A129654 -id:A129654 - cp's OEIS frontend

A355531 Minimal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

0, 1, 2, 1, 3, 1, 4, 1, 1, 1, 5, 1, 6, 1, 2, 1, 7, 1, 8, 1, 2, 1, 9, 1, 1, 1, 1, 1, 10, 1, 11, 1, 2, 1, 2, 1, 12, 1, 2, 1, 13, 1, 14, 1, 1, 1, 15, 1, 1, 1, 2, 1, 16, 1, 3, 1, 2, 1, 17, 1, 18, 1, 1, 1, 3, 1, 19, 1, 2, 1, 20, 1, 21, 1, 1, 1, 2, 1, 22, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 14 2022

nonn

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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 1.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are A008578.
Positions of 1's are 2 followed by A013929.
The non-augmented maximal version is A286470, also A355526.
The non-augmented version is A355524, also A355525.
Row minima of A355534, which has Heinz number A325351.
The maximal version is A355535.
A001222 counts prime indices.
A112798 lists prime indices, sum A056239.
Cf. A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355528, A355533, A355536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A320943 Numbers that have exactly 26 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

1559439365121, 2468046593376, 7760419091425

Offset: 1

Hugh Erling, Oct 24 2018

			a(1): 1559439365121 has representations P(n,k) = P(3, 519813121708)=P(6, 103962624343)=P(9, 43317760144)=P(11, 28353443004)=P(18, 10192414153)=P(27, 4442847196)=P(33, 2953483648)=P(57, 977092336)=P(66, 727011361)=P(69, 664722664)=P(81, 481308448)=P(86, 426659199)=P(129, 188885584)=P(131, 183140268)=P(171, 107288572)=P(209, 71744544)=P(237, 55761976)=P(414, 18240979)=P(473, 13969968)=P(513, 11874388)=P(711, 6178324)=P(729, 5876784)=P(1881, 881968)=P(3537, 249376)=P(16899, 10924)=P(720981, 8).
a(2): 2468046593376 has representations P(n,k) = P(3, 822682197793)=P(6, 164536439560)=P(12, 37394645356)=P(18, 16131023488)=P(24, 8942197804)=P(26, 7593989520)=P(39, 3330697159)=P(42, 2866488496)=P(56, 1602627660)=P(72, 965589436)=P(84, 707988124)=P(96, 541238290)=P(116, 370021980)=P(126, 313402744)=P(392, 32204796)=P(416, 28591830)=P(576, 14903665)=P(647, 11809911)=P(783, 8061483)=P(936, 5640220)=P(1827, 1479601)=P(2912, 582306)=P(4302, 266776)=P(5823, 145603)=P(7056, 99160)=P(145551, 235).
a(3): 7760419091425 has representations P(n,k) = P(5, 776041909144)=P(7, 369543766260)=P(10, 172453757589)=P(13, 99492552456)=P(19, 45382567788)=P(25, 25868063640)=P(35, 13042721164)=P(37, 11652280920)=P(49, 6598995828)=P(55, 5225871444)=P(65, 3730970719)=P(82, 2336771785)=P(143, 764347396)=P(145, 743335164)=P(154, 658723293)=P(205, 371134344)=P(290, 185190769)=P(325, 147396376)=P(475, 68935548)=P(1225, 10351368)=P(1378, 8179601)=P(1729, 5194893)=P(2755, 2045644)=P(7585, 269814)=P(1969825, 6)=P(3939649, 3).

Hugh Erling, Python program

Cf. A057145, A063778, A129654, A139601, A177029, A195527, A195528.

A subsequence of A090466.

PARI
```
\\ See Hugh Erling, Oct 29 2018 link.
```

A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 2, 1, 7, 2, 8, 3, 3, 5, 9, 2, 3, 6, 2, 4, 10, 2, 11, 1, 4, 7, 3, 2, 12, 8, 5, 3, 13, 3, 14, 5, 2, 9, 15, 2, 4, 3, 6, 6, 16, 2, 3, 4, 7, 10, 17, 2, 18, 11, 3, 1, 4, 4, 19, 7, 8, 3, 20, 2, 21, 12, 2, 8, 4, 5, 22, 3, 2

Offset: 1

Gus Wiseman, Jul 14 2022

nonn

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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Prepending 1 to the positions of 1's gives A000079.
Positions of first appearances are A008578.
Positions of 2's are A065119.
The non-augmented version is A286470, also A355526.
The non-augmented minimal version is A355524, also A355525.
The minimal version is A355531.
Row maxima of A355534, which has Heinz number A325351.
A001222 counts prime indices, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355533, A355536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A321156 Numbers that have exactly 5 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

561, 1485, 1701, 2016, 2556, 2601, 2850, 3025, 3060, 3256, 3321, 4186, 4761, 4851, 5226, 5320, 5565, 5841, 6175, 6216, 6336, 6525, 6670, 7425, 7821, 7840, 8001, 8100, 8625, 8646, 9730, 9856, 9945, 9976, 10116, 10296, 10450, 10585, 11025, 11305, 11340, 12025, 12090

Offset: 1

Hugh Erling, Oct 28 2018

nonn

n | 2*m where m is a term in this sequence. - David A. Corneth, Oct 29 2018

			561 has representations P(3, 188)=P(6, 39)=P(11, 12)=P(17, 6)=P(33, 3).
1485 has representations P(3, 496)=P(5, 150)=P(9, 43)=P(15, 16)=P(54, 3).
1701 has representations P(3, 568)=P(6, 115)=P(9, 49)=P(18, 13)=P(21, 10).

Hugh Erling, Table of n, a(n) for n = 1..100
Hugh Erling, Python code

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321157, A321158, A321159, A321160, A320943.

isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 5; \\ Michel Marcus, Oct 29 2018

is(n) = my(d=divisors(n<<1)); sum(i=2, #d, k=2*(d[i]^2 - 2 * d[i] + n) / (d[i] - 1) / d[i]; k == k\1 && min(d[i], k) >=3) == 5 \\ David A. Corneth, Oct 29 2018

A321157 Numbers that have exactly 7 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

11935, 12376, 21736, 24220, 41041, 45441, 51360, 52326, 53361, 54145, 54405, 58311, 58696, 73360, 82720, 89425, 90321, 96580, 101025, 102025, 108801, 113050, 117216, 118405, 122265, 122500, 122760, 123201, 123256, 127281, 128961, 135201, 144585, 152076, 165376, 166635, 169456, 174097

Offset: 1

Hugh Erling, Oct 29 2018

nonn

			11935 has representations P(n,k) = P(5, 1195) = P(7, 570) = P(10, 267) = P(14, 133) = P(35, 22) = P(55, 10) = P(154, 3).
12376 has representations P(n,k) = P(4, 2064) = P(7, 591) = P(16, 105) = P(26, 40) = P(34, 24) = P(56, 10) = P(91, 5).
21736 has representations P(n,k) = P(4, 3624) = P(8, 778) = P(11, 397) = P(16, 183) = P(19, 129) = P(22, 96) = P(208, 3).

Hugh Erling, Python program

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321158, A321159, A321160, A320943.

A321158 Numbers that have exactly 8 representations as a k-gonal number, P(m,k) = m((k-2)m - (k-4))/2, k and m >= 3.

Original entry on oeis.org

11781, 61776, 75141, 133056, 152361, 156520, 176176, 179740, 188650, 210925, 241605, 266085, 292825, 298936, 338625, 342585, 354025, 358281, 360801, 365365, 371925, 391392, 395200, 400960, 417340, 419805, 424270, 438516

Offset: 1

Hugh Erling, Oct 29 2018

nonn

			a(1) 11781 has representations P(m,k) = P(3, 3928)=P(6, 787)=P(9,329)=P(11, 216)=P(21, 58)=P(63, 8)=P(77, 6)=P(153, 3).
a(2) 61776 has representations P(m,k) = P(3, 20593)=P(6, 4120)=P(8,2208)=P(11, 1125)=P(26, 192)=P(36, 100)=P(176, 6)=P(351, 3).
a(3) 75141 has representations P(m,k) = P(3, 25048)=P(6, 5011)=P(9,2089)=P(11, 1368)=P(18, 493)=P(27, 216)=P(66, 37)=P(69, 34).

Hugh Erling, Python program

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321159, A321160, A320943.

r[n_] := Module[{k}, Sum[Boole[d >= 3 && (k = 2(d^2 - 2d + n)/(d^2 - d); IntegerQ[k] && k >= 3)], {d, Divisors[2n]}]];
Select[Range[500000], r[#] == 8&] (* Jean-François Alcover, Sep 23 2019, after Andrew Howroyd *)

r(n)={sumdiv(2*n, d, if(d>=3, my(k=2*(d^2 - 2*d + n)/(d^2 - d)); !frac(k) && k>=3))}
for(n=1, 5*10^5, if(r(n)==8, print1(n, ", "))) \\ Andrew Howroyd, Nov 26 2018

Python
```
# See link.
```

A321159 Numbers that have exactly 9 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

27405, 126225, 194481, 201825, 273105, 478401, 538461, 615681, 718641, 859600, 862785, 1056160, 1187145, 1257201, 1328481, 1413126, 1439361, 1532601, 1540540, 1619541, 1625625, 1708785, 1842400, 1849926, 1890945

Offset: 1

Hugh Erling, Oct 29 2018

nonn

			a(1) 27405 has representations P(n,k) = P(3, 9136)=P(5, 2742)=P(9, 763)=P(14, 303)=P(18, 181)=P(27, 80)=P(35, 48)=P(63, 16)=P(105, 7).
a(2) 126225 has representations P(n,k) = P(3, 42076)=P(5, 12624)=P(9, 3508)=P(15, 1204)=P(17, 930)=P(33, 241)=P(50, 105)=P(99, 28)=P(225, 7).
a(3) 194481 has representations P(n,k) = P(3, 64828)=P(6, 12967)=P(9, 5404)=P(14, 2139)=P(18, 1273)=P(21, 928)=P(27, 556)=P(81, 62)=P(441, 4).

Hugh Erling, Python program

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321158, A321160, A320943.

isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 9; \\ Michel Marcus, Nov 02 2018

Python
```
# See Erling link.
```

A321160 Numbers that have exactly 10 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

220780, 519156, 1079001, 1154440, 1324576, 1447551, 2429505, 2454705, 2491776, 2603601, 2665125, 2700621, 2772225, 2953665, 3000025, 3086721, 3316600, 3665376, 4488561, 4741660, 5142501, 5388201, 5785101, 6076225

Offset: 1

Hugh Erling, Oct 29 2018

nonn

			a(1) 220780 has representations P(n,k) = P(4, 36798) = P(7, 10515) = P(10, 4908) = P(14, 2428) = P(19, 1293) = P(28, 586) = P(35, 373) = P(38, 316) = P(40, 285) = P(664, 3).
a(2) 519156 has representations P(n,k) = P(3, 173053) = P(6, 34612) = P(8, 18543) = P(11, 9441) = P(27, 1481) = P(36, 826) = P(66, 244) = P(92, 126) = P(99, 109) = P(456, 7).
a(3) 1079001 has representations P(n,k) = P(3, 359668) = P(6, 71935) = P(9, 29974) = P(11, 19620) = P(14, 11859) = P(21, 5140) = P(27, 3076) = P(66, 505) = P(81, 335) = P(126, 139).

Hugh Erling, Python program

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321158, A321159, A320943.

isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 10; \\ Michel Marcus, Nov 02 2018

Python
```
# See links.
```

A333915 Number of ways to represent n as a pyramidal number.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2

Offset: 4

Ilya Gutkovskiy, Apr 09 2020

nonn

Frequency of n in the array A261720 of pyramidal numbers.

			a(10) = 2 because 10 is the third tetrahedral (or triangular pyramidal) number and also the second 9-gonal pyramidal number.
a(30) = 3 because 30 is the fourth square pyramidal number, the third octagonal pyramidal number and also the second 29-gonal pyramidal number.

Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A080851, A129654, A177025, A261720, A333916.

A322660 Numbers k > 1 for which the number of representations as an m-gonal number P(m,r) = r((m-2)r-(m-4))/2, with m>1, r>1, equals the number of divisors of k.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 111, 113, 121, 127, 131, 137, 139, 141, 145, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 201

Offset: 1

Daniel Suteu, Dec 22 2018

Numbers k > 1 such that A129654(k) = A000005(k).
Each prime number greater than 2 is a term of this sequence.
The first 20 composite terms are: 9, 15, 21, 25, 49, 51, 55, 81, 91, 111, 121, 141, 145, 169, 201, 235, 289, 291, 321, 325.

			15 is a term of this sequence, as it has 4 divisors and it can be represented in 4 different ways as an m-gonal number P(m,r) = r*((m-2)*r-(m-4))/2, with m>1, r>1, as following: 15 = P(15,2) = P(6,3) = P(3,5) = P(2,15).

Wikipedia, Polygonal number

Cf. A000005, A063778, A129654.

isok(k) = (k>1) && (sigma(k,0) == sumdiv(2*k, d, (d>1) && (2*k/d + 2*d - 4) % (d-1) == 0));

cp's OEIS Frontend

A355531 Minimal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

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A320943 Numbers that have exactly 26 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

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A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

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A321156 Numbers that have exactly 5 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

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A321157 Numbers that have exactly 7 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

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A321158 Numbers that have exactly 8 representations as a k-gonal number, P(m,k) = m*((k-2)*m - (k-4))/2, k and m >= 3.

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A321159 Numbers that have exactly 9 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

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A321160 Numbers that have exactly 10 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

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A333915 Number of ways to represent n as a pyramidal number.

A320943 Numbers that have exactly 26 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

A321156 Numbers that have exactly 5 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

A321157 Numbers that have exactly 7 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

A321158 Numbers that have exactly 8 representations as a k-gonal number, P(m,k) = m((k-2)m - (k-4))/2, k and m >= 3.

A321159 Numbers that have exactly 9 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

A321160 Numbers that have exactly 10 representations as a k-gonal number, P(n,k) = n((k-2)n - (k-4))/2, k and n >= 3.

A322660 Numbers k > 1 for which the number of representations as an m-gonal number P(m,r) = r((m-2)r-(m-4))/2, with m>1, r>1, equals the number of divisors of k.