1          1             1
   2     3     5         11           101
   3     7    27        111         11011
   4     5    39        101        100111
   5    15   119       1111       1110111
   6     9   287       1001     100011111
   7    31   495      11111     111101111
   8    21   671      10101    1010011111
   9    11   703       1011    1010111111
  10    13   831       1101    1100111111
  11    63  2015     111111   11111011111
  12    17  2175      10001  100001111111

Rémy Sigrist, PARI program
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A126646, A369317.

PARI
```
See Links section.
```

A382412 Numbers with no zeros in their base-7 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90

Offset: 1

Paolo Xausa, Mar 24 2025

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

Cf. zeroless numbers in other bases: A126646 (base 2), A032924 (base 3), A023705 (base 4), A023721 (base 5), A248910 (base 6), A255805 (base 8), A255808 (base 9), A052382 (base 10).

Cf. A007093, A043393, A249102, A382413 (complement).

Select[Range[100], DigitCount[#, 7, 0] == 0 &]

cp's OEIS Frontend

A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

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Formula

A330725 a(0) = 0; thereafter a(n) = a(n-1) + sigma(n) if sigma(n) > a(n-1), otherwise a(n) = a(n-1) - sigma(n), where sigma is the sum of divisors function A000203.

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Mathematica

A331165 a(n) = a(n-1) + p(n) if p(n) > a(n-1), otherwise a(n) = a(n-1) - p(n), where p is the partition function A000041 (assuming a(n) = 0 for n < 0).

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A334627 T(n,k) is the number of k's in the n-th row of Stern's triangle (A337277); triangle T(n,k), n >= 0, 1 <= k <= A000045(n+1), read by rows.

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A341746 If the runs in the binary expansion of n are (r_1, ..., r_k), then the runs in the binary expansion of a(n) are (r_1 + ... + r_k, r_1, ..., r_{k-1}).

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A367669 Number of degree 3 number fields unramified outside the first n prime numbers.

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A368057 Number of degree 4 number fields unramified outside the first n prime numbers.

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A368479 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} 2^j * j^k.

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A369277 Distinct values of A369317, in order of appearance.

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