A173390 -id:A173390 - cp's OEIS frontend

A379313 Positive integers whose prime indices are not all composite.

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

Or, positive integers whose prime indices include at least one 1 or prime number.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    17: {7}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}

Partitions of this type are counted by A000041 - A023895.
The "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
The complement is A320629, counted by A023895 (strict A204389).
For a unique prime we have A331915, counted by A379304 (strict A379305).
Positions of nonzeros in A379311.
For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
A377033 gives k-th differences of composite numbers, see A073445, A377034.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
`Cf. A038550, A087436, A173390, A379300, A379301.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@CompositeQ/@prix[#]&]

A173400 n-th difference between consecutive primes=n-th difference between consecutive nonnegative nonprimes.

Original entry on oeis.org

1, 3, 7, 20, 26, 33, 43, 49, 52, 81, 116, 140, 176, 265, 288, 313, 320, 323, 373, 377, 395, 398, 405, 408, 486, 492, 530, 555, 567, 592, 671, 681, 772, 805, 849, 874, 884, 931, 936, 1016, 1030, 1149, 1204, 1324, 1347, 1406, 1464, 1550, 1621, 1639, 1707, 1712

Offset: 1

Juri-Stepan Gerasimov, Feb 17 2010

Numbers n such that A001223(n)=A054546(n).

Cf. A001223, A075526, A173390, A173401, A029707.

A001223(a(n))=A054546(a(n)).

Extended by Charles R Greathouse IV, Mar 25 2010

A173401 Numbers k such that A075526(k-1) = A054546(k).

Original entry on oeis.org

1, 3, 4, 8, 11, 21, 29, 44, 53, 58, 61, 84, 105, 121, 149, 153, 179, 183, 213, 295, 308, 374, 461, 502, 535, 552, 609, 637, 659, 727, 730, 756, 850, 859, 865, 875, 885, 914, 1005, 1055, 1105, 1239, 1261, 1306, 1321, 1407, 1443, 1616, 1654, 1769, 1783, 1795, 1836

Offset: 1

Juri-Stepan Gerasimov, Feb 17 2010

Cf. A054546, A075526, A173390, A173400.

A008578 := proc(n) if n =1 then 1; else ithprime(n-1) ; end if; end proc:
A075526 := proc(n) A008578(n+2)-A008578(n+1) ; end proc:
A054546 := proc(n) if n <= 2 then op(n,[1,3]) ; else A002808(n-1)-A002808(n-2) ; end if; end proc:
for n from 1 to 2000 do if A075526(n-1) = A054546(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, Apr 24 2010

Formula index corrected, a(14) corrected and sequence extended beyond a(14) by R. J. Mathar, Apr 25 2010

A173891 Numbers n such that the n-th noncomposite number plus the (n+2)nd noncomposite number is an even semiprime.

Original entry on oeis.org

1, 3, 16, 37, 40, 47, 55, 56, 74, 103, 108, 111, 119, 130, 161, 165, 185, 188, 195, 200, 219, 240, 272, 273, 292, 340, 359, 388, 420, 427, 465, 466, 509, 521, 554, 600, 606, 622, 630, 634, 668, 683, 684, 703, 710, 711, 734, 762, 792, 814, 822, 823, 830, 831

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2010

nonn

Numbers n such that (A008578(n) + A008578(n+2))/2 is prime.

Is this the same as A173390? - R. J. Mathar, Jul 22 2010

Cf. A006562, A008578, A100484.

A008578 := proc(n) if n = 1 then 1; else ithprime(n-1) ; end if; end proc: for n from 1 to 1000 do sp := A008578(n)+A008578(n+2) ; if type(sp,'even') and numtheory[bigomega](sp) = 2 then printf("%d,",n) ; end if; end do: # R. J. Mathar, Apr 25 2010

prime(a(n+1)-1) = n-th balanced prime.

All but one value shifted by 1, and missing values (40, 634, 762) inserted, by R. J. Mathar, Apr 25 2010

A379542 Second term of the n-th differences of the prime numbers.

Original entry on oeis.org

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280

Offset: 0

Gus Wiseman, Jan 12 2025

sign

Also the inverse zero-based binomial transform of the odd prime numbers.

For all primes (not just odd) we have A007442.
Including 1 in the primes gives A030016.
Column n=2 of A095195.
The version for partitions is A320590 (first column A281425), see A175804, A053445.
For nonprime instead of prime we have A377036, see A377034-A377037.
Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.
A000040 lists the primes, differences A001223, A036263.
A002808 lists the composite numbers, differences A073783, A073445.
A008578 lists the noncomposite numbers, differences A075526.
Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

nn=40;Table[Differences[Prime[Range[nn+2]],n][[2]],{n,0,nn}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).

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A379313 Positive integers whose prime indices are not all composite.

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A173400 n-th difference between consecutive primes=n-th difference between consecutive nonnegative nonprimes.

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A173401 Numbers k such that A075526(k-1) = A054546(k).

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A173891 Numbers n such that the n-th noncomposite number plus the (n+2)nd noncomposite number is an even semiprime.

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A379542 Second term of the n-th differences of the prime numbers.

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