A006005 -id:A006005 - cp's OEIS frontend

A237589 Sum of first n odd noncomposite numbers.

1, 4, 9, 16, 27, 40, 57, 76, 99, 128, 159, 196, 237, 280, 327, 380, 439, 500, 567, 638, 711, 790, 873, 962, 1059, 1160, 1263, 1370, 1479, 1592, 1719, 1850, 1987, 2126, 2275, 2426, 2583, 2746, 2913, 3086, 3265, 3446, 3637, 3830, 4027, 4226, 4437, 4660, 4887

Offset: 1

Omar E. Pol, Feb 21 2014

Partial sums of A006005.

			For n = 5 the first five odd noncomposite numbers are 1, 3, 5, 7, 11, so a(5) = 1 + 3 + 5 + 7 + 11 = 27.

Cf. A000040, A006005, A007504, A008578, A014284.

a := proc(n) option remember; `if`(n=1, 1, a(n-1) + ithprime(n)) end:
seq(a(n), n=1..49); # Peter Luschny, Sep 20 2018

a[1]=1; a[n_]:=a[n]=a[n-1]+Prime[n]; Table[a[n], {n,1,49}] (* Robert P. P. McKone, Jan 18 2022 *)

terms(n) = my(s=1, i=0); forprime(p=3, , if(i >= n, break, print1(s, ", "); i++; s=s+p))
/* Print initial 50 terms as follows */
terms(50) \\ Felix Fröhlich, Sep 20 2018

a(n) = A007504(n) - 1 = A014284(n+1) - 2.

A327249 Expansion of Sum_{k>=1} x^k * (1 + k * x^k)^k.

Original entry on oeis.org

1, 2, 1, 5, 1, 14, 1, 17, 28, 26, 1, 160, 1, 50, 251, 321, 1, 622, 1, 1607, 1030, 122, 1, 6257, 3126, 170, 2917, 12202, 1, 27291, 1, 28929, 6656, 290, 84036, 117721, 1, 362, 13183, 407121, 1, 417881, 1, 220100, 850312, 530, 1, 2246465, 823544, 2100626

Offset: 1

Ilya Gutkovskiy, Sep 15 2019

nonn

Cf. A006005 (positions of 1's), A087909, A217668, A260180, A327238.

[&+[(n div d)^(d-1)*Binomial(n div d,d-1):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Sep 15 2019

nmax = 50; CoefficientList[Series[Sum[x^k (1 + k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^(# - 1) Binomial[n/#, # - 1] &], {n, 1, 50}]

a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d,d-1)); \\ Michel Marcus, Sep 15 2019

a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(n/d,d-1).

A348515 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1).

Original entry on oeis.org

1, -1, 1, -2, 1, 0, 1, -2, 2, 0, 1, -3, 1, 0, 2, -3, 1, -1, 1, -1, 2, 0, 1, -4, 2, 0, 2, -1, 1, -2, 1, -3, 2, 0, 2, -3, 1, 0, 2, -4, 1, 0, 1, -1, 3, 0, 1, -5, 2, -1, 2, -1, 1, 0, 2, -4, 2, 0, 1, -4, 1, 0, 3, -4, 2, 0, 1, -1, 2, -2, 1, -4, 1, 0, 3, -1, 2, 0, 1, -5

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A038548, A048272, A113652, A228441, A305152, A333781, A348660, A348951, A348952, A258998.

Table[DivisorSum[n, (-1)^(n/# + 1) &, # <= Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A348515(n) = sumdiv(n,d,if((d*d)<=n,(-1)^(1 + (n/d)),0)); \\ Antti Karttunen, Nov 05 2021

from sympy import divisors
def a(n): return sum((-1)**(n//d + 1) for d in divisors(n) if d*d <= n)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Nov 22 2021

G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k^2) / (1 + x^k).
a(n) = 1 iff n = 1 or n is an odd prime (A006005). - Bernard Schott, Nov 22 2021
a(n) = A258998(n) - A348951(n). - Ridouane Oudra, Aug 21 2025

A353605 Product_{n>=1} (1 + a(n)x^n) = 1 + Sum_{n>=1} prime(n+1)x^n.

Original entry on oeis.org

3, 5, -8, 35, -52, 118, -320, 1597, -2016, 6616, -16064, 40516, -122552, 381606, -903176, 4389807, -7597004, 22835416, -61172890, 188526110, -486889660, 1550995910, -4093173788, 11608277912, -33815484714, 105179650108, -279683446078, 883705997682, -2366564864546

Offset: 1

Ilya Gutkovskiy, May 07 2022

sign

Cf. A006005, A065091, A147557, A353606.

nn = 29; f[x_] := Product[(1 + a[n] x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - Sum[Prime[k + 1] x^k, {k, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A384232 Smallest number whose largest odd noncomposite divisor is its n-th divisor.

Original entry on oeis.org

1, 3, 6, 20, 42, 84, 156, 312, 684, 1020, 1380, 1860, 3480, 3720, 4920, 7320, 10980, 14640, 16920, 21960, 26280, 34920, 45720, 59640, 69840, 89880, 106680, 125160, 145320, 177240, 213360, 244440, 269640, 354480, 320040, 375480, 435960, 456120, 531720, 647640, 708120

Offset: 1

Omar E. Pol, May 23 2025

nonn

This coincide with A087134 except for the second term because here a(2) = 3 and there A087134(2) = 2.

			The divisors of 42 are [1, 2, 3, 6, 7, 14, 21, 42] and the largest odd noncomposite divisor is 7 and 7 is its 5th divisor, so a(5) = 42 because 42 the smallest number having that property.

Row 1 of A384233.
Companion of A383402.
Cf. A006005, A027750, A087134.

With[{t = Table[FirstPosition[Divisors[n], FactorInteger[n/2^IntegerExponent[n, 2]][[-1, 1]]][[1]], {n, 1, 10^6}]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* Amiram Eldar, May 23 2025 *)

A384233 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd noncomposite divisor is its k-th divisor, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 10, 20, 16, 9, 12, 28, 42, 32, 11, 14, 30, 60, 84, 64, 13, 15, 40, 66, 132, 156, 128, 17, 18, 44, 78, 168, 204, 312, 256, 19, 21, 52, 88, 198, 228, 408, 684, 512, 23, 22, 56, 102, 210, 264, 456, 696, 1020, 1024, 25, 24, 68, 104, 220, 276, 468, 744, 1140, 1380

Offset: 1

Omar E. Pol, May 22 2025

This is a permutation of the positive integers.

			The corner 15 X 15 of the square array is as follows:
      1,  3,  6,  20,  42,  84, 156, 312,  684, 1020, 1380, 1860, 3480, 3720,  4920, ...
      2,  5, 10,  28,  60, 132, 204, 408,  696, 1140, 1740, 2220, 3660, 4440,  5160, ...
      4,  7, 12,  30,  66, 168, 228, 456,  744, 1332, 2040, 2460, 4020, 5580,  5640, ...
      8,  9, 14,  40,  78, 198, 264, 468,  780, 1368, 2088, 2580, 4140, 6960,  6360, ...
     16, 11, 15,  44,  88, 210, 276, 510,  816, 1392, 2232, 2664, 4260, 7224,  6660, ...
     32, 13, 18,  52, 102, 220, 330, 552,  828, 1476, 2280, 2760, 4380, 7632,  7080, ...
     64, 17, 21,  56, 104, 234, 342, 570,  888, 1488, 2436, 2820, 4740, 7896,  7380, ...
    128, 19, 22,  68, 110, 252, 348, 612,  912, 1548, 2544, 2952, 4872, 8280,  7440, ...
    256, 23, 24,  70, 114, 260, 372, 624,  930, 1560, 2604, 3096, 4980, 8496,  7740, ...
    512, 25, 26,  76, 120, 272, 390, 660,  936, 1656, 2736, 3180, 5208, 8784,  8880, ...
   1024, 27, 33,  80, 126, 304, 396, 690,  984, 1692, 2790, 3384, 5220, 8904,  9912, ...
   2048, 29, 34,  90, 130, 306, 414, 792, 1032, 1710, 2832, 3420, 5256, 9030, 10248, ...
   4096, 31, 35,  92, 136, 336, 420, 870, 1044, 1776, 2928, 3540, 5328, 9324, 10440, ...
   8192, 37, 36,  99, 138, 340, 440, 920, 1104, 1908, 3060, 3612, 5340, 9648, 10512, ...
  16384, 41, 38, 100, 140, 368, 444, 966, 1110, 1932, 3108, 3816, 5520, 9660, 10836, ...
  ...
The divisors of 42 are [1, 2, 3, 6, 7, 14, 21, 42] and the largest odd noncomposite divisor is 7 and 7 is its 5th divisor, so T(1,5) = 42 because 42 the smallest number having that property.

Index entries for sequences that are permutations of the natural numbers.

Companion of A383961.
Row 1 gives A384232.
Column 1 gives A000079.
Cf. A006005, A027750, A061345, A065091, A087134.

f[n_] := FirstPosition[Divisors[n], FactorInteger[n/2^IntegerExponent[n, 2]][[-1, 1]]][[1]]; seq[m_] := Module[{t = Table[0, {m}, {m}], v = Table[0, {m}], c = 0, k = 1, i, j}, While[c < m*(m + 1)/2, i = f[k]; If[i <= m, j = v[[i]] + 1; If[j <= m - i + 1, t[[i]][[j]] = k; v[[i]]++; c++]]; k++]; Table[t[[j]][[i - j + 1]], {i, 1, m}, {j, 1, i}] // Flatten]; seq[11] (* Amiram Eldar, May 23 2025 *)

Conjecture: T(n,2) = A061345(n).

A130758 a(n) = n if n is not an odd prime number. Otherwise, a(n) = k, where k is the smallest integer such that n < 10^k.

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 6, 1, 8, 9, 10, 2, 12, 2, 14, 15, 16, 2, 18, 2, 20, 21, 22, 2, 24, 25, 26, 27, 28, 2, 30, 2, 32, 33, 34, 35, 36, 2, 38, 39, 40, 2, 42, 2, 44, 45, 46, 2, 48, 49, 50, 51, 52, 2, 54, 55, 56, 57, 58, 2, 60, 2, 62, 63, 64, 65, 66, 2, 68

Offset: 1

Mohammad K. Azarian, Aug 17 2007

nonn

Cf. A000027, A001477, A006005, A131516.

Table[If[OddQ[n]&&PrimeQ[n],IntegerLength[n],n],{n,0,70}] (* Harvey P. Dale, Apr 13 2016 *)

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A131195 Nonprime record values of Euler's totient function (A000010): 1 and composite n such that phi(n) is greater than all smaller composites.

Original entry on oeis.org

1, 4, 8, 9, 15, 21, 25, 35, 49, 65, 77, 85, 91, 115, 119, 121, 143, 161, 169, 187, 203, 209, 221, 247, 253, 287, 289, 319, 323, 341, 361, 391, 403, 437, 451, 473, 481, 493, 517, 527, 529, 583, 589, 611, 629, 649, 667, 689, 697, 703, 713, 731, 767, 779, 799, 817

Offset: 1

Alonso del Arte, Oct 20 2007

Since phi(p) = p - 1, allowing prime numbers in this sequence would make it A006005, the primes with a 1 replacing the initial 2.

Number of terms < 10^k, k=1,2,3,...: 4, 13, 61, 310, 1628, 9029, 51207, 295132, ..., . Robert G. Wilson v, Feb 19 2019

			a(3) = 8 because phi(8) = 4, which is greater than phi(4) = phi(6) = 2. (phi(5) = 4 and phi(7) = 6 are ignored because 5 and 7 are prime).

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A006005.

htcList = {1}; i = 4; currMax = 1; searchMax = 1000; While[i < searchMax, If[Not[PrimeQ[i]] && EulerPhi[i] > currMax, htcList = {htcList, i}; currMax = EulerPhi[i]]; i++ ]; Flatten[htcList]

A278287 Number of primes of the form 4k+3 such that 4(n - k) - 3 is 1 or prime where (4k+3)/3 >= n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 3, 2, 2, 4, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 2, 3, 1, 3, 5, 2, 2, 5, 2, 1, 4, 3, 2, 4, 3, 2, 4, 3, 2, 5, 3, 2, 5, 3, 1, 6, 3, 2, 4, 4, 2, 4, 4, 3, 4, 3, 2, 5, 2, 2, 6, 2, 2, 6, 1, 2, 4, 1, 2, 4, 2, 1, 5, 3, 3, 5, 2, 3, 6, 3

Offset: 1

Juri-Stepan Gerasimov, Dec 24 2016

nonn

Where 4(n - k) - 3 is a positive noncomposite number of the form 4m+1 (A280084).

			a(6) = 2 because 4*6 = 1 + 23 where (4*5+3)/3 = 23/3 > 6 and 4*6 = 5 + 19 where (4*4+3)/3 = 19/3 > 6.

Cf. A002144, A006005, A280084.

A298603 Number of partitions of n into odd prime parts (including 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 22, 26, 31, 36, 42, 49, 56, 65, 75, 86, 98, 112, 127, 144, 164, 185, 209, 235, 264, 297, 332, 372, 416, 463, 516, 574, 638, 708, 785, 869, 960, 1061, 1171, 1291, 1421, 1563, 1718, 1886, 2070, 2269, 2484, 2718, 2972, 3247, 3545, 3868, 4216, 4592

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

nonn

Partial sums of A099773.

			a(6) = 4 because we have [5, 1], [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

Index entries for related partition-counting sequences

Cf. A000607, A006005, A008578, A034891, A099773, A298604.

nmax = 62; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: (1/(1 - x))*Product_{k>=2} 1/(1 - x^prime(k)).

cp's OEIS Frontend

A237589 Sum of first n odd noncomposite numbers.

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A327249 Expansion of Sum_{k>=1} x^k * (1 + k * x^k)^k.

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A348515 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1).

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A353605 Product_{n>=1} (1 + a(n)*x^n) = 1 + Sum_{n>=1} prime(n+1)*x^n.

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A384232 Smallest number whose largest odd noncomposite divisor is its n-th divisor.

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A384233 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd noncomposite divisor is its k-th divisor, n >= 1, k >= 1.

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A130758 a(n) = n if n is not an odd prime number. Otherwise, a(n) = k, where k is the smallest integer such that n < 10^k.

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A131195 Nonprime record values of Euler's totient function (A000010): 1 and composite n such that phi(n) is greater than all smaller composites.

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A278287 Number of primes of the form 4k+3 such that 4(n - k) - 3 is 1 or prime where (4k+3)/3 >= n.

A353605 Product_{n>=1} (1 + a(n)x^n) = 1 + Sum_{n>=1} prime(n+1)x^n.