A001748 -id:A001748 - cp's OEIS frontend

A272470 7 times the primes.

14, 21, 35, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 371, 413, 427, 469, 497, 511, 553, 581, 623, 679, 707, 721, 749, 763, 791, 889, 917, 959, 973, 1043, 1057, 1099, 1141, 1169, 1211, 1253, 1267, 1337, 1351, 1379, 1393, 1477, 1561, 1589, 1603, 1631, 1673, 1687, 1757, 1799, 1841, 1883, 1897

Offset: 1

Omar E. Pol, Apr 30 2016

Column 4 of A272214.

k times the primes (k=1..6): A000040, A100484, A001748, A001749, A001750, A138636.

7 Prime@ Range@ 58 (* Michael De Vlieger, May 01 2016 *)

a(n) = 7*prime(n); \\ Michel Marcus, May 01 2016

from sympy import prime
for n in range(1,1000):print(7*prime(n),end=", ") # Soumil Mandal, May 08 2016

a(n) = 7*prime(n) = 7*A000040(n).

A306771 Numbers m such that m = i + j = i * k and phi(m) = phi(i) + phi(j) = phi(i) * phi(k) for some i, j, k, where phi is the Euler totient function A000010.

Original entry on oeis.org

3, 15, 21, 33, 39, 51, 57, 69, 75, 87, 93, 105, 111, 123, 129, 141, 147, 159, 165, 177, 183, 195, 201, 213, 219, 231, 237, 249, 255, 267, 273, 285, 291, 303, 309, 321, 327, 339, 345, 357, 363, 375, 381, 393, 399, 411, 417, 429, 435, 447, 453, 465, 471, 483, 489

Offset: 1

Michel Lagneau, Mar 09 2019

The 55 terms given in the data section are consistent with a definition "numbers congruent to 3 or 15 mod 18". - Peter Munn, May 12 2020
The observation above is true for the first 10^4 terms. - Amiram Eldar, Dec 08 2020
The observation above is true for every term; see link. - Flávio V. Fernandes, Apr 18 2022
A001748 \ {6, 9} is a subsequence because, for p prime >= 5, 3 * p = p + 2p = p * 3 and phi(3p) = phi(p) + phi(2p) = phi(p) * phi(3) = 2 * (p-1). - Bernard Schott, May 13 2022

			33 is in the sequence because:
phi(33) = phi(11 + 22) = phi(11) + phi(22) = 10 + 10 = 20, and
phi(33) = phi(3 * 11) = phi(3) * phi(11) = 2 * 10 = 20.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Flávio V. Fernandes, A306771(n) equals 3 times A007310(n)
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000010, A000040, A007310, A001748, A381671.

with(numtheory):
for n from 1 to 500 do:
ii:=0:
  for i from 1 to trunc(n/2) while(ii=0) do:
   if phi(i)+ phi(n-i)= phi(n) and n/i = floor(n/i)
      and phi(i)*phi(n/i)=phi(n)
      then
      ii:=1:printf(`%d, `,n):
      else
   fi:
  od:
od:

LinearRecurrence[{1, 1, -1}, {3, 15, 21}, 100] (* Paolo Xausa, Mar 07 2025 *)

isok(m) = {my(phim = eulerphi(m)); for (i=1, m\2, if ((eulerphi(i) + eulerphi(m-i) == phim) && !frac(m/i) && (eulerphi(m/i)*eulerphi(i) == phim), return (1)););} \\ Michel Marcus, Mar 09 2019

From Chai Wah Wu, Mar 07 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
G.f.: x*(3*x^2 + 12*x + 3)/((x - 1)^2*(x + 1)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) (A381671). - Amiram Eldar, Mar 08 2025

Incorrect comment deleted by Peter Munn, May 12 2020

Name corrected by Flávio V. Fernandes, Aug 26 2021 and Peter Munn, Sep 03 2021

A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes.

Original entry on oeis.org

1, 0, 2, 2, 3, 2, 5, 2, 5, 4, 7, 2, 11, 2, 9, 8, 9, 2, 17, 2, 17, 10, 13, 2, 23, 6, 15, 10, 23, 2, 29, 2, 17, 14, 19, 12, 35, 2, 21, 16, 37, 2, 41, 2, 35, 38, 25, 2, 47, 8, 47, 20, 41, 2, 53, 16, 51, 22, 31, 2, 59, 2, 33, 52, 33, 18, 65, 2, 53, 26, 67, 2, 71, 2, 39, 68, 59, 18, 77, 2, 77, 28, 43, 2, 83, 22, 45, 32, 79

Offset: 0

Alois P. Heinz, Dec 04 2018

Numbers where a(n) + A000010(n) != n + 1: A102467. - Robert G. Wilson v, Aug 23 2021

			a(6) = |{0,2,3,4,6}| = 5.
a(9) = |{0,3,6,9}| = 4.
a(10) = |{0,2,4,5,6,8,10}| = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Matt Baker, The Balanced Centrifuge Problem, Math Blog, 2018.
Holly Krieger and Brady Haran, The Centrifuge Problem, Numberphile video (2018)
T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, arXiv:math/9511209 [math.NT], 1995.
Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.
Robert G. Wilson v, Graph of the first 100001 terms.

Cf. A000040, A001248, A001748, A001750, A004280, A008588, A008864, A100484, A103306, A103314, A163300, A182986, A272470, A306275.

a:= proc(n) option remember; local f, b; f, b:=
       map(i-> i[1], ifactors(n)[2]),
       proc(m, i) option remember; m=0 or i>0 and
        (b(m, i-1) or f[i]<=m and b(m-f[i], i))
       end; forget(b); (t-> add(
      `if`(b(j, t) and b(n-j, t), 1, 0), j=0..n))(nops(f))
    end:
seq(a(n), n=0..100);

$RecursionLimit = 4096;
a[1] = 0;
a[n_] := a[n] = Module[{f, b}, f = FactorInteger[n][[All, 1]];
     b[m_, i_] := b[m, i] = m == 0 || i > 0 &&
     (b[m, i - 1] || f[[i]] <= m && b[m - f[[i]], i]);
     With[{t = Length[f]}, Sum[
     If[b[j, t] && b[n - j, t], 1, 0], {j, 0, n}]]];
Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Dec 13 2018, after Alois P. Heinz, corrected and updated Aug 07 2021 *)
f[n_] := Block[{c = 2, k = 2, p = First@# & /@ FactorInteger@ n}, While[k < n, If[ IntegerPartitions[k, All, p, 1] != {} && IntegerPartitions[n - k, All, p, 1] != {}, c++]; k++]; c]; f[0] = 1; f[1] = 0; Array[f, 75] (* Robert G. Wilson v, Aug 22 2021 *)

a(n) = |{ k : k and n-k can be written as a sum of prime factors of n }|.
a(n) = 2 <=> n is prime (A000040).
a(n) >= n-1 <=> n in {1,2,3,4} union { A008588 }.
a(n) = (n+4)/2 <=> n in { A100484 } minus { 4 }.
a(n) = (n+9)/3 <=> n in { A001748 } minus { 9 }.
a(n) = (n+25)/5 <=> n in { A001750 } minus { 25 }.
a(n) = (n+49)/7 <=> n in { A272470 } minus { 49 }.
a(n^2) = n+1 <=> n = 0 or n is prime <=> n in { A182986 }.
a(A001248(n)) = A008864(n).
a(n) is odd <=> n in { A163300 }.
a(n) is even <=> n in { A004280 }.

A164023 Smallest of largest parts in partitions of n into exactly three primes.

Original entry on oeis.org

2, 3, 3, 3, 5, 5, 5, 5, 7, 5, 7, 7, 11, 7, 11, 7, 13, 11, 11, 11, 13, 11, 13, 11, 17, 13, 17, 11, 19, 13, 17, 13, 19, 13, 19, 17, 23, 17, 23, 17, 31, 17, 23, 19, 29, 17, 31, 19, 29, 19, 31, 19, 37, 23, 29, 23, 31, 23, 31, 23, 41, 29, 37, 23, 37, 29, 41, 31, 41, 29, 37, 29, 47, 31

Offset: 6

Reinhard Zumkeller, Aug 08 2009

nonn

a(n) >= floor(n/3); a(A001748(n)) = A000040(n).

			a(16) = min{max(2,3,11),max(2,7,7)} = min{11,7} = 7;
a(17) = min{max(2,2,13),max(2,3,11),max(3,7,7),max(5,5,7)} = min{13,11,7,7} = 7.

R. Zumkeller, Table of n, a(n) for n = 6..2500
Index entries for sequences related to Goldbach conjecture

A164024, A068307.

A253106 Semiprimes with smallest factor <= 3.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 146, 158, 159, 166, 177, 178, 183, 194, 201, 202, 206, 213, 214, 218, 219, 226, 237, 249, 254, 262, 267, 274, 278

Offset: 1

Reinhard Zumkeller, Dec 26 2014

nonn

A253046(a(n)) != a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. Union of A001748 and A100484, A020639, A010051, A001358, A171156, A253046.

a253106 n = a253106_list !! (n-1)
a253106_list = filter f [1..] where
   f x = p <= 3 && a010051' (div x p) == 1  where p = a020639 x

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31

Offset: 1

Stefano Spezia, Jul 14 2019

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Wikipedia, Toeplitz matrix

Cf. A025581, A318173, A325516.

Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

[[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output

a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);

Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]

T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output

[[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

T(n, k) = A025581(n, k)*A000040(1 + k).

A351958 a(1) = 1, followed by numbers k for which the primorial inflation of k is equal to x * p#, where p# is the primorial (A034386) of some prime p, and 1 <= x < p.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 56, 57, 58, 59, 61, 62, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 97, 101, 102, 103, 104, 106, 107, 109, 111, 113, 114, 116, 118, 122, 123, 124

Offset: 1

Antti Karttunen, Apr 04 2022

nonn

Numbers k such that A108951(k) is in A060735.
Numbers k for which A324886(k) is a power of prime (in A000961).
Numbers k such that A108951(k) / A002110(A061395(k)) < A000040(1+A061395(k)), the next prime larger than the greatest prime dividing k.

Positions of 1's in A329040.
Cf. A000961, A002110, A061395, A034386, A060735, A108951, A324886, A351956 (characteristic function).
Subsequences: A000040, A008578, A100484, A001748 \ {9}, A001749 \ {8}.
Cf. also A344591.

isA351958(n) = A351956(n); \\ Uses the program given in A351956.

A370008 a(n) is the greatest prime less than 3*prime(n).

Original entry on oeis.org

5, 7, 13, 19, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, 139, 157, 173, 181, 199, 211, 211, 233, 241, 263, 283, 293, 307, 317, 317, 337, 379, 389, 409, 409, 443, 449, 467, 487, 499, 509, 523, 541, 571, 577, 587, 593, 631, 661, 677, 683, 691, 709, 719, 751

Offset: 1

Clark Kimberling, Feb 09 2024

nonn

			5 < 3*2 < 7 < 3*3 < 11 < 13 < 3*5, so (a(1), a(2), a(3)) = (5,7,13).

Cf. A000040, A001748, A059786, A059788, A370009.

Table[Prime[PrimePi[3*Prime[n]]], {n,1,200}]

a(n) = precprime(3*prime(n)); \\ Michel Marcus, Feb 10 2024

A063534 Numbers k such that C(k) = H(k) + d(k), where C(k) is Chowla's function A048050, H(k) is the half-totient function A023022 and d(k) is the number of divisors function A000005.

Original entry on oeis.org

6, 8, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753

Offset: 1

Jason Earls, Aug 02 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A048050, A023022, A000005, A001748.

Select[Range[1000], DivisorSigma[1, #] - 1 - # == EulerPhi[#]/2 + DivisorSigma[0, #] &] (* Paolo Xausa, Apr 17 2024 *)

C(n)=sigma(n)-n-1;
H(n)=eulerphi(n)/2;
j=[]; for(n=1,1200, if(C(n)==H(n)+numdiv(n),j=concat(j,n))); j

{ n=0; for (m=1, 10^9, if (sigma(m) - m - 1 == eulerphi(m)/2 + numdiv(m), write("b063534.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009

is(n) = {my(f = factor(n)); sigma(f) - n - 1 == eulerphi(f) / 2 + numdiv(f);} \\ Amiram Eldar, Apr 15 2024

Conjecture: a(n) = A001748(n), n <> 2. - R. J. Mathar, Dec 15 2008

The conjecture is false. The least counterexample is a(11546) = 368335 = 5 * 11 * 37 * 181. The next counterexample is 4922335, and there are no more below 10^10. - Amiram Eldar, Apr 15 2024

A171156 Numbers of the form 2p or 3p where p is a prime greater than 3.

Original entry on oeis.org

10, 14, 15, 21, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 146, 158, 159, 166, 177, 178, 183, 194, 201, 202, 206, 213, 214, 218, 219, 226, 237, 249, 254, 262, 267, 274, 278, 291, 298, 302

Offset: 1

Juri-Stepan Gerasimov, Dec 04 2009

Squarefree semiprimes (A006881) which have exactly one prime factor <=3 [R. J. Mathar, Dec 09 2009]

Cf. A253106, A001748, A100484.

isA171156 := proc(n) local f; f := numtheory[factorset](n) ; if numtheory[bigomega](n) = 2 and nops(f) = 2 then if f = {2,3} then false; else (n mod 2) = 0 or (n mod 3) = 0 ; fi; else false; fi; end:
for n from 10 to 500 do if isA171156(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Dec 09 2009

Edited by Charles R Greathouse IV, Mar 25 2010

cp's OEIS Frontend

A272470 7 times the primes.

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A306771 Numbers m such that m = i + j = i * k and phi(m) = phi(i) + phi(j) = phi(i) * phi(k) for some i, j, k, where phi is the Euler totient function A000010.

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A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes.

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A164023 Smallest of largest parts in partitions of n into exactly three primes.

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A253106 Semiprimes with smallest factor <= 3.

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A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

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A351958 a(1) = 1, followed by numbers k for which the primorial inflation of k is equal to x * p#, where p# is the primorial (A034386) of some prime p, and 1 <= x < p.

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A370008 a(n) is the greatest prime less than 3*prime(n).

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