cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Massimo Kofler

Massimo Kofler's wiki page.

Massimo Kofler has authored 116 sequences. Here are the ten most recent ones:

A383025 Centered pentagonal numbers that are deficient.

Original entry on oeis.org

1, 16, 31, 51, 76, 106, 141, 181, 226, 331, 391, 526, 601, 681, 766, 856, 951, 1051, 1156, 1381, 1501, 1756, 1891, 2031, 2326, 2481, 2641, 2806, 3151, 3331, 3706, 3901, 4101, 4306, 4516, 4731, 4951, 5176, 5641, 5881, 6376, 6631, 6891, 7156, 7426, 7701, 7981, 8266, 8851, 9151, 9766, 10081, 10401
Offset: 1

Views

Author

Massimo Kofler, Apr 13 2025

Keywords

Comments

The centered pentagonal numbers that are prime are terms (see A145838).

Examples

			16 = 2^4 is a term since it is the 3rd centered pentagonal number and larger than the sum of its proper divisors (1+2+4+8=15).
51 = 3*17 is a term since it is the 5th centered pentagonal number and larger than the sum of its proper divisors (1+3+17=21).
76 = 2^2*19 is a term since it is the 6th centered pentagonal number and larger than the sum of its proper divisors (1+2+4+19+38=64).

Links

Crossrefs

Intersection of A005891 and A005100.
Cf. A145838, A381488, A382696.

Programs

  • Maple
    select(t -> numtheory:-sigma(t) < 2*t, [seq( (5*n^2+5*n+2)/2, n=0..100)]); # Robert Israel, May 13 2025
  • Mathematica
    Select[Table[(5*n^2 + 5*n + 2)/2, {n, 0, 65}], DivisorSigma[-1, #] < 2 &] (* Amiram Eldar, Apr 13 2025 *)

Formula

a(n) == 1 (mod 5).

A381043 Centered pentagonal numbers which are squarefree semiprimes.

Original entry on oeis.org

6, 51, 106, 141, 226, 391, 526, 681, 766, 951, 1501, 1891, 2031, 2326, 2481, 2641, 3151, 3901, 4101, 4306, 6631, 6891, 7981, 8266, 8851, 10081, 10401, 11391, 13141, 14631, 15406, 16201, 20931, 23281, 24751, 27301, 27826, 28891, 29431, 30526, 32206, 33351, 35701, 36301, 38131, 38751, 41926
Offset: 1

Views

Author

Massimo Kofler, Apr 14 2025

Keywords

Comments

Numbers such as 22801=151^2 and 1666681=1291^2 are in A382132 but not here.

Examples

			A005891(1) = 6 = (5*1^2 + 5*1 + 2)/2 = 2*3.
A005891(4) = 51 = (5*4^2 + 5*4 + 2)/2 = 3*17.
A005891(6) = 106 = (5*6^2 + 5*6 + 2)/2 = 2*53.

Crossrefs

Intersection of A006881 and A005891.
Cf. A145838, A382132, A364610.

Programs

  • Mathematica
    Select[Table[5*n*(n + 1)/2 + 1, {n, 0, 150}], FactorInteger[#][[;; , 2]] == {1, 1} &] (* Amiram Eldar, Apr 14 2025 *)

A380937 Achilles numbers sandwiched between two semiprimes.

Original entry on oeis.org

288, 392, 1944, 4500, 4608, 7200, 9248, 13068, 14112, 14792, 16200, 18000, 19652, 21632, 26136, 26912, 28800, 31104, 32000, 34992, 38088, 38988, 41472, 42592, 45000, 48668, 49000, 52272, 55112, 56448, 60552, 69984, 78732, 79092, 87808, 88200, 95832, 98568
Offset: 1

Views

Author

Massimo Kofler, Apr 12 2025

Keywords

Comments

Achilles numbers are powerful but imperfect.
All the terms are divisible by 4.

Examples

			288 = 2^5 * 3^2 (between 287 = 7 * 41 and 289 = 17^2).
392 = 2^3 * 7^2 (between 391 = 17 * 23 and 393 = 3 * 131).
1944 = 2^3 * 3^5 (between 1943 = 29 * 67 and 1945 = 5 * 389).

Crossrefs

Cf. A001358, A052486, A382831.

Programs

  • Mathematica
    achQ[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Min[e] > 1 && GCD @@ e == 1]; semiQ[n_] := PrimeOmega[n] == 2; Select[4*Range[25000], achQ[#] && And @@ semiQ /@ (# + {-1, 1}) &] (* Amiram Eldar, Apr 12 2025 *)

A382696 Centered pentagonal numbers that are abundant.

Original entry on oeis.org

276, 456, 1266, 1626, 2176, 2976, 3516, 5406, 6126, 8556, 9456, 12426, 13506, 17016, 18276, 22326, 23766, 28356, 29976, 35106, 36906, 39376, 42576, 44556, 50766, 52926, 59676, 62016, 69306, 71826, 79656, 82356, 89776, 90726, 93606, 94576, 102516, 105576, 115026, 118266, 128256, 131676, 142206, 145806
Offset: 1

Views

Author

Massimo Kofler, Apr 03 2025

Keywords

Comments

The sequence is infinite, e.g. A005891(n) is a term when 1 < n == 1 or 10 (mod 12). - Robert Israel, Apr 06 2025

Examples

			276 = 2^2*3*23 is a term since it is a centered pentagonal number and less than the sum of its proper divisors (1+2+3+4+6+12+23+46+69+92+138=396).
456 = 2^3*3*19 is a term since it is a centered pentagonal number and less than the sum of its proper divisors  (1+2+3+4+6+8+12+19+24+38+ 57+ 76+114+152+228=744).

Crossrefs

Intersection of A005891 and A005101.
Cf. A379264.

Programs

  • Maple
    select(t -> numtheory:-sigma(t) > 2*t, [seq((5*n^2+5*n+2)/2, n=1..500)]); # Robert Israel, Apr 06 2025
  • Mathematica
    Select[Table[(5*n^2 + 5*n + 2)/2, {n, 1, 250}], DivisorSigma[-1, #] > 2 &] (* Amiram Eldar, Apr 03 2025 *)

A382628 Centered hexagonal numbers that are sphenic numbers.

Original entry on oeis.org

3367, 4921, 8911, 9919, 10621, 14911, 18487, 21931, 25669, 27937, 37297, 41419, 55081, 63511, 66157, 72541, 80197, 106597, 108871, 113491, 117019, 130417, 134197, 136747, 139321, 174967, 195841, 198919, 203581, 219511, 226051, 232687, 236041, 244531, 247969, 256669, 258427, 269101, 272707, 287371
Offset: 1

Views

Author

Massimo Kofler, Apr 01 2025

Keywords

Comments

All terms are odd.

Examples

			3367 is the 33rd centered hexagonal number and 3367 = 7*13*37 is the product of 3 distinct primes.
8911 is the 54th centered hexagonal number and 8911 = 7*19*67 is the product of 3 distinct primes.

Crossrefs

Intersection of A007304 and A003215.
Cf. A113530.

Programs

  • Mathematica
    Select[Table[3*n*(n+1) + 1, {n, 0, 400}], FactorInteger[#][[;; , 2]] == {1, 1, 1} &] (* Amiram Eldar, Apr 01 2025 *)
  • PARI
    select(x->((omega(x)==3) && (bigomega(x)==3)), vector(100, n, 3*n*(n+1) + 1)) \\ Michel Marcus, Apr 02 2025

Formula

a(n) == 1 (mod 6).

A382451 Centered pentagonal numbers which are the products of four distinct primes.

Original entry on oeis.org

5406, 12426, 20026, 23766, 40641, 55131, 83266, 115026, 118266, 136306, 142206, 145806, 176226, 184281, 205206, 209526, 245706, 279726, 284766, 315951, 326706, 371526, 387106, 407031, 413106, 419226, 425391, 498406, 505126, 553426, 623751, 638826, 672106, 685131
Offset: 1

Views

Author

Massimo Kofler, Mar 26 2025

Keywords

Examples

			A005891(46) = 5406 = (5*46^2 + 5*46 + 2)/2 = 2*3*17*53.
A005891(70) = 12426 = (5*70^2 + 5*70 + 2)/2 = 2*3*19*109.
A005891(127) = 40641 = (5*127^2 + 5*127 + 2)/2 = 3*19*23*31.

Crossrefs

Intersection of A005891 and A046386.
Cf. A364610.

Programs

  • Mathematica
    Select[Table[5*n*(n+1)/2+1, {n, 0, 600}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Mar 26 2025 *)

A382231 Octagonal numbers that are the product of three distinct primes.

Original entry on oeis.org

645, 1045, 1281, 2465, 2821, 3201, 3605, 7701, 8965, 12545, 15841, 17633, 18565, 20501, 23585, 24661, 25761, 26885, 30401, 34133, 36741, 45141, 51221, 52801, 57685, 59361, 62785, 66305, 68101, 71765, 73633, 89441, 95765, 100101, 116033, 120801, 123221, 125665, 138245
Offset: 1

Views

Author

Massimo Kofler, Mar 19 2025

Keywords

Comments

All terms are odd numbers.

Examples

			645 is a term because 645=3*5*43 is a sphenic number and is the 15th octagonal number.
1045 is a term because 1045=5*11*19 is a sphenic number and is the 19th octagonal number.
1281 is a term because 1281=3*7*61 is a sphenic number and is the 21st octagonal number.

Crossrefs

Intersection of A007304 and A000567.
Cf. A259677.

Programs

  • Maple
    N:= 10^6: # for terms <= N
isoct:= proc(n) issqr(1+3*n) and sqrt(1+3*n) mod 3 = 2 end proc:
P:= select(isprime,[seq(i,i=3..N/15,2)]): nP:= nops(P):
R:= NULL:
for i from 1 to nP while P[i]*P[i+1]*P[i+2] <= N do
  for j from i+1 to nP while P[i]*P[j]*P[j+1] <= N do
    for k from j+1 to nP  do
      v:= P[i]*P[j]*P[k];
      if v > N then break fi;
      if isoct(v) then R:= R,v fi;
od od od:
sort([R]); # Robert Israel, Mar 19 2025
  • Mathematica
    Select[Table[n*(3*n-2), {n, 1, 220}], FactorInteger[#][[;;, 2]] == {1, 1, 1} &] (* Amiram Eldar, Mar 19 2025 *)

A382133 Products of 4 distinct primes that are the average of two consecutive primes.

Original entry on oeis.org

462, 570, 714, 858, 870, 1190, 1230, 1254, 1290, 1302, 1482, 1590, 1722, 1785, 1806, 1995, 2046, 2130, 2170, 2210, 2470, 2490, 2870, 3030, 3045, 3255, 3390, 3410, 3705, 3774, 3795, 3885, 3930, 4002, 4218, 4242, 4278, 4422, 4510, 4515, 4641, 4785, 4935, 5010, 5110
Offset: 1

Views

Author

Massimo Kofler, Mar 17 2025

Keywords

Examples

			462 is a term because 462=2*3*7*11 is the product of four distinct primes and 462 = (461+463)/2.
714 is a term because 714=2*3*7*17 is the product of four distinct primes and 714 = (709+719)/2.
210 is not a term because although 210=2*3*5*7 is the product of four distinct primes 210 != (199 + 211)/2.

Crossrefs

Intersection of A024675 and A046386.
Cf. A078443, A130178.

Programs

  • Mathematica
    Select[Range[5200], 2*# == Plus @@ NextPrime[#, {-1, 1}] && FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Mar 17 2025 *)

A382132 Centered pentagonal numbers which are semiprimes.

Original entry on oeis.org

6, 51, 106, 141, 226, 391, 526, 681, 766, 951, 1501, 1891, 2031, 2326, 2481, 2641, 3151, 3901, 4101, 4306, 6631, 6891, 7981, 8266, 8851, 10081, 10401, 11391, 13141, 14631, 15406, 16201, 20931, 22801, 23281, 24751, 27301, 27826, 28891, 29431, 30526, 32206, 33351, 35701, 36301, 38131, 38751
Offset: 1

Views

Author

Massimo Kofler, Mar 17 2025

Keywords

Examples

			A005891(1) = 6 = (5*1^2 + 5*1 + 2)/2 = 2*3.
A005891(4) = 51 = (5*4^2 + 5*4 + 2)/2 = 3*17.
A005891(6) = 106 = (5*6^2 + 5*6 + 2)/2 = 2*53.

Crossrefs

Intersection of A001358 and A005891.
Cf. A364610.

Programs

  • Mathematica
    Select[Table[(5*n^2 + 5*n + 2)/2, {n, 1, 125}], PrimeOmega[#] == 2 &] (* Amiram Eldar, Mar 17 2025 *)

A381960 Centered heptagonal numbers which are semiprime.

Original entry on oeis.org

22, 106, 253, 386, 841, 1198, 1618, 2101, 2458, 3046, 3473, 4166, 4411, 5461, 6623, 6931, 7246, 7897, 8926, 9647, 10018, 12811, 13238, 14113, 15947, 16423, 17893, 19951, 22121, 22681, 24403, 24991, 26797, 27413, 30598, 31921, 32593, 33958, 38221, 40447, 41966, 43513
Offset: 1

Views

Author

Massimo Kofler, Mar 11 2025

Keywords

Examples

			A069099(3) = 22 = 21*(3-1)/2 + 1 = 2*11.
A069099(6) = 106 = 42*(6-1)/2 + 1 = 2*53.
A069099(9) = 253 = 63*(9-1)/2 + 1 = 11*23.

Crossrefs

Intersection of A069099 and A001358.
Cf. A360183.

Programs

  • Mathematica
    Select[Table[(7*n^2 - 7*n + 2)/2, {n, 1, 120}], PrimeOmega[#] == 2 &] (* Amiram Eldar, Mar 11 2025 *)
  • PARI
    select(x->bigomega(x)==2, vector(120, n, (7*n^2-7*n+2)/2)) \\ Michel Marcus, Mar 11 2025

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